Optimal. Leaf size=124 \[ \frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{4} i b^3 \text {PolyLog}\left (4,e^{2 i \csc ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 124, 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 = {5331, 3798,
2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {3}{2} b^2 \text {Li}_3\left (e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac {3}{2} i b \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2+\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^3-\frac {3}{4} i b^3 \text {Li}_4\left (e^{2 i \csc ^{-1}(c x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3798
Rule 5331
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx &=-\text {Subst}\left (\int (a+b x)^3 \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {i \left (a+b \csc ^{-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,\csc ^{-1}(c x)\right )\\ &=\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+(3 b) \text {Subst}\left (\int (a+b x)^2 \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 i \csc ^{-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,\csc ^{-1}(c x)\right )\\ &=\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_3\left (e^{2 i \csc ^{-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,\csc ^{-1}(c x)\right )\\ &=\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_3\left (e^{2 i \csc ^{-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 \csc ^{-1}(c x)}\right )\\ &=\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_3\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{4} i b^3 \text {Li}_4\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 242, normalized size = 1.95 \begin {gather*} a^3 \log (c x)+\frac {3}{2} i a^2 b \left (\csc ^{-1}(c x) \left (\csc ^{-1}(c x)+2 i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right )+\text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )+\frac {1}{8} i a b^2 \left (\pi ^3-8 \csc ^{-1}(c x)^3+24 i \csc ^{-1}(c x)^2 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )-24 \csc ^{-1}(c x) \text {PolyLog}\left (2,e^{-2 i \csc ^{-1}(c x)}\right )+12 i \text {PolyLog}\left (3,e^{-2 i \csc ^{-1}(c x)}\right )\right )+\frac {1}{64} i b^3 \left (\pi ^4-16 \csc ^{-1}(c x)^4+64 i \csc ^{-1}(c x)^3 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )-96 \csc ^{-1}(c x)^2 \text {PolyLog}\left (2,e^{-2 i \csc ^{-1}(c x)}\right )+96 i \csc ^{-1}(c x) \text {PolyLog}\left (3,e^{-2 i \csc ^{-1}(c x)}\right )+48 \text {PolyLog}\left (4,e^{-2 i \csc ^{-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. 665 vs. \(2 (169 ) = 338\).
time = 0.40, size = 666, normalized size = 5.37
method | result | size |
derivativedivides | \(a^{3} \ln \left (c x \right )+\frac {3 i a^{2} b \mathrm {arccsc}\left (c x \right )^{2}}{2}-b^{3} \mathrm {arccsc}\left (c x \right )^{3} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i a^{2} b \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 b^{3} \mathrm {arccsc}\left (c x \right ) \polylog \left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i b^{3} \polylog \left (4, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b^{3} \mathrm {arccsc}\left (c x \right )^{3} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i a \,b^{2} \mathrm {arccsc}\left (c x \right )^{3}-6 b^{3} \mathrm {arccsc}\left (c x \right ) \polylog \left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {i b^{3} \mathrm {arccsc}\left (c x \right )^{4}}{4}+3 i b^{3} \mathrm {arccsc}\left (c x \right )^{2} \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-3 a \,b^{2} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-3 a \,b^{2} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i b^{3} \polylog \left (4, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 i a \,b^{2} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 a \,b^{2} \polylog \left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 a \,b^{2} \polylog \left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 i a \,b^{2} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-3 a^{2} b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-3 a^{2} b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i b^{3} \mathrm {arccsc}\left (c x \right )^{2} \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i a^{2} b \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\) | \(666\) |
default | \(a^{3} \ln \left (c x \right )+\frac {3 i a^{2} b \mathrm {arccsc}\left (c x \right )^{2}}{2}-b^{3} \mathrm {arccsc}\left (c x \right )^{3} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i a^{2} b \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 b^{3} \mathrm {arccsc}\left (c x \right ) \polylog \left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i b^{3} \polylog \left (4, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b^{3} \mathrm {arccsc}\left (c x \right )^{3} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i a \,b^{2} \mathrm {arccsc}\left (c x \right )^{3}-6 b^{3} \mathrm {arccsc}\left (c x \right ) \polylog \left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {i b^{3} \mathrm {arccsc}\left (c x \right )^{4}}{4}+3 i b^{3} \mathrm {arccsc}\left (c x \right )^{2} \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-3 a \,b^{2} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-3 a \,b^{2} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i b^{3} \polylog \left (4, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 i a \,b^{2} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 a \,b^{2} \polylog \left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 a \,b^{2} \polylog \left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 i a \,b^{2} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-3 a^{2} b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-3 a^{2} b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i b^{3} \mathrm {arccsc}\left (c x \right )^{2} \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i a^{2} b \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\) | \(666\) |
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 {acsc}{\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 {asin}\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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