\(\int \frac {x^6 \sec ^{-1}(x)}{(-1+x^2)^{5/2}} \, dx\) [689]

Optimal result

Integrand size = 15, antiderivative size = 175 \[ \int \frac {x^6 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx=\frac {\sqrt {x^2} \left (2-3 x^2\right )}{6 \left (-1+x^2\right )}-\frac {13}{6} \coth ^{-1}\left (\sqrt {x^2}\right )-\frac {5 x^3 \sec ^{-1}(x)}{6 \left (-1+x^2\right )^{3/2}}+\frac {x^5 \sec ^{-1}(x)}{2 \left (-1+x^2\right )^{3/2}}-\frac {5 x \sec ^{-1}(x)}{2 \sqrt {-1+x^2}}-\frac {5 i \sqrt {x^2} \sec ^{-1}(x) \arctan \left (e^{i \sec ^{-1}(x)}\right )}{x}+\frac {5 i \sqrt {x^2} \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(x)}\right )}{2 x}-\frac {5 i \sqrt {x^2} \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(x)}\right )}{2 x} \] Output:

-13/6*arccoth((x^2)^(1/2))-5/6*x^3*arcsec(x)/(x^2-1)^(3/2)+1/2*x^5*arcsec( 
x)/(x^2-1)^(3/2)+1/6*(-3*x^2+2)*(x^2)^(1/2)/(x^2-1)-5*I*arcsec(x)*arctan(1 
/x+I*(1-1/x^2)^(1/2))*(x^2)^(1/2)/x+5/2*I*polylog(2,-I*(1/x+I*(1-1/x^2)^(1 
/2)))*(x^2)^(1/2)/x-5/2*I*polylog(2,I*(1/x+I*(1-1/x^2)^(1/2)))*(x^2)^(1/2) 
/x-5/2*x*arcsec(x)/(x^2-1)^(1/2)
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(383\) vs. \(2(175)=350\).

Time = 1.24 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.19 \[ \int \frac {x^6 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx=-\frac {x^5 \left (22 \sec ^{-1}(x)+40 \sec ^{-1}(x) \cos \left (2 \sec ^{-1}(x)\right )-30 \sec ^{-1}(x) \cos \left (4 \sec ^{-1}(x)\right )-30 \sqrt {1-\frac {1}{x^2}} \sec ^{-1}(x) \log \left (1-i e^{i \sec ^{-1}(x)}\right )+30 \sqrt {1-\frac {1}{x^2}} \sec ^{-1}(x) \log \left (1+i e^{i \sec ^{-1}(x)}\right )+26 \sqrt {1-\frac {1}{x^2}} \log \left (\cos \left (\frac {1}{2} \sec ^{-1}(x)\right )\right )-26 \sqrt {1-\frac {1}{x^2}} \log \left (\sin \left (\frac {1}{2} \sec ^{-1}(x)\right )\right )+16 \sin \left (2 \sec ^{-1}(x)\right )-60 i \sqrt {1-\frac {1}{x^2}} \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(x)}\right ) \sin ^2\left (2 \sec ^{-1}(x)\right )+60 i \sqrt {1-\frac {1}{x^2}} \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(x)}\right ) \sin ^2\left (2 \sec ^{-1}(x)\right )-15 \sec ^{-1}(x) \log \left (1-i e^{i \sec ^{-1}(x)}\right ) \sin \left (3 \sec ^{-1}(x)\right )+15 \sec ^{-1}(x) \log \left (1+i e^{i \sec ^{-1}(x)}\right ) \sin \left (3 \sec ^{-1}(x)\right )+13 \log \left (\cos \left (\frac {1}{2} \sec ^{-1}(x)\right )\right ) \sin \left (3 \sec ^{-1}(x)\right )-13 \log \left (\sin \left (\frac {1}{2} \sec ^{-1}(x)\right )\right ) \sin \left (3 \sec ^{-1}(x)\right )-4 \sin \left (4 \sec ^{-1}(x)\right )+15 \sec ^{-1}(x) \log \left (1-i e^{i \sec ^{-1}(x)}\right ) \sin \left (5 \sec ^{-1}(x)\right )-15 \sec ^{-1}(x) \log \left (1+i e^{i \sec ^{-1}(x)}\right ) \sin \left (5 \sec ^{-1}(x)\right )-13 \log \left (\cos \left (\frac {1}{2} \sec ^{-1}(x)\right )\right ) \sin \left (5 \sec ^{-1}(x)\right )+13 \log \left (\sin \left (\frac {1}{2} \sec ^{-1}(x)\right )\right ) \sin \left (5 \sec ^{-1}(x)\right )\right )}{96 \left (-1+x^2\right )^{3/2}} \] Input:

Integrate[(x^6*ArcSec[x])/(-1 + x^2)^(5/2),x]
 

Output:

-1/96*(x^5*(22*ArcSec[x] + 40*ArcSec[x]*Cos[2*ArcSec[x]] - 30*ArcSec[x]*Co 
s[4*ArcSec[x]] - 30*Sqrt[1 - x^(-2)]*ArcSec[x]*Log[1 - I*E^(I*ArcSec[x])] 
+ 30*Sqrt[1 - x^(-2)]*ArcSec[x]*Log[1 + I*E^(I*ArcSec[x])] + 26*Sqrt[1 - x 
^(-2)]*Log[Cos[ArcSec[x]/2]] - 26*Sqrt[1 - x^(-2)]*Log[Sin[ArcSec[x]/2]] + 
 16*Sin[2*ArcSec[x]] - (60*I)*Sqrt[1 - x^(-2)]*PolyLog[2, (-I)*E^(I*ArcSec 
[x])]*Sin[2*ArcSec[x]]^2 + (60*I)*Sqrt[1 - x^(-2)]*PolyLog[2, I*E^(I*ArcSe 
c[x])]*Sin[2*ArcSec[x]]^2 - 15*ArcSec[x]*Log[1 - I*E^(I*ArcSec[x])]*Sin[3* 
ArcSec[x]] + 15*ArcSec[x]*Log[1 + I*E^(I*ArcSec[x])]*Sin[3*ArcSec[x]] + 13 
*Log[Cos[ArcSec[x]/2]]*Sin[3*ArcSec[x]] - 13*Log[Sin[ArcSec[x]/2]]*Sin[3*A 
rcSec[x]] - 4*Sin[4*ArcSec[x]] + 15*ArcSec[x]*Log[1 - I*E^(I*ArcSec[x])]*S 
in[5*ArcSec[x]] - 15*ArcSec[x]*Log[1 + I*E^(I*ArcSec[x])]*Sin[5*ArcSec[x]] 
 - 13*Log[Cos[ArcSec[x]/2]]*Sin[5*ArcSec[x]] + 13*Log[Sin[ArcSec[x]/2]]*Si 
n[5*ArcSec[x]]))/(-1 + x^2)^(3/2)
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.13, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5765, 5205, 253, 264, 219, 5209, 215, 219, 5209, 219, 5219, 3042, 4669, 2715, 2838}

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.

\(\displaystyle \int \frac {x^6 \sec ^{-1}(x)}{\left (x^2-1\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5765

\(\displaystyle -\frac {\sqrt {x^2} \int \frac {x^3 \arccos \left (\frac {1}{x}\right )}{\left (1-\frac {1}{x^2}\right )^{5/2}}d\frac {1}{x}}{x}\)

\(\Big \downarrow \) 5205

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \int \frac {x \arccos \left (\frac {1}{x}\right )}{\left (1-\frac {1}{x^2}\right )^{5/2}}d\frac {1}{x}-\frac {1}{2} \int \frac {x^2}{\left (1-\frac {1}{x^2}\right )^2}d\frac {1}{x}-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}\right )}{x}\)

\(\Big \downarrow \) 253

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \int \frac {x \arccos \left (\frac {1}{x}\right )}{\left (1-\frac {1}{x^2}\right )^{5/2}}d\frac {1}{x}+\frac {1}{2} \left (-\frac {3}{2} \int \frac {x^2}{1-\frac {1}{x^2}}d\frac {1}{x}-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}\right )}{x}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \int \frac {x \arccos \left (\frac {1}{x}\right )}{\left (1-\frac {1}{x^2}\right )^{5/2}}d\frac {1}{x}+\frac {1}{2} \left (-\frac {3}{2} \left (\int \frac {1}{1-\frac {1}{x^2}}d\frac {1}{x}-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}\right )}{x}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \int \frac {x \arccos \left (\frac {1}{x}\right )}{\left (1-\frac {1}{x^2}\right )^{5/2}}d\frac {1}{x}-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{2} \left (-\frac {3}{2} \left (\text {arctanh}\left (\frac {1}{x}\right )-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )\right )}{x}\)

\(\Big \downarrow \) 5209

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \left (\int \frac {x \arccos \left (\frac {1}{x}\right )}{\left (1-\frac {1}{x^2}\right )^{3/2}}d\frac {1}{x}+\frac {1}{3} \int \frac {1}{\left (1-\frac {1}{x^2}\right )^2}d\frac {1}{x}+\frac {\arccos \left (\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{2} \left (-\frac {3}{2} \left (\text {arctanh}\left (\frac {1}{x}\right )-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )\right )}{x}\)

\(\Big \downarrow \) 215

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \left (\int \frac {x \arccos \left (\frac {1}{x}\right )}{\left (1-\frac {1}{x^2}\right )^{3/2}}d\frac {1}{x}+\frac {1}{3} \left (\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {1}{x}+\frac {1}{2 \left (1-\frac {1}{x^2}\right ) x}\right )+\frac {\arccos \left (\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{2} \left (-\frac {3}{2} \left (\text {arctanh}\left (\frac {1}{x}\right )-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )\right )}{x}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \left (\int \frac {x \arccos \left (\frac {1}{x}\right )}{\left (1-\frac {1}{x^2}\right )^{3/2}}d\frac {1}{x}+\frac {\arccos \left (\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{3} \left (\frac {1}{2} \text {arctanh}\left (\frac {1}{x}\right )+\frac {1}{2 \left (1-\frac {1}{x^2}\right ) x}\right )\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{2} \left (-\frac {3}{2} \left (\text {arctanh}\left (\frac {1}{x}\right )-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )\right )}{x}\)

\(\Big \downarrow \) 5209

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \left (\int \frac {x \arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}+\int \frac {1}{1-\frac {1}{x^2}}d\frac {1}{x}+\frac {\arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}+\frac {\arccos \left (\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{3} \left (\frac {1}{2} \text {arctanh}\left (\frac {1}{x}\right )+\frac {1}{2 \left (1-\frac {1}{x^2}\right ) x}\right )\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{2} \left (-\frac {3}{2} \left (\text {arctanh}\left (\frac {1}{x}\right )-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )\right )}{x}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \left (\int \frac {x \arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}d\frac {1}{x}+\frac {\arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}+\frac {\arccos \left (\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{3} \left (\frac {1}{2} \text {arctanh}\left (\frac {1}{x}\right )+\frac {1}{2 \left (1-\frac {1}{x^2}\right ) x}\right )+\text {arctanh}\left (\frac {1}{x}\right )\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{2} \left (-\frac {3}{2} \left (\text {arctanh}\left (\frac {1}{x}\right )-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )\right )}{x}\)

\(\Big \downarrow \) 5219

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \left (-\int x \arccos \left (\frac {1}{x}\right )d\arccos \left (\frac {1}{x}\right )+\frac {\arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}+\frac {\arccos \left (\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{3} \left (\frac {1}{2} \text {arctanh}\left (\frac {1}{x}\right )+\frac {1}{2 \left (1-\frac {1}{x^2}\right ) x}\right )+\text {arctanh}\left (\frac {1}{x}\right )\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{2} \left (-\frac {3}{2} \left (\text {arctanh}\left (\frac {1}{x}\right )-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )\right )}{x}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \left (-\int \arccos \left (\frac {1}{x}\right ) \csc \left (\arccos \left (\frac {1}{x}\right )+\frac {\pi }{2}\right )d\arccos \left (\frac {1}{x}\right )+\frac {\arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}+\frac {\arccos \left (\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{3} \left (\frac {1}{2} \text {arctanh}\left (\frac {1}{x}\right )+\frac {1}{2 \left (1-\frac {1}{x^2}\right ) x}\right )+\text {arctanh}\left (\frac {1}{x}\right )\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{2} \left (-\frac {3}{2} \left (\text {arctanh}\left (\frac {1}{x}\right )-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )\right )}{x}\)

\(\Big \downarrow \) 4669

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \left (\int \log \left (1-i e^{i \arccos \left (\frac {1}{x}\right )}\right )d\arccos \left (\frac {1}{x}\right )-\int \log \left (1+i e^{i \arccos \left (\frac {1}{x}\right )}\right )d\arccos \left (\frac {1}{x}\right )+2 i \arccos \left (\frac {1}{x}\right ) \arctan \left (e^{i \arccos \left (\frac {1}{x}\right )}\right )+\frac {\arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}+\frac {\arccos \left (\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{3} \left (\frac {1}{2} \text {arctanh}\left (\frac {1}{x}\right )+\frac {1}{2 \left (1-\frac {1}{x^2}\right ) x}\right )+\text {arctanh}\left (\frac {1}{x}\right )\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{2} \left (-\frac {3}{2} \left (\text {arctanh}\left (\frac {1}{x}\right )-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )\right )}{x}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \left (-i \int x \log \left (1-i e^{i \arccos \left (\frac {1}{x}\right )}\right )de^{i \arccos \left (\frac {1}{x}\right )}+i \int x \log \left (1+i e^{i \arccos \left (\frac {1}{x}\right )}\right )de^{i \arccos \left (\frac {1}{x}\right )}+2 i \arccos \left (\frac {1}{x}\right ) \arctan \left (e^{i \arccos \left (\frac {1}{x}\right )}\right )+\frac {\arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}+\frac {\arccos \left (\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{3} \left (\frac {1}{2} \text {arctanh}\left (\frac {1}{x}\right )+\frac {1}{2 \left (1-\frac {1}{x^2}\right ) x}\right )+\text {arctanh}\left (\frac {1}{x}\right )\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{2} \left (-\frac {3}{2} \left (\text {arctanh}\left (\frac {1}{x}\right )-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )\right )}{x}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {\sqrt {x^2} \left (\frac {5}{2} \left (2 i \arccos \left (\frac {1}{x}\right ) \arctan \left (e^{i \arccos \left (\frac {1}{x}\right )}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos \left (\frac {1}{x}\right )}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos \left (\frac {1}{x}\right )}\right )+\frac {\arccos \left (\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}+\frac {\arccos \left (\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{3} \left (\frac {1}{2} \text {arctanh}\left (\frac {1}{x}\right )+\frac {1}{2 \left (1-\frac {1}{x^2}\right ) x}\right )+\text {arctanh}\left (\frac {1}{x}\right )\right )-\frac {x^2 \arccos \left (\frac {1}{x}\right )}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{2} \left (-\frac {3}{2} \left (\text {arctanh}\left (\frac {1}{x}\right )-x\right )-\frac {x}{2 \left (1-\frac {1}{x^2}\right )}\right )\right )}{x}\)

Input:

Int[(x^6*ArcSec[x])/(-1 + x^2)^(5/2),x]
 

Output:

-((Sqrt[x^2]*(-1/2*(x^2*ArcCos[x^(-1)])/(1 - x^(-2))^(3/2) + (-1/2*x/(1 - 
x^(-2)) - (3*(-x + ArcTanh[x^(-1)]))/2)/2 + (5*(ArcCos[x^(-1)]/(3*(1 - x^( 
-2))^(3/2)) + ArcCos[x^(-1)]/Sqrt[1 - x^(-2)] + (2*I)*ArcCos[x^(-1)]*ArcTa 
n[E^(I*ArcCos[x^(-1)])] + (1/(2*(1 - x^(-2))*x) + ArcTanh[x^(-1)]/2)/3 + A 
rcTanh[x^(-1)] - I*PolyLog[2, (-I)*E^(I*ArcCos[x^(-1)])] + I*PolyLog[2, I* 
E^(I*ArcCos[x^(-1)])]))/2))/x)
 

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5205

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcCos[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Simp[b* 
c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*( 
1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, 
 c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 5209

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a 
+ b*ArcCos[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) 
   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Simp[b*c 
*(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)* 
(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b 
, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
 

rule 5219

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* 
(x_)^2], x_Symbol] :> Simp[(-(c^(m + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[ 
d + e*x^2]]   Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]], x] /; Fr 
eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
 

rule 5765

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[-Sqrt[x^2]/x   Subst[Int[(e + d*x^2)^p*((a + b* 
ArcCos[x/c])^n/x^(m + 2*(p + 1))), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, 
 n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1 
/2] && GtQ[e, 0] && LtQ[d, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.97 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.41

method	result	size
default	\(\frac {\sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{2} \operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (3 \,\operatorname {arcsec}\left (x \right ) x^{4}-3 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-20 x^{2} \operatorname {arcsec}\left (x \right )+2 x \sqrt {\frac {x^{2}-1}{x^{2}}}+15 \,\operatorname {arcsec}\left (x \right )\right )}{6 \left (x^{2}-1\right )^{2}}+\frac {i \operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (15 i \operatorname {arcsec}\left (x \right ) \ln \left (1+i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )-15 i \operatorname {arcsec}\left (x \right ) \ln \left (1-i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )+13 i \ln \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}+1\right )-13 i \ln \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}-1\right )+15 \operatorname {dilog}\left (1+i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )-15 \operatorname {dilog}\left (1-i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )\right )}{6}\)	\(246\)

Input:

int(x^6*arcsec(x)/(x^2-1)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/6*((x^2-1)/x^2)^(1/2)*x^2/(x^2-1)^2*csgn(x*(1-1/x^2)^(1/2))*(3*arcsec(x) 
*x^4-3*((x^2-1)/x^2)^(1/2)*x^3-20*x^2*arcsec(x)+2*x*((x^2-1)/x^2)^(1/2)+15 
*arcsec(x))+1/6*I*csgn(x*(1-1/x^2)^(1/2))*(15*I*arcsec(x)*ln(1+I*(1/x+I*(1 
-1/x^2)^(1/2)))-15*I*arcsec(x)*ln(1-I*(1/x+I*(1-1/x^2)^(1/2)))+13*I*ln(1/x 
+I*(1-1/x^2)^(1/2)+1)-13*I*ln(1/x+I*(1-1/x^2)^(1/2)-1)+15*dilog(1+I*(1/x+I 
*(1-1/x^2)^(1/2)))-15*dilog(1-I*(1/x+I*(1-1/x^2)^(1/2))))
 

Fricas [F]

\[ \int \frac {x^6 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx=\int { \frac {x^{6} \operatorname {arcsec}\left (x\right )}{{\left (x^{2} - 1\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^6*arcsec(x)/(x^2-1)^(5/2),x, algorithm="fricas")
 

Output:

integral(sqrt(x^2 - 1)*x^6*arcsec(x)/(x^6 - 3*x^4 + 3*x^2 - 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^6 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**6*asec(x)/(x**2-1)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^6 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx=\int { \frac {x^{6} \operatorname {arcsec}\left (x\right )}{{\left (x^{2} - 1\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^6*arcsec(x)/(x^2-1)^(5/2),x, algorithm="maxima")
 

Output:

integrate(x^6*arcsec(x)/(x^2 - 1)^(5/2), x)
 

Giac [F]

\[ \int \frac {x^6 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx=\int { \frac {x^{6} \operatorname {arcsec}\left (x\right )}{{\left (x^{2} - 1\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^6*arcsec(x)/(x^2-1)^(5/2),x, algorithm="giac")
 

Output:

integrate(x^6*arcsec(x)/(x^2 - 1)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx=\int \frac {x^6\,\mathrm {acos}\left (\frac {1}{x}\right )}{{\left (x^2-1\right )}^{5/2}} \,d x \] Input:

int((x^6*acos(1/x))/(x^2 - 1)^(5/2),x)
 

Output:

int((x^6*acos(1/x))/(x^2 - 1)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^6 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx=\int \frac {\mathit {asec} \left (x \right ) x^{6}}{\sqrt {x^{2}-1}\, x^{4}-2 \sqrt {x^{2}-1}\, x^{2}+\sqrt {x^{2}-1}}d x \] Input:

int(x^6*asec(x)/(x^2-1)^(5/2),x)
 

Output:

int((asec(x)*x**6)/(sqrt(x**2 - 1)*x**4 - 2*sqrt(x**2 - 1)*x**2 + sqrt(x** 
2 - 1)),x)