Optimal. Leaf size=207 \[ \frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^4}+\frac {i b^3 \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^4} \]
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Rubi [A]
time = 0.16, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5331, 4495,
4271, 3852, 8, 4269, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {b^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^4}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {i b^3 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 c^4}+\frac {b^3 x \sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 3852
Rule 4269
Rule 4271
Rule 4495
Rule 5331
Rubi steps
\begin {align*} \int x^3 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx &=-\frac {\text {Subst}\left (\int (a+b x)^3 \cot (x) \csc ^4(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^4}\\ &=\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {(3 b) \text {Subst}\left (\int (a+b x)^2 \csc ^4(x) \, dx,x,\csc ^{-1}(c x)\right )}{4 c^4}\\ &=\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b \text {Subst}\left (\int (a+b x)^2 \csc ^2(x) \, dx,x,\csc ^{-1}(c x)\right )}{2 c^4}-\frac {b^3 \text {Subst}\left (\int \csc ^2(x) \, dx,x,\csc ^{-1}(c x)\right )}{4 c^4}\\ &=\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b^2 \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\csc ^{-1}(c x)\right )}{c^4}+\frac {b^3 \text {Subst}\left (\int 1 \, dx,x,c \sqrt {1-\frac {1}{c^2 x^2}} x\right )}{4 c^4}\\ &=\frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )}{c^4}\\ &=\frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^4}+\frac {b^3 \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^4}\\ &=\frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^4}-\frac {\left (i b^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^4}\\ &=\frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^4}+\frac {i b^3 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 285, normalized size = 1.38 \begin {gather*} \frac {2 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x+b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x+a b^2 c^2 x^2+a^2 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3+a^3 c^4 x^4+b^2 \left (3 a c^4 x^4+b \left (2 i+2 c \sqrt {1-\frac {1}{c^2 x^2}} x+c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3\right )\right ) \csc ^{-1}(c x)^2+b^3 c^4 x^4 \csc ^{-1}(c x)^3+b \csc ^{-1}(c x) \left (c x \left (b^2 c x+3 a^2 c^3 x^3+2 a b \sqrt {1-\frac {1}{c^2 x^2}} \left (2+c^2 x^2\right )\right )-4 b^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right )-4 a b^2 \log \left (\frac {1}{c x}\right )+2 i b^3 \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{4 c^4} \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. 457 vs. \(2 (213 ) = 426\).
time = 0.82, size = 458, normalized size = 2.21
method | result | size |
derivativedivides | \(\frac {\frac {c^{4} x^{4} a^{3}}{4}+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{3} c^{4} x^{4}}{4}+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{4}+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}{2}+i b^{3} \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {b^{3} \mathrm {arccsc}\left (c x \right ) c^{2} x^{2}}{4}+\frac {b^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}{4}+\frac {i b^{3} \mathrm {arccsc}\left (c x \right )^{2}}{2}-b^{3} \mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b^{3} \mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b^{3} \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\frac {i b^{3}}{4}+\frac {3 a \,b^{2} \mathrm {arccsc}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {a \,b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{2}+\frac {a \,b^{2} c^{2} x^{2}}{4}+a \,b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x -a \,b^{2} \ln \left (\frac {1}{c x}\right )+3 a^{2} b \left (\frac {c^{4} x^{4} \mathrm {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\) | \(458\) |
default | \(\frac {\frac {c^{4} x^{4} a^{3}}{4}+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{3} c^{4} x^{4}}{4}+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{4}+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}{2}+i b^{3} \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {b^{3} \mathrm {arccsc}\left (c x \right ) c^{2} x^{2}}{4}+\frac {b^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}{4}+\frac {i b^{3} \mathrm {arccsc}\left (c x \right )^{2}}{2}-b^{3} \mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b^{3} \mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b^{3} \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\frac {i b^{3}}{4}+\frac {3 a \,b^{2} \mathrm {arccsc}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {a \,b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{2}+\frac {a \,b^{2} c^{2} x^{2}}{4}+a \,b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x -a \,b^{2} \ln \left (\frac {1}{c x}\right )+3 a^{2} b \left (\frac {c^{4} x^{4} \mathrm {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\) | \(458\) |
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^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}\, 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^3\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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