Integrand size = 21, antiderivative size = 914 \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {4 b \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}{15 c e}-\frac {4 b d \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}{5 e \sqrt {c^2 d^2+e^2} \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right )}+\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 {16 b d^{5/2} \sqrt {1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d} \sqrt {1+c^2 x^2}}\right )}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {4 b d \left (c^2 d^2+e^2\right )^{3/4} \sqrt {\frac {1+c^2 x^2}{\left (1+\frac {c^2 d^2}{e^2}\right ) \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right )^2}} \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right ) 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 (1+\frac {c d}{\sqrt {c^2 d^2+e^2}}\right )\right )}{5 c^{5/2} e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 b \sqrt [4]{c^2 d^2+e^2} \left (8 c^4 d^4+7 c^2 d^2 e^2-e^4-c d \sqrt {c^2 d^2+e^2} \left (8 c^2 d^2+3 e^2\right )\right ) \sqrt {\frac {1+c^2 x^2}{\left (1+\frac {c^2 d^2}{e^2}\right ) \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right )^2}} \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right ) \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 (1+\frac {c d}{\sqrt {c^2 d^2+e^2}}\right )\right )}{15 c^{7/2} e^5 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {8 b d^2 \left (c d-\sqrt {c^2 d^2+e^2}\right )^2 \sqrt {\frac {e^2 \left (1+c^2 x^2\right )}{\left (\sqrt {c^2 d^2+e^2}+c (d+e x)\right )^2}} \left (\sqrt {c^2 d^2+e^2}+c (d+e x)\right ) \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 (1+\frac {c d}{\sqrt {c^2 d^2+e^2}}\right )\right )}{15 c^{3/2} e^5 \sqrt [4]{c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}} x} \] Output:
4/15*b*(1+1/c^2/x^2)^(1/2)*x*(e*x+d)^(1/2)/c/e-4/5*b*d*(1+1/c^2/x^2)^(1/2) *x*(e*x+d)^(1/2)/e/(c^2*d^2+e^2)^(1/2)/(1+c*(e*x+d)/(c^2*d^2+e^2)^(1/2))+2 *d^2*(e*x+d)^(1/2)*(a+b*arccsch(c*x))/e^3-4/3*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^3-16/15*b*d^(5/2)*(c^2*x ^2+1)^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)/(c^2*x^2+1)^(1/2))/c/e^3/(1+1/c^ 2/x^2)^(1/2)/x+4/5*b*d*(c^2*d^2+e^2)^(3/4)*((c^2*x^2+1)/(1+c^2*d^2/e^2)/(1 +c*(e*x+d)/(c^2*d^2+e^2)^(1/2))^2)^(1/2)*(1+c*(e*x+d)/(c^2*d^2+e^2)^(1/2)) *EllipticE(sin(2*arctan(c^(1/2)*(e*x+d)^(1/2)/(c^2*d^2+e^2)^(1/4))),1/2*(2 +2*c*d/(c^2*d^2+e^2)^(1/2))^(1/2))/c^(5/2)/e^3/(1+1/c^2/x^2)^(1/2)/x+2/15* b*(c^2*d^2+e^2)^(1/4)*(8*c^4*d^4+7*c^2*d^2*e^2-e^4-c*d*(c^2*d^2+e^2)^(1/2) *(8*c^2*d^2+3*e^2))*((c^2*x^2+1)/(1+c^2*d^2/e^2)/(1+c*(e*x+d)/(c^2*d^2+e^2 )^(1/2))^2)^(1/2)*(1+c*(e*x+d)/(c^2*d^2+e^2)^(1/2))*InverseJacobiAM(2*arct an(c^(1/2)*(e*x+d)^(1/2)/(c^2*d^2+e^2)^(1/4)),1/2*(2+2*c*d/(c^2*d^2+e^2)^( 1/2))^(1/2))/c^(7/2)/e^5/(1+1/c^2/x^2)^(1/2)/x-8/15*b*d^2*(c*d-(c^2*d^2+e^ 2)^(1/2))^2*((c^2*x^2+1)*e^2/(c*(e*x+d)+(c^2*d^2+e^2)^(1/2))^2)^(1/2)*(c*( e*x+d)+(c^2*d^2+e^2)^(1/2))*EllipticPi(sin(2*arctan(c^(1/2)*(e*x+d)^(1/2)/ (c^2*d^2+e^2)^(1/4))),1/4*(c*d+(c^2*d^2+e^2)^(1/2))^2/c/d/(c^2*d^2+e^2)^(1 /2),1/2*(2+2*c*d/(c^2*d^2+e^2)^(1/2))^(1/2))/c^(3/2)/e^5/(c^2*d^2+e^2)^(1/ 4)/(1+1/c^2/x^2)^(1/2)/x
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 34.15 (sec) , antiderivative size = 1012, normalized size of antiderivative = 1.11 \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx =\text {Too large to display} \] Input:
Integrate[(x^2*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x],x]
Output:
-((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*...
Time = 3.21 (sec) , antiderivative size = 1458, normalized size of antiderivative = 1.60, 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}\) |
Input:
Int[(x^2*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x],x]
Output:
(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))...
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 = 13.11 (sec) , antiderivative size = 1991, normalized size of antiderivative = 2.18
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1991\) |
| default | \(\text {Expression too large to display}\) | \(1991\) |
| parts | \(\text {Expression too large to display}\) | \(1994\) |
Input:
int(x^2*(a+b*arccsch(c*x))/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
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*(I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^2*e*(e*x+ d)^(5/2)-((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^3*d*(e*x+d)^(5/2)-I*(-(I*c*(e *x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-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)*((I* e+c*d)*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+I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*e^3*(e*x+d)^(1/2)-2*I*((I*e+c* d)*c/(c^2*d^2+e^2))^(1/2)*c^2*d*e*(e*x+d)^(3/2)-4*(-(I*c*(e*x+d)*e+c^2*d*( e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-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)*((I*e+c*d)*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*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x +d)*e-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)*((I*e+c*d)*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+8*(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2 +e^2))^(1/2)*((I*c*(e*x+d)*e-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)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),1/(I*e+c*d)/ c*(c^2*d^2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((I*e+c*d)*c/(c^2*d^2 +e^2))^(1/2))*c^3*d^3+2*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^2*(e*x+...
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{\sqrt {e x + d}} \,d x } \] Input:
integrate(x^2*(a+b*arccsch(c*x))/(e*x+d)^(1/2),x, algorithm="fricas")
Output:
integral((b*x^2*arccsch(c*x) + a*x^2)/sqrt(e*x + d), x)
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {d + e x}}\, dx \] Input:
integrate(x**2*(a+b*acsch(c*x))/(e*x+d)**(1/2),x)
Output:
Integral(x**2*(a + b*acsch(c*x))/sqrt(d + e*x), x)
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{\sqrt {e x + d}} \,d x } \] Input:
integrate(x^2*(a+b*arccsch(c*x))/(e*x+d)^(1/2),x, algorithm="maxima")
Output:
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))
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{\sqrt {e x + d}} \,d x } \] Input:
integrate(x^2*(a+b*arccsch(c*x))/(e*x+d)^(1/2),x, algorithm="giac")
Output:
integrate((b*arccsch(c*x) + a)*x^2/sqrt(e*x + d), x)
Timed out. \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {d+e\,x}} \,d x \] Input:
int((x^2*(a + b*asinh(1/(c*x))))/(d + e*x)^(1/2),x)
Output:
int((x^2*(a + b*asinh(1/(c*x))))/(d + e*x)^(1/2), x)
\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {16 \sqrt {e x +d}\, a \,d^{2}-8 \sqrt {e x +d}\, a d e x +6 \sqrt {e x +d}\, a \,e^{2} x^{2}+15 \left (\int \frac {\mathit {acsch} \left (c x \right ) x^{2}}{\sqrt {e x +d}}d x \right ) b \,e^{3}}{15 e^{3}} \] Input:
int(x^2*(a+b*acsch(c*x))/(e*x+d)^(1/2),x)
Output:
(16*sqrt(d + e*x)*a*d**2 - 8*sqrt(d + e*x)*a*d*e*x + 6*sqrt(d + e*x)*a*e** 2*x**2 + 15*int((acsch(c*x)*x**2)/sqrt(d + e*x),x)*b*e**3)/(15*e**3)