Integrand size = 21, antiderivative size = 499 \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {4 b c \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {20 b c d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {32 b d^2 \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
2/3*(e*x+d)^(3/2)*(a+b*arccsch(c*x))/e^3-2*d^2*(a+b*arccsch(c*x))/e^3/(e*x +d)^(1/2)-4*d*(a+b*arccsch(c*x))*(e*x+d)^(1/2)/e^3+32/3*b*d^2*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)+4/3*b*c*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)-20/3*b*c*d*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)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 35.03 (sec) , antiderivative size = 979, normalized size of antiderivative = 1.96 \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=-\frac {a d^3 \left (1+\frac {e x}{d}\right )^{3/2} B_{-\frac {e x}{d}}\left (3,-\frac {1}{2}\right )}{e^3 (d+e x)^{3/2}}+\frac {b \left (-\frac {c^2 \left (e+\frac {d}{x}\right )^2 x^2 \left (-\frac {4 \sqrt {1+\frac {1}{c^2 x^2}}}{3 e^2}+\frac {16 c d \text {csch}^{-1}(c x)}{3 e^3}-\frac {2 c d \text {csch}^{-1}(c x)}{e^2 \left (e+\frac {d}{x}\right )}-\frac {2 c x \text {csch}^{-1}(c x)}{3 e^2}\right )}{(d+e x)^{3/2}}+\frac {2 \left (e+\frac {d}{x}\right )^{3/2} (c x)^{3/2} \left (-\frac {5 \sqrt {2} c d e \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^2 d^2-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 {2 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 )}{3 e^3 (d+e x)^{3/2}}\right )}{c^3} \]
-((a*d^3*(1 + (e*x)/d)^(3/2)*Beta[-((e*x)/d), 3, -1/2])/(e^3*(d + e*x)^(3/ 2))) + (b*(-((c^2*(e + d/x)^2*x^2*((-4*Sqrt[1 + 1/(c^2*x^2)])/(3*e^2) + (1 6*c*d*ArcCsch[c*x])/(3*e^3) - (2*c*d*ArcCsch[c*x])/(e^2*(e + d/x)) - (2*c* x*ArcCsch[c*x])/(3*e^2)))/(d + e*x)^(3/2)) + (2*(e + d/x)^(3/2)*(c*x)^(3/2 )*((-5*Sqrt[2]*c*d*e*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^2*d^2 - 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)) + (2*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)]*EllipticF[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[S qrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*S qrt[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]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*(I + c*x))/(c...
Leaf count is larger than twice the leaf count of optimal. \(1415\) vs. \(2(499)=998\).
Time = 3.01 (sec) , antiderivative size = 1415, normalized size of antiderivative = 2.84, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6864, 27, 7272, 2351, 599, 25, 27, 631, 1511, 1416, 1509, 1540, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6864 |
| \(\displaystyle \frac {b \int -\frac {2 \left (8 d^2+4 e x d-e^2 x^2\right )}{3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{c}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \int \frac {8 d^2+4 e x d-e^2 x^2}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{3 c e^3}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \int \frac {8 d^2+4 e x d-e^2 x^2}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (8 d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx+\int \frac {4 d e-e^2 x}{\sqrt {d+e x} \sqrt {c^2 x^2+1}}dx\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (8 d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx-\frac {2 \int -\frac {e^2 (4 d-e x)}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e^2}\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {2 \int \frac {e^2 (4 d-e x)}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e^2}+8 d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 \int \frac {4 d-e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}+8 d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 631 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 \int \frac {4 d-e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-16 d^2 \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 \left (\frac {\left (5 c d-\sqrt {c^2 d^2+e^2}\right ) \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}+\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}\right )-16 d^2 \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 \left (\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}+\frac {\sqrt [4]{c^2 d^2+e^2} \left (5 c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}\right )-16 d^2 \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 \left (\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}+\frac {\sqrt [4]{c^2 d^2+e^2} \left (5 c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}\right )-16 d^2 \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 1540 |
| \(\displaystyle -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) d^2}{e^3 \sqrt {d+e x}}-\frac {4 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) d}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 \left (\frac {\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}+\frac {\sqrt [4]{c^2 d^2+e^2} \left (5 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 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}}\right )-16 d^2 \left (\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 {c \left (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}}{e^2}\right )\right )}{3 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^2}{e^3 \sqrt {d+e x}}-\frac {4 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) d}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 \left (\frac {\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}+\frac {\sqrt [4]{c^2 d^2+e^2} \left (5 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 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}}\right )-16 d^2 \left (\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 {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (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 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}}\right )\right )}{3 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^2}{e^3 \sqrt {d+e x}}-\frac {4 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) d}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 \left (\frac {\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}+\frac {\sqrt [4]{c^2 d^2+e^2} \left (5 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 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}}\right )-16 d^2 \left (\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 {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (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 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}}\right )\right )}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
(-2*d^2*(a + b*ArcCsch[c*x]))/(e^3*Sqrt[d + e*x]) - (4*d*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^3 + (2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) - (2*b*Sqrt[1 + c^2*x^2]*(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)]*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])/(Sqr t[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)^(1/4)*(5*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))/Sq rt[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*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])) - 16*d^2* (-1/2*(Sqrt[c]*(c^2*d^2 + e^2)^(1/4)*(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))/Sq rt[c^2*d^2 + e^2])^2)]*EllipticF[2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*...
3.1.64.3.1 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[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[-2 Subst[Int[1/((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[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[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2) Int[1 /Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2) I nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
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 = 9.03 (sec) , antiderivative size = 896, normalized size of antiderivative = 1.80
| method | result | size |
| derivativedivides | \(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-2 d \sqrt {e x +d}-\frac {d^{2}}{\sqrt {e x +d}}\right )+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsch}\left (c x \right )}{3}-2 \,\operatorname {arccsch}\left (c x \right ) d \sqrt {e x +d}-\frac {\operatorname {arccsch}\left (c x \right ) d^{2}}{\sqrt {e x +d}}-\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (5 i \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c d e -8 i \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c d e -4 \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}+8 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c^{2} d^{2}+\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}\right )}{3 c^{2} \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}\, \left (-c d +i e \right )}\right )}{e^{3}}\) | \(896\) |
| default | \(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-2 d \sqrt {e x +d}-\frac {d^{2}}{\sqrt {e x +d}}\right )+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsch}\left (c x \right )}{3}-2 \,\operatorname {arccsch}\left (c x \right ) d \sqrt {e x +d}-\frac {\operatorname {arccsch}\left (c x \right ) d^{2}}{\sqrt {e x +d}}-\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (5 i \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c d e -8 i \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c d e -4 \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}+8 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c^{2} d^{2}+\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}\right )}{3 c^{2} \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}\, \left (-c d +i e \right )}\right )}{e^{3}}\) | \(896\) |
| parts | \(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-2 d \sqrt {e x +d}-\frac {d^{2}}{\sqrt {e x +d}}\right )}{e^{3}}+\frac {2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsch}\left (c x \right )}{3}-2 \,\operatorname {arccsch}\left (c x \right ) d \sqrt {e x +d}-\frac {\operatorname {arccsch}\left (c x \right ) d^{2}}{\sqrt {e x +d}}-\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (5 i \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c d e -8 i \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c d e -4 \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}+8 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c^{2} d^{2}+\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}\right )}{3 c^{2} \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}\, \left (-c d +i e \right )}\right )}{e^{3}}\) | \(899\) |
2/e^3*(a*(1/3*(e*x+d)^(3/2)-2*d*(e*x+d)^(1/2)-d^2/(e*x+d)^(1/2))+b*(1/3*(e *x+d)^(3/2)*arccsch(c*x)-2*arccsch(c*x)*d*(e*x+d)^(1/2)-arccsch(c*x)*d^2/( e*x+d)^(1/2)-2/3/c^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)*((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 )*(5*I*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-8*I*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+e^2)/d,(-(I*e-c*d )*c/(c^2*d^2+e^2))^(1/2)/((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2))*c*d*e-4*Ellipt icF((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-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^2*d^2+ 8*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+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((c*d+I*e)*c/(c^2*d^2+ e^2))^(1/2))*c^2*d^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^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))*e^2)/((c^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2+e^2)/c^2/e^2/x ^2)^(1/2)/x/((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)/(I*e-c*d)))
Timed out. \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
2/3*a*((e*x + d)^(3/2)/e^3 - 6*sqrt(e*x + d)*d/e^3 - 3*d^2/(sqrt(e*x + d)* e^3)) + 1/3*b*(2*(e^2*x^2 - 4*d*e*x - 8*d^2)*log(sqrt(c^2*x^2 + 1) + 1)/(s qrt(e*x + d)*e^3) + 3*integrate(2/3*(c^2*e^2*x^3 - 4*c^2*d*e*x^2 - 8*c^2*d ^2*x)/((c^2*e^3*x^2 + e^3)*sqrt(c^2*x^2 + 1)*sqrt(e*x + d) + (c^2*e^3*x^2 + e^3)*sqrt(e*x + d)), x) - 3*integrate(-1/3*(6*c^2*d*e^2*x^3 - (3*e^3*log (c) + 2*e^3)*c^2*x^4 + 16*c^2*d^3*x + 3*(8*c^2*d^2*e - e^3*log(c))*x^2 - 3 *(c^2*e^3*x^4 + e^3*x^2)*log(x))/((c^2*e^4*x^3 + c^2*d*e^3*x^2 + e^4*x + d *e^3)*sqrt(e*x + d)), x))
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]