Integrand size = 21, antiderivative size = 939 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt {1+\frac {1}{c^2 x^2}}}-\frac {4 b d \sqrt {d+e x} \left (1+c^2 x^2\right )}{21 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}+\frac {24 b c d^2 \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{35 \left (-c^2\right )^{3/2} e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b c \left (2 c^2 d^2+9 e^2\right ) \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{105 \left (-c^2\right )^{5/2} e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {64 b c d^3 \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{35 \left (-c^2\right )^{3/2} e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b c d \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{105 \left (-c^2\right )^{5/2} e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {64 b d^4 \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{35 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
2*d^2*(e*x+d)^(3/2)*(a+b*arccsch(c*x))/e^4-6/5*d*(e*x+d)^(5/2)*(a+b*arccsc h(c*x))/e^4+2/7*(e*x+d)^(7/2)*(a+b*arccsch(c*x))/e^4-2*d^3*(a+b*arccsch(c* x))*(e*x+d)^(1/2)/e^4+4/35*b*(c^2*x^2+1)*(e*x+d)^(1/2)/c^3/e/(1+1/c^2/x^2) ^(1/2)-4/21*b*d*(c^2*x^2+1)*(e*x+d)^(1/2)/c^3/e^2/x/(1+1/c^2/x^2)^(1/2)+64 /35*b*d^4*EllipticPi(1/2*(1-x*(-c^2)^(1/2))^(1/2)*2^(1/2),2,2^(1/2)*(e/(d* (-c^2)^(1/2)+e))^(1/2))*(c^2*x^2+1)^(1/2)*((e*x+d)*(-c^2)^(1/2)/(d*(-c^2)^ (1/2)+e))^(1/2)/c/e^4/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+24/35*b*c*d^2*El lipticE(1/2*(1-x*(-c^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-c^2)^(1/2)/(c^2*d-e*( -c^2)^(1/2)))^(1/2))*(e*x+d)^(1/2)*(c^2*x^2+1)^(1/2)/(-c^2)^(3/2)/e^3/x/(1 +1/c^2/x^2)^(1/2)/(c^2*(e*x+d)/(c^2*d-e*(-c^2)^(1/2)))^(1/2)+4/105*b*c*(2* c^2*d^2+9*e^2)*EllipticE(1/2*(1-x*(-c^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-c^2) ^(1/2)/(c^2*d-e*(-c^2)^(1/2)))^(1/2))*(e*x+d)^(1/2)*(c^2*x^2+1)^(1/2)/(-c^ 2)^(5/2)/e^3/x/(1+1/c^2/x^2)^(1/2)/(c^2*(e*x+d)/(c^2*d-e*(-c^2)^(1/2)))^(1 /2)-64/35*b*c*d^3*EllipticF(1/2*(1-x*(-c^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-c ^2)^(1/2)/(c^2*d-e*(-c^2)^(1/2)))^(1/2))*(c^2*x^2+1)^(1/2)*(c^2*(e*x+d)/(c ^2*d-e*(-c^2)^(1/2)))^(1/2)/(-c^2)^(3/2)/e^3/x/(1+1/c^2/x^2)^(1/2)/(e*x+d) ^(1/2)-32/105*b*c*d*(c^2*d^2+e^2)*EllipticF(1/2*(1-x*(-c^2)^(1/2))^(1/2)*2 ^(1/2),(-2*e*(-c^2)^(1/2)/(c^2*d-e*(-c^2)^(1/2)))^(1/2))*(c^2*x^2+1)^(1/2) *(c^2*(e*x+d)/(c^2*d-e*(-c^2)^(1/2)))^(1/2)/(-c^2)^(5/2)/e^3/x/(1+1/c^2/x^ 2)^(1/2)/(e*x+d)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 34.76 (sec) , antiderivative size = 1098, normalized size of antiderivative = 1.17 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {a d^4 \sqrt {1+\frac {e x}{d}} B_{-\frac {e x}{d}}\left (4,\frac {1}{2}\right )}{e^4 \sqrt {d+e x}}+\frac {b \left (-\frac {c \left (e+\frac {d}{x}\right ) x \left (\frac {4 \left (-16 c^2 d^2+9 e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}}}{105 e^3}+\frac {32 c^3 d^3 \text {csch}^{-1}(c x)}{35 e^4}-\frac {2 c^3 x^3 \text {csch}^{-1}(c x)}{7 e}-\frac {4 c^2 x^2 \left (e \sqrt {1+\frac {1}{c^2 x^2}}-3 c d \text {csch}^{-1}(c x)\right )}{35 e^2}+\frac {4 c x \left (5 c d e \sqrt {1+\frac {1}{c^2 x^2}}-12 c^2 d^2 \text {csch}^{-1}(c x)\right )}{105 e^3}\right )}{\sqrt {d+e x}}+\frac {2 \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (-\frac {\sqrt {2} \left (40 c^3 d^3 e-8 c d e^3\right ) \sqrt {1+i c x} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2} \sqrt {\frac {e (1-i c x)}{i c d+e}}}+\frac {i \sqrt {2} (c d-i e) \left (48 c^4 d^4-16 c^2 d^2 e^2+9 e^4\right ) \sqrt {1+i c x} \sqrt {\frac {e (i+c x) (c d+c e x)}{(i c d+e)^2}} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )}{e \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}-\frac {2 \left (-16 c^3 d^3 e+9 c d e^3\right ) \cosh \left (2 \text {csch}^{-1}(c x)\right ) \left (-\left ((c d+c e x) \left (1+c^2 x^2\right )\right )+\frac {c x \left (c d \sqrt {2+2 i c x} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )+2 \sqrt {-\frac {e (-i+c x)}{c d+i e}} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )-i e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-i e}}\right ),\frac {c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt {2+2 i c x} \sqrt {-\frac {e (i+c x)}{c d-i e}} \sqrt {\frac {e (i+c x) (c d+c e x)}{(i c d+e)^2}} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )\right )}{2 \sqrt {-\frac {e (i+c x)}{c d-i e}}}\right )}{c d \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (2+c^2 x^2\right )}\right )}{105 e^4 \sqrt {d+e x}}\right )}{c^4} \]
(a*d^4*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 4, 1/2])/(e^4*Sqrt[d + e*x]) + ( b*(-((c*(e + d/x)*x*((4*(-16*c^2*d^2 + 9*e^2)*Sqrt[1 + 1/(c^2*x^2)])/(105* e^3) + (32*c^3*d^3*ArcCsch[c*x])/(35*e^4) - (2*c^3*x^3*ArcCsch[c*x])/(7*e) - (4*c^2*x^2*(e*Sqrt[1 + 1/(c^2*x^2)] - 3*c*d*ArcCsch[c*x]))/(35*e^2) + ( 4*c*x*(5*c*d*e*Sqrt[1 + 1/(c^2*x^2)] - 12*c^2*d^2*ArcCsch[c*x]))/(105*e^3) ))/Sqrt[d + e*x]) + (2*Sqrt[e + d/x]*Sqrt[c*x]*(-((Sqrt[2]*(40*c^3*d^3*e - 8*c*d*e^3)*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*Elli pticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqr t[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x))/(I*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(48*c^4*d^4 - 16*c^2*d^2*e^2 + 9*e^4)*Sqrt [1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] )/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*(-16*c^3*d^3*e + 9*c*d*e^3)*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*( c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*Elliptic F[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[ -((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*( (c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e) /(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c* d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I +...
Time = 3.96 (sec) , antiderivative size = 1548, normalized size of antiderivative = 1.65, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6864, 27, 7272, 2351, 630, 1656, 1416, 2185, 27, 687, 27, 599, 25, 27, 1511, 1416, 1509, 2222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 6864 |
| \(\displaystyle \frac {b \int -\frac {2 \sqrt {d+e x} \left (16 d^3-8 e x d^2+6 e^2 x^2 d-5 e^3 x^3\right )}{35 e^4 \sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{c}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \int \frac {\sqrt {d+e x} \left (16 d^3-8 e x d^2+6 e^2 x^2 d-5 e^3 x^3\right )}{\sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{35 c e^4}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \int \frac {\sqrt {d+e x} \left (16 d^3-8 e x d^2+6 e^2 x^2 d-5 e^3 x^3\right )}{x \sqrt {c^2 x^2+1}}dx}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (16 d^3 \int \frac {\sqrt {d+e x}}{x \sqrt {c^2 x^2+1}}dx+\int \frac {\sqrt {d+e x} \left (-5 x^2 e^3+6 d x e^2-8 d^2 e\right )}{\sqrt {c^2 x^2+1}}dx\right )}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 630 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (\int \frac {\sqrt {d+e x} \left (-5 x^2 e^3+6 d x e^2-8 d^2 e\right )}{\sqrt {c^2 x^2+1}}dx-32 d^3 \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 1656 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (\int \frac {\sqrt {d+e x} \left (-5 x^2 e^3+6 d x e^2-8 d^2 e\right )}{\sqrt {c^2 x^2+1}}dx-32 d^3 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )\right )}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (\int \frac {\sqrt {d+e x} \left (-5 x^2 e^3+6 d x e^2-8 d^2 e\right )}{\sqrt {c^2 x^2+1}}dx-32 d^3 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}\right )\right )}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (-32 d^3 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}\right )+\frac {2 \int -\frac {5 e^3 \sqrt {d+e x} \left (8 d^2 c^2-8 d e x c^2-3 e^2\right )}{2 \sqrt {c^2 x^2+1}}dx}{5 c^2 e^2}-\frac {2 e^2 \sqrt {c^2 x^2+1} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (-32 d^3 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}\right )-\frac {e \int \frac {\sqrt {d+e x} \left (8 d^2 c^2-8 d e x c^2-3 e^2\right )}{\sqrt {c^2 x^2+1}}dx}{c^2}-\frac {2 e^2 \sqrt {c^2 x^2+1} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (-32 d^3 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}\right )-\frac {e \left (\frac {2 \int \frac {c^2 \left (d \left (24 c^2 d^2-e^2\right )+e \left (16 c^2 d^2-9 e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {16}{3} d e \sqrt {c^2 x^2+1} \sqrt {d+e x}\right )}{c^2}-\frac {2 e^2 \sqrt {c^2 x^2+1} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (-32 d^3 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}\right )-\frac {e \left (\frac {1}{3} \int \frac {d \left (24 c^2 d^2-e^2\right )+e \left (16 c^2 d^2-9 e^2\right ) x}{\sqrt {d+e x} \sqrt {c^2 x^2+1}}dx-\frac {16}{3} d e \sqrt {c^2 x^2+1} \sqrt {d+e x}\right )}{c^2}-\frac {2 e^2 \sqrt {c^2 x^2+1} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (-32 d^3 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}\right )-\frac {e \left (-\frac {2 \int -\frac {e \left (8 d \left (c^2 d^2+e^2\right )+\left (16 c^2 d^2-9 e^2\right ) (d+e x)\right )}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{3 e^2}-\frac {16}{3} d e \sqrt {c^2 x^2+1} \sqrt {d+e x}\right )}{c^2}-\frac {2 e^2 \sqrt {c^2 x^2+1} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (-32 d^3 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}\right )-\frac {e \left (\frac {2 \int \frac {e \left (8 d \left (c^2 d^2+e^2\right )+\left (16 c^2 d^2-9 e^2\right ) (d+e x)\right )}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{3 e^2}-\frac {16}{3} d e \sqrt {c^2 x^2+1} \sqrt {d+e x}\right )}{c^2}-\frac {2 e^2 \sqrt {c^2 x^2+1} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (-32 d^3 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}\right )-\frac {e \left (\frac {2 \int \frac {8 d \left (c^2 d^2+e^2\right )+\left (16 c^2 d^2-9 e^2\right ) (d+e x)}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{3 e}-\frac {16}{3} d e \sqrt {c^2 x^2+1} \sqrt {d+e x}\right )}{c^2}-\frac {2 e^2 \sqrt {c^2 x^2+1} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{7/2}}{7 e^4}-\frac {6 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{5/2}}{5 e^4}+\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{e^4}-\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right ) \sqrt {d+e x}}{e^4}-\frac {2 b \sqrt {c^2 x^2+1} \left (-32 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right ) d^3-\frac {e \left (\frac {2 \left (\frac {\sqrt {c^2 d^2+e^2} \left (16 c^2 d^2+8 c \sqrt {c^2 d^2+e^2} d-9 e^2\right ) \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}-\frac {\left (16 c^2 d^2-9 e^2\right ) \sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}\right )}{3 e}-\frac {16}{3} d e \sqrt {d+e x} \sqrt {c^2 x^2+1}\right )}{c^2}-\frac {2 e^2 (d+e x)^{3/2} \sqrt {c^2 x^2+1}}{c^2}\right )}{35 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{7/2}}{7 e^4}-\frac {6 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{5/2}}{5 e^4}+\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{e^4}-\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right ) \sqrt {d+e x}}{e^4}-\frac {2 b \sqrt {c^2 x^2+1} \left (-32 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right ) d^3-\frac {e \left (\frac {2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (16 c^2 d^2+8 c \sqrt {c^2 d^2+e^2} d-9 e^2\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\left (16 c^2 d^2-9 e^2\right ) \sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}\right )}{3 e}-\frac {16}{3} d e \sqrt {d+e x} \sqrt {c^2 x^2+1}\right )}{c^2}-\frac {2 e^2 (d+e x)^{3/2} \sqrt {c^2 x^2+1}}{c^2}\right )}{35 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{7/2}}{7 e^4}-\frac {6 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{5/2}}{5 e^4}+\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{e^4}-\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right ) \sqrt {d+e x}}{e^4}-\frac {2 b \sqrt {c^2 x^2+1} \left (-32 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right ) d^3-\frac {e \left (\frac {2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (16 c^2 d^2+8 c \sqrt {c^2 d^2+e^2} d-9 e^2\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\left (16 c^2 d^2-9 e^2\right ) \sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )}{3 e}-\frac {16}{3} d e \sqrt {d+e x} \sqrt {c^2 x^2+1}\right )}{c^2}-\frac {2 e^2 (d+e x)^{3/2} \sqrt {c^2 x^2+1}}{c^2}\right )}{35 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
| \(\Big \downarrow \) 2222 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{7/2}}{7 e^4}-\frac {6 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{5/2}}{5 e^4}+\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{e^4}-\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right ) \sqrt {d+e x}}{e^4}-\frac {2 b \sqrt {c^2 x^2+1} \left (-32 \left (d \left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {\left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right )}{2 \sqrt {d}}+\frac {\sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c d+\sqrt {c^2 d^2+e^2}\right )^2}{4 c d \sqrt {c^2 d^2+e^2}},2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{4 \sqrt {c} d \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right )-\frac {\left (\frac {c^2 d^2}{e^2}+1\right ) \sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right ) d^3-\frac {e \left (\frac {2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (16 c^2 d^2+8 c \sqrt {c^2 d^2+e^2} d-9 e^2\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\left (16 c^2 d^2-9 e^2\right ) \sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )}{3 e}-\frac {16}{3} d e \sqrt {d+e x} \sqrt {c^2 x^2+1}\right )}{c^2}-\frac {2 e^2 (d+e x)^{3/2} \sqrt {c^2 x^2+1}}{c^2}\right )}{35 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
(-2*d^3*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^4 + (2*d^2*(d + e*x)^(3/2)*( a + b*ArcCsch[c*x]))/e^4 - (6*d*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e ^4) + (2*(d + e*x)^(7/2)*(a + b*ArcCsch[c*x]))/(7*e^4) - (2*b*Sqrt[1 + c^2 *x^2]*((-2*e^2*(d + e*x)^(3/2)*Sqrt[1 + c^2*x^2])/c^2 - (e*((-16*d*e*Sqrt[ d + e*x]*Sqrt[1 + c^2*x^2])/3 + (2*(-(((16*c^2*d^2 - 9*e^2)*Sqrt[c^2*d^2 + e^2]*(-((Sqrt[d + e*x]*Sqrt[1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2])/((1 + (c^2*d^2)/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2* d^2 + e^2]))) + ((c^2*d^2 + e^2)^(1/4)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e ^2])*Sqrt[(1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2) /e^2)/((1 + (c^2*d^2)/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])^2)]*Ell ipticE[2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^(1/4)], (1 + (c*d) /Sqrt[c^2*d^2 + e^2])/2])/(Sqrt[c]*Sqrt[1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2])))/c) + ((c^2*d^2 + e^2)^(3/4)*(16*c^2* d^2 - 9*e^2 + 8*c*d*Sqrt[c^2*d^2 + e^2])*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])*Sqrt[(1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^ 2)/e^2)/((1 + (c^2*d^2)/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])^2)]*E llipticF[2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^(1/4)], (1 + (c* d)/Sqrt[c^2*d^2 + e^2])/2])/(2*c^(3/2)*Sqrt[1 + (c^2*d^2)/e^2 - (2*c^2*d*( d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2])))/(3*e)))/c^2 - 32*d^3*(-1/2*((1 + (c^2*d^2)/e^2)*(c^2*d^2 + e^2)^(1/4)*(1 - (c*d)/Sqrt[c^2*d^2 + e^2])*(...
3.1.57.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[Sqrt[(c_) + (d_.)*(x_)]/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[-2 Subst[Int[x^2/((c - x^2)*Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^ 2) + b*(x^4/d^2)]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[(x_)^2/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(-a)*((e + d*q)/(c*d^2 - a*e^2)) Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[a*d*((e + d*q)/(c*d^2 - a*e ^2)) Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, x, x] Int[(c + d*x)^n*((a + b*x^2)^p/x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHid e[u, x]}, Simp[(a + b*ArcCsch[c*x]) v, x] + Simp[b/c Int[SimplifyIntegr and[v/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x] ] /; FreeQ[{a, b, c}, x]
Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[b^IntPart[p]*(( a + b*x^n)^FracPart[p]/(x^(n*FracPart[p])*(1 + a*(1/(x^n*b)))^FracPart[p])) Int[u*x^(n*p)*(1 + a*(1/(x^n*b)))^p, x], x] /; FreeQ[{a, b, p}, x] && ! IntegerQ[p] && ILtQ[n, 0] && !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]
Result contains complex when optimal does not.
Time = 10.79 (sec) , antiderivative size = 2545, normalized size of antiderivative = 2.71
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2545\) |
| default | \(\text {Expression too large to display}\) | \(2545\) |
| parts | \(\text {Expression too large to display}\) | \(2546\) |
2/e^4*(-a*(-1/7*(e*x+d)^(7/2)+3/5*d*(e*x+d)^(5/2)-d^2*(e*x+d)^(3/2)+d^3*(e *x+d)^(1/2))-b*(-1/7*arccsch(c*x)*(e*x+d)^(7/2)+3/5*arccsch(c*x)*d*(e*x+d) ^(5/2)-arccsch(c*x)*d^2*(e*x+d)^(3/2)+arccsch(c*x)*d^3*(e*x+d)^(1/2)+2/105 /c^4*(14*I*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^3*d*e*(e*x+d)^(5/2)+3*((c*d +I*e)*c/(c^2*d^2+e^2))^(1/2)*c^4*d*(e*x+d)^(7/2)+40*I*(-(I*c*e*(e*x+d)+c^2 *d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-c^2*d*(e*x+d) +c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticF((e*x+d)^(1/2)*((c*d+I*e)*c/(c ^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c^3*d^3 *e-14*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^4*d^2*(e*x+d)^(5/2)-3*I*((c*d+I* e)*c/(c^2*d^2+e^2))^(1/2)*c*e^3*(e*x+d)^(3/2)-3*I*((c*d+I*e)*c/(c^2*d^2+e^ 2))^(1/2)*c^3*e*(e*x+d)^(7/2)-8*I*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e ^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d ^2+e^2))^(1/2)*EllipticF((e*x+d)^(1/2)*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2),( -(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c*d*e^3+19*((c*d+I*e)*c/(c^ 2*d^2+e^2))^(1/2)*c^4*d^3*(e*x+d)^(3/2)-24*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)- c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-c^2*d*(e*x+d)+c^2*d^2+e^ 2)/(c^2*d^2+e^2))^(1/2)*EllipticF((e*x+d)^(1/2)*((c*d+I*e)*c/(c^2*d^2+e^2) )^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c^4*d^4-16*(-(I*c* e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)- c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticE((e*x+d)^(1/2)...
\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{\sqrt {e x + d}} \,d x } \]
\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {d + e x}}\, dx \]
\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{\sqrt {e x + d}} \,d x } \]
2/35*a*(5*(e*x + d)^(7/2)/e^4 - 21*(e*x + d)^(5/2)*d/e^4 + 35*(e*x + d)^(3 /2)*d^2/e^4 - 35*sqrt(e*x + d)*d^3/e^4) + 1/35*b*(2*(5*e^4*x^4 - d*e^3*x^3 + 2*d^2*e^2*x^2 - 8*d^3*e*x - 16*d^4)*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(e* x + d)*e^4) + 35*integrate(2/35*(5*c^2*e^4*x^5 - c^2*d*e^3*x^4 + 2*c^2*d^2 *e^2*x^3 - 8*c^2*d^3*e*x^2 - 16*c^2*d^4*x)/((c^2*e^4*x^2 + e^4)*sqrt(c^2*x ^2 + 1)*sqrt(e*x + d) + (c^2*e^4*x^2 + e^4)*sqrt(e*x + d)), x) - 35*integr ate(-1/35*(2*c^2*d*e^3*x^4 + 16*c^2*d^3*e*x^2 - 5*(7*e^4*log(c) + 2*e^4)*c ^2*x^5 + 32*c^2*d^4*x - (4*c^2*d^2*e^2 + 35*e^4*log(c))*x^3 - 35*(c^2*e^4* x^5 + e^4*x^3)*log(x))/((c^2*e^4*x^2 + e^4)*sqrt(e*x + d)), x))
\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{\sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {d+e\,x}} \,d x \]