Integrand size = 10, antiderivative size = 92 \[ \int \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=x \left (a+b \sec ^{-1}(c x)\right )^2+\frac {4 i b \left (a+b \sec ^{-1}(c x)\right ) \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{c}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{c}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{c} \]
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Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5324, 4494, 4266, 2317, 2438} \[ \int \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\frac {4 i b \arctan \left (e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{c}+x \left (a+b \sec ^{-1}(c x)\right )^2-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{c}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{c} \]
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Rule 2317
Rule 2438
Rule 4266
Rule 4494
Rule 5324
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+b x)^2 \sec (x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c} \\ & = x \left (a+b \sec ^{-1}(c x)\right )^2-\frac {(2 b) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sec ^{-1}(c x)\right )}{c} \\ & = x \left (a+b \sec ^{-1}(c x)\right )^2+\frac {4 i b \left (a+b \sec ^{-1}(c x)\right ) \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{c}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c} \\ & = x \left (a+b \sec ^{-1}(c x)\right )^2+\frac {4 i b \left (a+b \sec ^{-1}(c x)\right ) \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{c}-\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{c}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{c} \\ & = x \left (a+b \sec ^{-1}(c x)\right )^2+\frac {4 i b \left (a+b \sec ^{-1}(c x)\right ) \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{c}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{c}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{c} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.77 \[ \int \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\frac {a^2 c x+2 a b \left (c x \sec ^{-1}(c x)+\log \left (\cos \left (\frac {1}{2} \sec ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sec ^{-1}(c x)\right )\right )-\log \left (\cos \left (\frac {1}{2} \sec ^{-1}(c x)\right )+\sin \left (\frac {1}{2} \sec ^{-1}(c x)\right )\right )\right )+b^2 \left (\sec ^{-1}(c x) \left (c x \sec ^{-1}(c x)-2 \log \left (1-i e^{i \sec ^{-1}(c x)}\right )+2 \log \left (1+i e^{i \sec ^{-1}(c x)}\right )\right )-2 i \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )+2 i \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )\right )}{c} \]
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Time = 0.50 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.11
method | result | size |
derivativedivides | \(\frac {a^{2} c x +b^{2} \left (\operatorname {arcsec}\left (c x \right )^{2} c x +2 \,\operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-2 \,\operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-2 i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\right )+2 a b \left (c x \,\operatorname {arcsec}\left (c x \right )-\ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}\) | \(194\) |
default | \(\frac {a^{2} c x +b^{2} \left (\operatorname {arcsec}\left (c x \right )^{2} c x +2 \,\operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-2 \,\operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-2 i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\right )+2 a b \left (c x \,\operatorname {arcsec}\left (c x \right )-\ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}\) | \(194\) |
parts | \(a^{2} x +\frac {b^{2} \left (\operatorname {arcsec}\left (c x \right )^{2} c x +2 \,\operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-2 \,\operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-2 i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\right )}{c}+2 a b x \,\operatorname {arcsec}\left (c x \right )-\frac {2 a b \ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c}\) | \(194\) |
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\[ \int \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} \,d x } \]
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\[ \int \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}\, dx \]
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\[ \int \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} \,d x } \]
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\[ \int \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int \left (a+b \sec ^{-1}(c x)\right )^2 \, dx=\int {\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
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