Integrand size = 14, antiderivative size = 128 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-\left (a+b \sec ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \sec ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \sec ^{-1}(c x)}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5330, 3800, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=-\frac {3}{2} b^2 \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )+\frac {3}{2} i b \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^2+\frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-\log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^3-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \sec ^{-1}(c x)}\right ) \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 5330
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int (a+b x)^3 \tan (x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-2 i \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^3}{1+e^{2 i x}} \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-\left (a+b \sec ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )+(3 b) \text {Subst}\left (\int (a+b x)^2 \log \left (1+e^{2 i x}\right ) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-\left (a+b \sec ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \sec ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )-\left (3 i b^2\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-\left (a+b \sec ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \sec ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right )+\frac {1}{2} \left (3 b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{2 i x}\right ) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-\left (a+b \sec ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \sec ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {1}{4} \left (3 i b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 i \sec ^{-1}(c x)}\right ) \\ & = \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-\left (a+b \sec ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \sec ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \sec ^{-1}(c x)}\right ) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\frac {1}{4} \left (6 i a^2 b \sec ^{-1}(c x)^2+4 i a b^2 \sec ^{-1}(c x)^3+i b^3 \sec ^{-1}(c x)^4-12 a^2 b \sec ^{-1}(c x) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )-12 a b^2 \sec ^{-1}(c x)^2 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )-4 b^3 \sec ^{-1}(c x)^3 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )+4 a^3 \log (c x)+6 i b \left (a+b \sec ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )-6 b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right )-3 i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \sec ^{-1}(c x)}\right )\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (177 ) = 354\).
Time = 0.86 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.84
method | result | size |
parts | \(a^{3} \ln \left (x \right )+b^{3} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{4}}{4}-\operatorname {arcsec}\left (c x \right )^{3} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {3 i \operatorname {arcsec}\left (c x \right )^{2} \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 \,\operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 i \operatorname {polylog}\left (4, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{4}\right )+3 a \,b^{2} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{3}}{3}-\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )-\frac {\operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )+3 a^{2} b \left (\frac {i \operatorname {arcsec}\left (c x \right )^{2}}{2}-\operatorname {arcsec}\left (c x \right ) \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {i \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )\) | \(364\) |
derivativedivides | \(a^{3} \ln \left (c x \right )+b^{3} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{4}}{4}-\operatorname {arcsec}\left (c x \right )^{3} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {3 i \operatorname {arcsec}\left (c x \right )^{2} \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 \,\operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 i \operatorname {polylog}\left (4, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{4}\right )+3 a \,b^{2} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{3}}{3}-\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )-\frac {\operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )+3 a^{2} b \left (\frac {i \operatorname {arcsec}\left (c x \right )^{2}}{2}-\operatorname {arcsec}\left (c x \right ) \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {i \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )\) | \(366\) |
default | \(a^{3} \ln \left (c x \right )+b^{3} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{4}}{4}-\operatorname {arcsec}\left (c x \right )^{3} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {3 i \operatorname {arcsec}\left (c x \right )^{2} \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 \,\operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}-\frac {3 i \operatorname {polylog}\left (4, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{4}\right )+3 a \,b^{2} \left (\frac {i \operatorname {arcsec}\left (c x \right )^{3}}{3}-\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )-\frac {\operatorname {polylog}\left (3, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )+3 a^{2} b \left (\frac {i \operatorname {arcsec}\left (c x \right )^{2}}{2}-\operatorname {arcsec}\left (c x \right ) \ln \left (1+{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )+\frac {i \operatorname {polylog}\left (2, -{\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}\right )}{2}\right )\) | \(366\) |
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\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
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\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}}{x}\, dx \]
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\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
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\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3}{x} \,d x \]
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