Integrand size = 21, antiderivative size = 707 \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 b c d \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{5 \left (-c^2\right )^{3/2} e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {32 b c d^2 \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b c \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d^3 \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
-4/3*d*(e*x+d)^(3/2)*(a+b*arccsch(c*x))/e^3+2/5*(e*x+d)^(5/2)*(a+b*arccsch (c*x))/e^3+2*d^2*(a+b*arccsch(c*x))*(e*x+d)^(1/2)/e^3+4/15*b*(c^2*x^2+1)*( e*x+d)^(1/2)/c^3/e/x/(1+1/c^2/x^2)^(1/2)-32/15*b*d^3*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^3/x/(1+1/c^ 2/x^2)^(1/2)/(e*x+d)^(1/2)-4/5*b*c*d*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)^(3/2)/e^2/x/(1+1/c^2/x^2)^(1/2)/(c^2*(e*x+d)/(c^2 *d-e*(-c^2)^(1/2)))^(1/2)+32/15*b*c*d^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)^(3/2)/e^2/x/(1+1/ c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/15*b*c*(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^2/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 = 34.98 (sec) , antiderivative size = 1012, normalized size of antiderivative = 1.43 \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=-\frac {a d^3 \sqrt {1+\frac {e x}{d}} B_{-\frac {e x}{d}}\left (3,\frac {1}{2}\right )}{e^3 \sqrt {d+e x}}+\frac {b \left (-\frac {c \left (e+\frac {d}{x}\right ) x \left (\frac {4 c d \sqrt {1+\frac {1}{c^2 x^2}}}{5 e^2}-\frac {16 c^2 d^2 \text {csch}^{-1}(c x)}{15 e^3}-\frac {2 c^2 x^2 \text {csch}^{-1}(c x)}{5 e}-\frac {4 c x \left (e \sqrt {1+\frac {1}{c^2 x^2}}-2 c d \text {csch}^{-1}(c x)\right )}{15 e^2}\right )}{\sqrt {d+e x}}-\frac {2 \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (-\frac {\sqrt {2} \left (7 c^2 d^2 e-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 (8 c^3 d^3-3 c d e^2\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 {6 c d e \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 )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (2+c^2 x^2\right )}\right )}{15 e^3 \sqrt {d+e x}}\right )}{c^3} \]
-((a*d^3*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 3, 1/2])/(e^3*Sqrt[d + e*x])) + (b*(-((c*(e + d/x)*x*((4*c*d*Sqrt[1 + 1/(c^2*x^2)])/(5*e^2) - (16*c^2*d^ 2*ArcCsch[c*x])/(15*e^3) - (2*c^2*x^2*ArcCsch[c*x])/(5*e) - (4*c*x*(e*Sqrt [1 + 1/(c^2*x^2)] - 2*c*d*ArcCsch[c*x]))/(15*e^2)))/Sqrt[d + e*x]) - (2*Sq rt[e + d/x]*Sqrt[c*x]*(-((Sqrt[2]*(7*c^2*d^2*e - e^3)*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x) )/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqrt[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)* (8*c^3*d^3 - 3*c*d*e^2)*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)) + (6*c*d*e*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)]*El lipticF[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 + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*El lipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*...
Leaf count is larger than twice the leaf count of optimal. \(1458\) vs. \(2(707)=1414\).
Time = 3.11 (sec) , antiderivative size = 1458, normalized size of antiderivative = 2.06, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {6864, 27, 7272, 2351, 630, 687, 27, 599, 25, 27, 1511, 1416, 1509, 1656, 1416, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 6864 |
| \(\displaystyle \frac {b \int \frac {2 \sqrt {d+e x} \left (8 d^2-4 e x d+3 e^2 x^2\right )}{15 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{c}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \int \frac {\sqrt {d+e x} \left (8 d^2-4 e x d+3 e^2 x^2\right )}{\sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{15 c e^3}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \int \frac {\sqrt {d+e x} \left (8 d^2-4 e x d+3 e^2 x^2\right )}{x \sqrt {c^2 x^2+1}}dx}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (8 d^2 \int \frac {\sqrt {d+e x}}{x \sqrt {c^2 x^2+1}}dx+\int \frac {\sqrt {d+e x} \left (3 e^2 x-4 d e\right )}{\sqrt {c^2 x^2+1}}dx\right )}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 630 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (\int \frac {\sqrt {d+e x} \left (3 e^2 x-4 d e\right )}{\sqrt {c^2 x^2+1}}dx-16 d^2 \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (\frac {2 \int -\frac {3 e \left (4 d^2 c^2+3 d e x c^2+e^2\right )}{2 \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{3 c^2}-16 d^2 \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}+\frac {2 e^2 \sqrt {c^2 x^2+1} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {e \int \frac {4 d^2 c^2+3 d e x c^2+e^2}{\sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{c^2}-16 d^2 \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}+\frac {2 e^2 \sqrt {c^2 x^2+1} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-16 d^2 \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}+\frac {2 \int -\frac {e \left (d^2 c^2+3 d (d+e x) c^2+e^2\right )}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c^2 e}+\frac {2 e^2 \sqrt {c^2 x^2+1} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-16 d^2 \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {2 \int \frac {e \left (d^2 c^2+3 d (d+e x) c^2+e^2\right )}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c^2 e}+\frac {2 e^2 \sqrt {c^2 x^2+1} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-16 d^2 \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {2 \int \frac {d^2 c^2+3 d (d+e x) c^2+e^2}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c^2}+\frac {2 e^2 \sqrt {c^2 x^2+1} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-16 d^2 \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {2 \left (\sqrt {c^2 d^2+e^2} \left (\sqrt {c^2 d^2+e^2}+3 c d\right ) \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-3 c d \sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{c^2}+\frac {2 e^2 \sqrt {c^2 x^2+1} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\sqrt {c^2 d^2+e^2}+3 c d\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}-3 c d \sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{c^2}-16 d^2 \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}+\frac {2 e^2 \sqrt {c^2 x^2+1} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{5/2}}{5 e^3}-\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{3 e^3}+\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \sqrt {d+e x}}{e^3}+\frac {2 b \sqrt {c^2 x^2+1} \left (-16 \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x} d^2-\frac {2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (3 c d+\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 \sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-3 c d \sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )\right )}{c^2}+\frac {2 e^2 \sqrt {d+e x} \sqrt {c^2 x^2+1}}{c^2}\right )}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
| \(\Big \downarrow \) 1656 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{5/2}}{5 e^3}-\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{3 e^3}+\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \sqrt {d+e x}}{e^3}+\frac {2 b \sqrt {c^2 x^2+1} \left (-16 \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 ) d^2-\frac {2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (3 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}}-3 c d \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 )\right )}{c^2}+\frac {2 e^2 \sqrt {d+e x} \sqrt {c^2 x^2+1}}{c^2}\right )}{15 c e^3 \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)^{5/2}}{5 e^3}-\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{3 e^3}+\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \sqrt {d+e x}}{e^3}+\frac {2 b \sqrt {c^2 x^2+1} \left (-16 \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^2-\frac {2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (3 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}}-3 c d \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 )\right )}{c^2}+\frac {2 e^2 \sqrt {d+e x} \sqrt {c^2 x^2+1}}{c^2}\right )}{15 c e^3 \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)^{5/2}}{5 e^3}-\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{3 e^3}+\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \sqrt {d+e x}}{e^3}+\frac {2 b \sqrt {c^2 x^2+1} \left (-16 \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^2-\frac {2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (3 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}}-3 c d \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 )\right )}{c^2}+\frac {2 e^2 \sqrt {d+e x} \sqrt {c^2 x^2+1}}{c^2}\right )}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
(2*d^2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^3 - (4*d*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) + (2*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^ 3) + (2*b*Sqrt[1 + c^2*x^2]*((2*e^2*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2])/c^2 - (2*(-3*c*d*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))/Sq rt[c^2*d^2 + e^2])^2)]*EllipticE[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^2*d^2 + e^2)^(3/4)*(3*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) ]*EllipticF[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*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^2 - 16*d^2*(-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])*(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))...
3.1.58.3.1 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[((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 = 9.41 (sec) , antiderivative size = 1991, normalized size of antiderivative = 2.82
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1991\) |
| default | \(\text {Expression too large to display}\) | \(1991\) |
| parts | \(\text {Expression too large to display}\) | \(1994\) |
2/e^3*(a*(1/5*(e*x+d)^(5/2)-2/3*(e*x+d)^(3/2)*d+d^2*(e*x+d)^(1/2))+b*(1/5* arccsch(c*x)*(e*x+d)^(5/2)-2/3*arccsch(c*x)*(e*x+d)^(3/2)*d+arccsch(c*x)*d ^2*(e*x+d)^(1/2)+2/15/c^3*(7*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^2*d^2*e-((c*d+I*e)*c/(c^2*d^ 2+e^2))^(1/2)*c^3*d*(e*x+d)^(5/2)+I*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*(e*x +d)^(5/2)*c^2*e-4*(-(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)*E llipticF((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^3-3*(-(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)*((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-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 ))*e^3+8*(-(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)*EllipticPi ((e*x+d)^(1/2)*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2),1/(c*d+I*e)/c*(c^2*d^2...
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{\sqrt {e x + d}} \,d x } \]
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {d + e x}}\, dx \]
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{\sqrt {e x + d}} \,d x } \]
2/15*a*(3*(e*x + d)^(5/2)/e^3 - 10*(e*x + d)^(3/2)*d/e^3 + 15*sqrt(e*x + d )*d^2/e^3) + 1/15*b*(2*(3*e^3*x^3 - d*e^2*x^2 + 4*d^2*e*x + 8*d^3)*log(sqr t(c^2*x^2 + 1) + 1)/(sqrt(e*x + d)*e^3) + 15*integrate(2/15*(3*c^2*e^3*x^4 - c^2*d*e^2*x^3 + 4*c^2*d^2*e*x^2 + 8*c^2*d^3*x)/((c^2*e^3*x^2 + e^3)*sqr t(c^2*x^2 + 1)*sqrt(e*x + d) + (c^2*e^3*x^2 + e^3)*sqrt(e*x + d)), x) - 15 *integrate(-1/15*(2*c^2*d*e^2*x^3 - 3*(5*e^3*log(c) + 2*e^3)*c^2*x^4 - 16* c^2*d^3*x - (8*c^2*d^2*e + 15*e^3*log(c))*x^2 - 15*(c^2*e^3*x^4 + e^3*x^2) *log(x))/((c^2*e^3*x^2 + e^3)*sqrt(e*x + d)), x))
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{\sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {d+e\,x}} \,d x \]