Optimal. Leaf size=128 \[ \frac {i \left (a+b \sec ^{-1}(c x)\right )^4}{4 b}-\left (a+b \sec ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \sec ^{-1}(c x)\right )^2 \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{4} i b^3 \text {PolyLog}\left (4,-e^{2 i \sec ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5330, 3800,
2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {3}{2} b^2 \text {Li}_3\left (-e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )+\frac {3}{2} i b \text {Li}_2\left (-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 \text {Li}_4\left (-e^{2 i \sec ^{-1}(c x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 5330
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x} \, dx &=\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 \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )-\left (3 i b^2\right ) \text {Subst}\left (\int (a+b x) \text {Li}_2\left (-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 \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {Li}_3\left (-e^{2 i \sec ^{-1}(c x)}\right )+\frac {1}{2} \left (3 b^3\right ) \text {Subst}\left (\int \text {Li}_3\left (-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 \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {Li}_3\left (-e^{2 i \sec ^{-1}(c x)}\right )-\frac {1}{4} \left (3 i b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_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 \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {Li}_3\left (-e^{2 i \sec ^{-1}(c x)}\right )-\frac {3}{4} i b^3 \text {Li}_4\left (-e^{2 i \sec ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 204, normalized size = 1.59 \begin {gather*} \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 \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )-6 b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {PolyLog}\left (3,-e^{2 i \sec ^{-1}(c x)}\right )-3 i b^3 \text {PolyLog}\left (4,-e^{2 i \sec ^{-1}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 389 vs. \(2 (177 ) = 354\).
time = 0.23, size = 390, normalized size = 3.05
method | result | size |
derivativedivides | \(a^{3} \ln \left (c x \right )+\frac {i b^{3} \mathrm {arcsec}\left (c x \right )^{4}}{4}-b^{3} \mathrm {arcsec}\left (c x \right )^{3} \ln \left (1+\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )+\frac {3 i b^{3} \mathrm {arcsec}\left (c x \right )^{2} \polylog \left (2, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2}-\frac {3 b^{3} \mathrm {arcsec}\left (c x \right ) \polylog \left (3, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2}-\frac {3 i b^{3} \polylog \left (4, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{4}+i a \,b^{2} \mathrm {arcsec}\left (c x \right )^{3}-3 a \,b^{2} \mathrm {arcsec}\left (c x \right )^{2} \ln \left (1+\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )+3 i a \,b^{2} \mathrm {arcsec}\left (c x \right ) \polylog \left (2, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )-\frac {3 a \,b^{2} \polylog \left (3, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2}+\frac {3 i a^{2} b \mathrm {arcsec}\left (c x \right )^{2}}{2}-3 a^{2} b \,\mathrm {arcsec}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )+\frac {3 i a^{2} b \polylog \left (2, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2}\) | \(390\) |
default | \(a^{3} \ln \left (c x \right )+\frac {i b^{3} \mathrm {arcsec}\left (c x \right )^{4}}{4}-b^{3} \mathrm {arcsec}\left (c x \right )^{3} \ln \left (1+\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )+\frac {3 i b^{3} \mathrm {arcsec}\left (c x \right )^{2} \polylog \left (2, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2}-\frac {3 b^{3} \mathrm {arcsec}\left (c x \right ) \polylog \left (3, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2}-\frac {3 i b^{3} \polylog \left (4, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{4}+i a \,b^{2} \mathrm {arcsec}\left (c x \right )^{3}-3 a \,b^{2} \mathrm {arcsec}\left (c x \right )^{2} \ln \left (1+\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )+3 i a \,b^{2} \mathrm {arcsec}\left (c x \right ) \polylog \left (2, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )-\frac {3 a \,b^{2} \polylog \left (3, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2}+\frac {3 i a^{2} b \mathrm {arcsec}\left (c x \right )^{2}}{2}-3 a^{2} b \,\mathrm {arcsec}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )+\frac {3 i a^{2} b \polylog \left (2, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2}\) | \(390\) |
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 \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}}{x}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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