Optimal. Leaf size=96 \[ -\frac {6 \sqrt [3]{c \sin ^3(a+b x)}}{b^4}+\frac {3 x^2 \sqrt [3]{c \sin ^3(a+b x)}}{b^2}+\frac {6 x \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b^3}-\frac {x^3 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]
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Rubi [A]
time = 0.14, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6852, 3377,
2717} \begin {gather*} -\frac {6 \sqrt [3]{c \sin ^3(a+b x)}}{b^4}+\frac {6 x \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b^3}+\frac {3 x^2 \sqrt [3]{c \sin ^3(a+b x)}}{b^2}-\frac {x^3 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3377
Rule 6852
Rubi steps
\begin {align*} \int x^3 \sqrt [3]{c \sin ^3(a+b x)} \, dx &=\left (\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int x^3 \sin (a+b x) \, dx\\ &=-\frac {x^3 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b}+\frac {\left (3 \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int x^2 \cos (a+b x) \, dx}{b}\\ &=\frac {3 x^2 \sqrt [3]{c \sin ^3(a+b x)}}{b^2}-\frac {x^3 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b}-\frac {\left (6 \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int x \sin (a+b x) \, dx}{b^2}\\ &=\frac {3 x^2 \sqrt [3]{c \sin ^3(a+b x)}}{b^2}+\frac {6 x \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b^3}-\frac {x^3 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b}-\frac {\left (6 \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \cos (a+b x) \, dx}{b^3}\\ &=-\frac {6 \sqrt [3]{c \sin ^3(a+b x)}}{b^4}+\frac {3 x^2 \sqrt [3]{c \sin ^3(a+b x)}}{b^2}+\frac {6 x \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b^3}-\frac {x^3 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 47, normalized size = 0.49 \begin {gather*} -\frac {\left (6-3 b^2 x^2+b x \left (-6+b^2 x^2\right ) \cot (a+b x)\right ) \sqrt [3]{c \sin ^3(a+b x)}}{b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.11, size = 151, normalized size = 1.57
method | result | size |
risch | \(-\frac {i \left (b^{3} x^{3}+3 i b^{2} x^{2}-6 b x -6 i\right ) \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{2 i \left (b x +a \right )}}{2 b^{4} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} \left (b^{3} x^{3}-3 i b^{2} x^{2}-6 b x +6 i\right )}{2 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) b^{4}}\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 146, normalized size = 1.52 \begin {gather*} \frac {3 \, {\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} a^{2} c^{\frac {1}{3}} - 3 \, {\left ({\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) - 2 \, {\left (b x + a\right )} \sin \left (b x + a\right )\right )} a c^{\frac {1}{3}} + \frac {4 \, a^{3} c^{\frac {1}{3}}}{\frac {\sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1} + {\left ({\left ({\left (b x + a\right )}^{3} - 6 \, b x - 6 \, a\right )} \cos \left (b x + a\right ) - 3 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} c^{\frac {1}{3}}}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 74, normalized size = 0.77 \begin {gather*} -\frac {{\left ({\left (b^{3} x^{3} - 6 \, b x\right )} \cos \left (b x + a\right ) - 3 \, {\left (b^{2} x^{2} - 2\right )} \sin \left (b x + a\right )\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {1}{3}}}{b^{4} \sin \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.49, size = 129, normalized size = 1.34 \begin {gather*} \begin {cases} \frac {x^{4} \sqrt [3]{c \sin ^{3}{\left (a \right )}}}{4} & \text {for}\: b = 0 \\0 & \text {for}\: a = - b x \vee a = - b x + \pi \\- \frac {x^{3} \sqrt [3]{c \sin ^{3}{\left (a + b x \right )}} \cos {\left (a + b x \right )}}{b \sin {\left (a + b x \right )}} + \frac {3 x^{2} \sqrt [3]{c \sin ^{3}{\left (a + b x \right )}}}{b^{2}} + \frac {6 x \sqrt [3]{c \sin ^{3}{\left (a + b x \right )}} \cos {\left (a + b x \right )}}{b^{3} \sin {\left (a + b x \right )}} - \frac {6 \sqrt [3]{c \sin ^{3}{\left (a + b x \right )}}}{b^{4}} & \text {otherwise} \end {cases} \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 [B]
time = 5.71, size = 109, normalized size = 1.14 \begin {gather*} \frac {2^{1/3}\,{\left (c\,\left (3\,\sin \left (a+b\,x\right )-\sin \left (3\,a+3\,b\,x\right )\right )\right )}^{1/3}\,\left (3\,b^2\,x^2-12\,{\sin \left (a+b\,x\right )}^2+6\,b\,x\,\sin \left (2\,a+2\,b\,x\right )+3\,b^2\,x^2\,\left (2\,{\sin \left (a+b\,x\right )}^2-1\right )-b^3\,x^3\,\sin \left (2\,a+2\,b\,x\right )\right )}{4\,b^4\,{\sin \left (a+b\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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