Integrand size = 21, antiderivative size = 462 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {4 b c d^2 \sqrt {1-\frac {1}{c^2 x^2}} x}{3 e^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {32 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {64 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \] Output:
4/3*b*c*d^2*(1-1/c^2/x^2)^(1/2)*x/e^2/(c^2*d^2-e^2)/(e*x+d)^(1/2)+2/3*d^3* (a+b*arccsc(c*x))/e^4/(e*x+d)^(3/2)-6*d^2*(a+b*arccsc(c*x))/e^4/(e*x+d)^(1 /2)-6*d*(e*x+d)^(1/2)*(a+b*arccsc(c*x))/e^4+2/3*(e*x+d)^(3/2)*(a+b*arccsc( c*x))/e^4+4/3*b*(e*x+d)^(1/2)*(-c^2*x^2+1)^(1/2)*EllipticE(1/2*(-c*x+1)^(1 /2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))/c^2/e/(c^2*d^2-e^2)/(1-1/c^2/x^2)^( 1/2)/x/(c*(e*x+d)/(c*d+e))^(1/2)+32/3*b*d*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2* x^2+1)^(1/2)*EllipticF(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2 ))/c^2/e^3/(1-1/c^2/x^2)^(1/2)/x/(e*x+d)^(1/2)+64/3*b*d^2*(c*(e*x+d)/(c*d+ e))^(1/2)*(-c^2*x^2+1)^(1/2)*EllipticPi(1/2*(-c*x+1)^(1/2)*2^(1/2),2,2^(1/ 2)*(e/(c*d+e))^(1/2))/c/e^4/(1-1/c^2/x^2)^(1/2)/x/(e*x+d)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 33.62 (sec) , antiderivative size = 887, normalized size of antiderivative = 1.92 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x)^(5/2),x]
Output:
(a*d^4*(1 + (e*x)/d)^(5/2)*Beta[-((e*x)/d), 4, -3/2])/(e^4*(d + e*x)^(5/2) ) + (b*(-((c^3*(e + d/x)^3*x^3*((-4*Sqrt[1 - 1/(c^2*x^2)])/(3*e*(-(c^2*d^2 ) + e^2)) + (32*c*d*ArcCsc[c*x])/(3*e^4) - (2*c*d*ArcCsc[c*x])/(3*e^2*(e + d/x)^2) - (2*c*x*ArcCsc[c*x])/(3*e^3) - (2*(-2*c^2*d^2*e*Sqrt[1 - 1/(c^2* x^2)] - 7*c^3*d^3*ArcCsc[c*x] + 7*c*d*e^2*ArcCsc[c*x]))/(3*e^3*(-(c^2*d^2) + e^2)*(e + d/x))))/(d + e*x)^(5/2)) + (2*(e + d/x)^(5/2)*(c*x)^(5/2)*((2 *(8*c^3*d^3*e - 8*c*d*e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2] *EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c ^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(16*c^4*d^4 - 16*c^2*d^2*e^2 - e^ 4)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sq rt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/ x]*(c*x)^(3/2)) + (2*e^3*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin [Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x )/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqr t[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqr t[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-( c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*Ellipti cPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(Sqrt[1 - 1/(c^2* x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(3*(c*d - e)*e^4*(c*d +...
Time = 2.59 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.21, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.048, Rules used = {5770, 27, 7272, 2351, 635, 25, 27, 498, 27, 508, 327, 632, 186, 413, 412, 2182, 27, 600, 508, 327, 511, 321}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5770 |
| \(\displaystyle \frac {b \int -\frac {2 \left (16 d^3+24 e x d^2+6 e^2 x^2 d-e^3 x^3\right )}{3 e^4 \sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}}dx}{c}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \int \frac {16 d^3+24 e x d^2+6 e^2 x^2 d-e^3 x^3}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}}dx}{3 c e^4}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {16 d^3+24 e x d^2+6 e^2 x^2 d-e^3 x^3}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (16 d^3 \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 635 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (16 d^3 \left (\int -\frac {e}{d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}\right )+\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (16 d^3 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\int \frac {e}{d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )+\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (16 d^3 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx}{d}\right )+\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+16 d^3 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c^2 \int -\frac {\sqrt {d+e x}}{2 \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}\right )}{d}\right )\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+16 d^3 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {c^2 \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}+\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{d}\right )\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+16 d^3 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (16 d^3 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (16 d^3 \left (\frac {\int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (16 d^3 \left (-\frac {2 \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (16 d^3 \left (-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {-x^2 e^3+6 d x e^2+24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+16 d^3 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {2 \int \frac {e \left (d \left (24 c^2 d^2-7 e^2\right )+e \left (16 c^2 d^2+e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}+16 d^3 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {34 d^2 e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \int \frac {d \left (24 c^2 d^2-7 e^2\right )+e \left (16 c^2 d^2+e^2\right ) x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}+16 d^3 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {34 d^2 e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \left (8 d \left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+\left (16 c^2 d^2+e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx\right )}{c^2 d^2-e^2}+16 d^3 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {34 d^2 e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \left (8 d \left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 \left (16 c^2 d^2+e^2\right ) \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2 d^2-e^2}+16 d^3 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {34 d^2 e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \left (8 d \left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 \left (16 c^2 d^2+e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2 d^2-e^2}+16 d^3 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {34 d^2 e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \left (-\frac {16 d \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^2+e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2 d^2-e^2}+16 d^3 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {34 d^2 e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \left (-\frac {16 d \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^2+e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2 d^2-e^2}+16 d^3 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {34 d^2 e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
Input:
Int[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x)^(5/2),x]
Output:
(2*d^3*(a + b*ArcCsc[c*x]))/(3*e^4*(d + e*x)^(3/2)) - (6*d^2*(a + b*ArcCsc [c*x]))/(e^4*Sqrt[d + e*x]) - (6*d*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^4 + (2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^4) - (2*b*Sqrt[1 - c^2*x^2] *((34*d^2*e^2*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*Sqrt[d + e*x]) + (e*((-2 *(16*c^2*d^2 + e^2)*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[(c*(d + e*x))/(c*d + e)]) - (16*d*(c^2*d^2 - e^ 2)*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x])))/(c^2*d^2 - e^2) + 16*d^3*(-((e*((2*e *Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*Sqrt[d + e*x]) - (2*c*Sqrt[d + e*x]*E llipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/((c^2*d^2 - e^2) *Sqrt[(c*(d + e*x))/(c*d + e)])))/d) - (2*Sqrt[1 - (e*(1 - c*x))/(c*d + e) ]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(d*Sqrt[d + e/c - (e*(1 - c*x))/c]))))/(3*c*e^4*Sqrt[1 - 1/(c^2*x^2)]*x)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[((c_) + (d_.)*(x_))^(n_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[c^(n + 1/2) Int[1/(x*Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] + Int[( (c + d*x)^n/Sqrt[a + b*x^2])*ExpandToSum[(1 - c^(n + 1/2)*(c + d*x)^(-n - 1 /2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n + 1/2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
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_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide [u, x]}, Simp[(a + b*ArcCsc[c*x]) v, x] + Simp[b/c Int[SimplifyIntegran d[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]
Leaf count of result is larger than twice the leaf count of optimal. \(1064\) vs. \(2(417)=834\).
Time = 12.39 (sec) , antiderivative size = 1065, normalized size of antiderivative = 2.31
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1065\) |
| default | \(\text {Expression too large to display}\) | \(1065\) |
| parts | \(\text {Expression too large to display}\) | \(1080\) |
Input:
int(x^3*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/e^4*(-a*(-1/3*(e*x+d)^(3/2)+3*d*(e*x+d)^(1/2)-1/3*d^3/(e*x+d)^(3/2)+3*d^ 2/(e*x+d)^(1/2))-b*(-1/3*(e*x+d)^(3/2)*arccsc(c*x)+3*arccsc(c*x)*d*(e*x+d) ^(1/2)-1/3*arccsc(c*x)*d^3/(e*x+d)^(3/2)+3*arccsc(c*x)*d^2/(e*x+d)^(1/2)-2 /3/c^2*((c/(c*d-e))^(1/2)*c^3*d^2*(e*x+d)^2-8*((-c*(e*x+d)+c*d-e)/(c*d-e)) ^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d- e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^3*d^3*(e*x+d)^(1/2)+16*((-c*(e*x+d)+c *d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticPi((e*x+d) ^(1/2)*(c/(c*d-e))^(1/2),1/c*(c*d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d-e))^(1/2) )*c^3*d^3*(e*x+d)^(1/2)-2*(c/(c*d-e))^(1/2)*c^3*d^3*(e*x+d)+(c/(c*d-e))^(1 /2)*c^3*d^4+7*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+ e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2 ))*c*d*e^2*(e*x+d)^(1/2)+((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c *d+e)/(c*d+e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c *d+e))^(1/2))*c*d*e^2*(e*x+d)^(1/2)-16*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)* ((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticPi((e*x+d)^(1/2)*(c/(c*d-e))^(1 /2),1/c*(c*d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d-e))^(1/2))*c*d*e^2*(e*x+d)^(1/ 2)-((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*E llipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*e^3*(e*x +d)^(1/2)+((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^ (1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2)...
\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^3*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="fricas")
Output:
integral((b*x^3*arccsc(c*x) + a*x^3)*sqrt(e*x + d)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)
Timed out. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(x**3*(a+b*acsc(c*x))/(e*x+d)**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e+c*d>0)', see `assume?` for mor e details)
\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^3*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="giac")
Output:
integrate((b*arccsc(c*x) + a)*x^3/(e*x + d)^(5/2), x)
Timed out. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:
int((x^3*(a + b*asin(1/(c*x))))/(d + e*x)^(5/2),x)
Output:
int((x^3*(a + b*asin(1/(c*x))))/(d + e*x)^(5/2), x)
\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {3 \sqrt {e x +d}\, \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{3}}{\sqrt {e x +d}\, d^{2}+2 \sqrt {e x +d}\, d e x +\sqrt {e x +d}\, e^{2} x^{2}}d x \right ) b d \,e^{4}+3 \sqrt {e x +d}\, \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{3}}{\sqrt {e x +d}\, d^{2}+2 \sqrt {e x +d}\, d e x +\sqrt {e x +d}\, e^{2} x^{2}}d x \right ) b \,e^{5} x -32 a \,d^{3}-48 a \,d^{2} e x -12 a d \,e^{2} x^{2}+2 a \,e^{3} x^{3}}{3 \sqrt {e x +d}\, e^{4} \left (e x +d \right )} \] Input:
int(x^3*(a+b*acsc(c*x))/(e*x+d)^(5/2),x)
Output:
(3*sqrt(d + e*x)*int((acsc(c*x)*x**3)/(sqrt(d + e*x)*d**2 + 2*sqrt(d + e*x )*d*e*x + sqrt(d + e*x)*e**2*x**2),x)*b*d*e**4 + 3*sqrt(d + e*x)*int((acsc (c*x)*x**3)/(sqrt(d + e*x)*d**2 + 2*sqrt(d + e*x)*d*e*x + sqrt(d + e*x)*e* *2*x**2),x)*b*e**5*x - 32*a*d**3 - 48*a*d**2*e*x - 12*a*d*e**2*x**2 + 2*a* e**3*x**3)/(3*sqrt(d + e*x)*e**4*(d + e*x))