Integrand size = 14, antiderivative size = 147 \[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\frac {b^2 x}{3 c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {2 i b \left (a+b \sec ^{-1}(c x)\right ) \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{3 c^3} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5330, 4494, 4270, 4266, 2317, 2438} \[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\frac {2 i b \arctan \left (e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{3 c^3}-\frac {b x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2-\frac {i b^2 \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 x}{3 c^2} \]
[In]
[Out]
Rule 2317
Rule 2438
Rule 4266
Rule 4270
Rule 4494
Rule 5330
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+b x)^2 \sec ^3(x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^3} \\ & = \frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2-\frac {(2 b) \text {Subst}\left (\int (a+b x) \sec ^3(x) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3} \\ & = \frac {b^2 x}{3 c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2-\frac {b \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3} \\ & = \frac {b^2 x}{3 c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {2 i b \left (a+b \sec ^{-1}(c x)\right ) \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3}-\frac {b^2 \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3} \\ & = \frac {b^2 x}{3 c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {2 i b \left (a+b \sec ^{-1}(c x)\right ) \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{3 c^3}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{3 c^3} \\ & = \frac {b^2 x}{3 c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {2 i b \left (a+b \sec ^{-1}(c x)\right ) \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{3 c^3} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.54 \[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\frac {1}{3} \left (a^2 x^3+\frac {a b \left (2 x^4 \sec ^{-1}(c x)+\frac {c x-c^3 x^3+\sqrt {-1+c^2 x^2} \log \left (-c x+\sqrt {-1+c^2 x^2}\right )}{c^4 \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{x}+\frac {b^2 \left (c x-c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \sec ^{-1}(c x)+c^3 x^3 \sec ^{-1}(c x)^2-\sec ^{-1}(c x) \log \left (1-i e^{i \sec ^{-1}(c x)}\right )+\sec ^{-1}(c x) \log \left (1+i e^{i \sec ^{-1}(c x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )\right )}{c^3}\right ) \]
[In]
[Out]
Time = 1.17 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.94
method | result | size |
parts | \(\frac {a^{2} x^{3}}{3}+\frac {b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}-\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}+\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}\right )}{c^{3}}+\frac {2 a b \left (\frac {c^{3} x^{3} \operatorname {arcsec}\left (c x \right )}{3}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\) | \(285\) |
derivativedivides | \(\frac {\frac {a^{2} c^{3} x^{3}}{3}+b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}-\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}+\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}\right )+2 a b \left (\frac {c^{3} x^{3} \operatorname {arcsec}\left (c x \right )}{3}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\) | \(286\) |
default | \(\frac {\frac {a^{2} c^{3} x^{3}}{3}+b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}-\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}+\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}\right )+2 a b \left (\frac {c^{3} x^{3} \operatorname {arcsec}\left (c x \right )}{3}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\) | \(286\) |
[In]
[Out]
\[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
[In]
[Out]
\[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int x^{2} \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}\, dx \]
[In]
[Out]
\[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
[In]
[Out]
\[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
[In]
[Out]