Optimal. Leaf size=79 \[ \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^2}-\frac {x \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b}+\frac {1}{4} x^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \]
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Rubi [A]
time = 0.08, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6852, 3391, 30}
\begin {gather*} \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^2}+\frac {1}{4} x^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac {x \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3391
Rule 6852
Rubi steps
\begin {align*} \int x \left (c \sin ^3(a+b x)\right )^{2/3} \, dx &=\left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int x \sin ^2(a+b x) \, dx\\ &=\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^2}-\frac {x \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b}+\frac {1}{2} \left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int x \, dx\\ &=\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^2}-\frac {x \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b}+\frac {1}{4} x^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 55, normalized size = 0.70 \begin {gather*} -\frac {\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} (\cos (2 (a+b x))+2 b x (-b x+\sin (2 (a+b x))))}{8 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.16, size = 174, normalized size = 2.20
method | result | size |
risch | \(-\frac {x^{2} \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b x +a \right )}}{4 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {i \left (2 b x +i\right ) \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{4 i \left (b x +a \right )}}{16 b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} \left (2 b x -i\right )}{16 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2} b^{2}}\) | \(174\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs.
\(2 (67) = 134\).
time = 0.53, size = 162, normalized size = 2.05 \begin {gather*} -\frac {16 \, {\left (c^{\frac {2}{3}} \arctan \left (\frac {\sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1}\right ) - \frac {\frac {c^{\frac {2}{3}} \sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1} - \frac {c^{\frac {2}{3}} \sin \left (b x + a\right )^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}}}{\frac {2 \, \sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {\sin \left (b x + a\right )^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + 1}\right )} a + {\left (2 \, {\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right )\right )} c^{\frac {2}{3}}}{16 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 82, normalized size = 1.04 \begin {gather*} -\frac {{\left (2 \, b^{2} x^{2} - 4 \, b x \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {2}{3}}}{8 \, {\left (b^{2} \cos \left (b x + a\right )^{2} - b^{2}\right )}} \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 \left (c \sin ^{3}{\left (a + b x \right )}\right )^{\frac {2}{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 = -0.01 \begin {gather*} \int x\,{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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