Optimal. Leaf size=278 \[ \frac {3 i \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \sec ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \sec ^{-1}(a+b x)^3-\frac {6 i a \sec ^{-1}(a+b x)^2 \text {ArcTan}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {3 \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \sec ^{-1}(a+b x) \text {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \sec ^{-1}(a+b x) \text {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {3 i \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\right )}{2 b^2}-\frac {6 a \text {PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {6 a \text {PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )}{b^2} \]
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Rubi [A]
time = 0.20, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 12, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {5366, 4511,
4275, 4266, 2611, 2320, 6724, 4269, 3800, 2221, 2317, 2438} \begin {gather*} -\frac {a^2 \sec ^{-1}(a+b x)^3}{2 b^2}-\frac {6 i a \sec ^{-1}(a+b x)^2 \text {ArcTan}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \sec ^{-1}(a+b x) \text {Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \sec ^{-1}(a+b x) \text {Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {3 i \text {Li}_2\left (-e^{2 i \sec ^{-1}(a+b x)}\right )}{2 b^2}-\frac {6 a \text {Li}_3\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {6 a \text {Li}_3\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^2}+\frac {3 i \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {3 \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {1}{2} x^2 \sec ^{-1}(a+b x)^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3800
Rule 4266
Rule 4269
Rule 4275
Rule 4511
Rule 5366
Rule 6724
Rubi steps
\begin {align*} \int x \sec ^{-1}(a+b x)^3 \, dx &=\frac {\text {Subst}\left (\int x^3 \sec (x) (-a+\sec (x)) \tan (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {1}{2} x^2 \sec ^{-1}(a+b x)^3-\frac {3 \text {Subst}\left (\int x^2 (-a+\sec (x))^2 \, dx,x,\sec ^{-1}(a+b x)\right )}{2 b^2}\\ &=\frac {1}{2} x^2 \sec ^{-1}(a+b x)^3-\frac {3 \text {Subst}\left (\int \left (a^2 x^2-2 a x^2 \sec (x)+x^2 \sec ^2(x)\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{2 b^2}\\ &=-\frac {a^2 \sec ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \sec ^{-1}(a+b x)^3-\frac {3 \text {Subst}\left (\int x^2 \sec ^2(x) \, dx,x,\sec ^{-1}(a+b x)\right )}{2 b^2}+\frac {(3 a) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^2}\\ &=-\frac {3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \sec ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \sec ^{-1}(a+b x)^3-\frac {6 i a \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {3 \text {Subst}\left (\int x \tan (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^2}-\frac {(6 a) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^2}+\frac {(6 a) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {3 i \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \sec ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \sec ^{-1}(a+b x)^3-\frac {6 i a \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \sec ^{-1}(a+b x) \text {Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \sec ^{-1}(a+b x) \text {Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {(6 i) \text {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\sec ^{-1}(a+b x)\right )}{b^2}-\frac {(6 i a) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^2}+\frac {(6 i a) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {3 i \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \sec ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \sec ^{-1}(a+b x)^3-\frac {6 i a \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {3 \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \sec ^{-1}(a+b x) \text {Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \sec ^{-1}(a+b x) \text {Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {3 \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^2}-\frac {(6 a) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {(6 a) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \sec ^{-1}(a+b x)}\right )}{b^2}\\ &=\frac {3 i \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \sec ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \sec ^{-1}(a+b x)^3-\frac {6 i a \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {3 \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \sec ^{-1}(a+b x) \text {Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \sec ^{-1}(a+b x) \text {Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {6 a \text {Li}_3\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {6 a \text {Li}_3\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {(3 i) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sec ^{-1}(a+b x)}\right )}{2 b^2}\\ &=\frac {3 i \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \sec ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \sec ^{-1}(a+b x)^3-\frac {6 i a \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {3 \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \sec ^{-1}(a+b x) \text {Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \sec ^{-1}(a+b x) \text {Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {3 i \text {Li}_2\left (-e^{2 i \sec ^{-1}(a+b x)}\right )}{2 b^2}-\frac {6 a \text {Li}_3\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}+\frac {6 a \text {Li}_3\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 248, normalized size = 0.89 \begin {gather*} \frac {1}{2} \left (x^2 \sec ^{-1}(a+b x)^3-\frac {3 \left (-i \sec ^{-1}(a+b x)^2+(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2+\frac {1}{3} a^2 \sec ^{-1}(a+b x)^3+4 i a \sec ^{-1}(a+b x)^2 \text {ArcTan}\left (e^{i \sec ^{-1}(a+b x)}\right )+2 \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )+4 i a \sec ^{-1}(a+b x) \text {PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )-i \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\right )+4 a \left (-i \sec ^{-1}(a+b x) \text {PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )+\text {PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )\right )-4 a \text {PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )\right )}{b^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.01, size = 379, normalized size = 1.36
method | result | size |
derivativedivides | \(\frac {-\frac {\mathrm {arcsec}\left (b x +a \right )^{2} \left (2 \,\mathrm {arcsec}\left (b x +a \right ) a \left (b x +a \right )-\mathrm {arcsec}\left (b x +a \right ) \left (b x +a \right )^{2}+3 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )+3 i\right )}{2}+6 i a \,\mathrm {arcsec}\left (b x +a \right ) \polylog \left (2, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )-3 a \mathrm {arcsec}\left (b x +a \right )^{2} \ln \left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )-6 a \polylog \left (3, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )-6 i a \,\mathrm {arcsec}\left (b x +a \right ) \polylog \left (2, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )+3 a \mathrm {arcsec}\left (b x +a \right )^{2} \ln \left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )+6 a \polylog \left (3, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )+3 i \mathrm {arcsec}\left (b x +a \right )^{2}-3 \,\mathrm {arcsec}\left (b x +a \right ) \ln \left (1+\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )^{2}\right )+\frac {3 i \polylog \left (2, -\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )^{2}\right )}{2}}{b^{2}}\) | \(379\) |
default | \(\frac {-\frac {\mathrm {arcsec}\left (b x +a \right )^{2} \left (2 \,\mathrm {arcsec}\left (b x +a \right ) a \left (b x +a \right )-\mathrm {arcsec}\left (b x +a \right ) \left (b x +a \right )^{2}+3 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )+3 i\right )}{2}+6 i a \,\mathrm {arcsec}\left (b x +a \right ) \polylog \left (2, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )-3 a \mathrm {arcsec}\left (b x +a \right )^{2} \ln \left (1+i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )-6 a \polylog \left (3, -i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )-6 i a \,\mathrm {arcsec}\left (b x +a \right ) \polylog \left (2, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )+3 a \mathrm {arcsec}\left (b x +a \right )^{2} \ln \left (1-i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )+6 a \polylog \left (3, i \left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )\right )+3 i \mathrm {arcsec}\left (b x +a \right )^{2}-3 \,\mathrm {arcsec}\left (b x +a \right ) \ln \left (1+\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )^{2}\right )+\frac {3 i \polylog \left (2, -\left (\frac {1}{b x +a}+i \sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )^{2}\right )}{2}}{b^{2}}\) | \(379\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x \operatorname {asec}^{3}{\left (a + b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\mathrm {acos}\left (\frac {1}{a+b\,x}\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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