Optimal. Leaf size=126 \[ -\frac {3 b^2 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {3 b x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {3 i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {3 i b^3 \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )}{2 c^2} \]
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Rubi [A] time = 0.14, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5222, 4409, 4184, 3719, 2190, 2279, 2391} \[ \frac {3 i b^3 \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{2 c^2}-\frac {3 b^2 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {3 b x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {3 i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^3 \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4184
Rule 4409
Rule 5222
Rubi steps
\begin {align*} \int x \left (a+b \sec ^{-1}(c x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x)^3 \sec ^2(x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^2}\\ &=\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {(3 b) \operatorname {Subst}\left (\int (a+b x)^2 \sec ^2(x) \, dx,x,\sec ^{-1}(c x)\right )}{2 c^2}\\ &=-\frac {3 b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^2}\\ &=\frac {3 i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {\left (6 i b^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sec ^{-1}(c x)\right )}{c^2}\\ &=\frac {3 i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{c^2}+\frac {\left (3 b^3\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c^2}\\ &=\frac {3 i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sec ^{-1}(c x)}\right )}{2 c^2}\\ &=\frac {3 i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{c^2}+\frac {3 i b^3 \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 184, normalized size = 1.46 \[ \frac {a \left (a c x \left (a c x-3 b \sqrt {1-\frac {1}{c^2 x^2}}\right )-6 b^2 \log \left (\frac {1}{c x}\right )\right )-3 b^2 \sec ^{-1}(c x)^2 \left (-a c^2 x^2+b \left (c x \sqrt {1-\frac {1}{c^2 x^2}}-i\right )\right )-3 b \sec ^{-1}(c x) \left (a c x \left (2 b \sqrt {1-\frac {1}{c^2 x^2}}-a c x\right )+2 b^2 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )\right )+b^3 c^2 x^2 \sec ^{-1}(c x)^3+3 i b^3 \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )}{2 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x \operatorname {arcsec}\left (c x\right )^{3} + 3 \, a b^{2} x \operatorname {arcsec}\left (c x\right )^{2} + 3 \, a^{2} b x \operatorname {arcsec}\left (c x\right ) + a^{3} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.80, size = 285, normalized size = 2.26 \[ \frac {x^{2} a^{3}}{2}+\frac {x^{2} b^{3} \mathrm {arcsec}\left (c x \right )^{3}}{2}-\frac {3 b^{3} \mathrm {arcsec}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}{2 c}+\frac {3 i b^{3} \mathrm {arcsec}\left (c x \right )^{2}}{2 c^{2}}-\frac {3 b^{3} \mathrm {arcsec}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{c^{2}}+\frac {3 i b^{3} \polylog \left (2, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2 c^{2}}+\frac {3 x^{2} a \,b^{2} \mathrm {arcsec}\left (c x \right )^{2}}{2}-\frac {3 a \,b^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \mathrm {arcsec}\left (c x \right ) x}{c}-\frac {3 a \,b^{2} \ln \left (\frac {1}{c x}\right )}{c^{2}}+\frac {3 a^{2} b \,x^{2} \mathrm {arcsec}\left (c x \right )}{2}-\frac {3 a^{2} b x}{2 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 a^{2} b}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3}{2} \, a b^{2} x^{2} \operatorname {arcsec}\left (c x\right )^{2} + \frac {1}{2} \, a^{3} x^{2} + \frac {3}{2} \, {\left (x^{2} \operatorname {arcsec}\left (c x\right ) - \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} a^{2} b - 3 \, {\left (\frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arcsec}\left (c x\right )}{c} - \frac {\log \relax (x)}{c^{2}}\right )} a b^{2} + \frac {1}{8} \, {\left (4 \, x^{2} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )^{3} - 3 \, x^{2} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right ) \log \left (c^{2} x^{2}\right )^{2} - 8 \, \int \frac {3 \, {\left ({\left (4 \, x \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} - x \log \left (c^{2} x^{2}\right )^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1} + 4 \, {\left (2 \, c^{2} x^{3} \log \relax (c)^{2} - 2 \, x \log \relax (c)^{2} + 2 \, {\left (c^{2} x^{3} - x\right )} \log \relax (x)^{2} - {\left ({\left (2 \, c^{2} \log \relax (c) + c^{2}\right )} x^{3} - x {\left (2 \, \log \relax (c) + 1\right )} + 2 \, {\left (c^{2} x^{3} - x\right )} \log \relax (x)\right )} \log \left (c^{2} x^{2}\right ) + 4 \, {\left (c^{2} x^{3} \log \relax (c) - x \log \relax (c)\right )} \log \relax (x)\right )} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )\right )}}{8 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x}\right )} b^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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