Optimal. Leaf size=119 \[ b^2 \cos (2 a) \text {Ci}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-b^2 \sin (2 a) \text {Si}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac {b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x} \]
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Rubi [A] time = 0.23, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6720, 3314, 29, 3312, 3303, 3299, 3302} \[ b^2 \cos (2 a) \text {CosIntegral}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-b^2 \sin (2 a) \text {Si}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac {b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 3299
Rule 3302
Rule 3303
Rule 3312
Rule 3314
Rule 6720
Rubi steps
\begin {align*} \int \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x^3} \, dx &=\left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac {\sin ^2(a+b x)}{x^3} \, dx\\ &=-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac {b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x}+\left (b^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac {1}{x} \, dx-\left (2 b^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac {\sin ^2(a+b x)}{x} \, dx\\ &=-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac {b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x}+b^2 \csc ^2(a+b x) \log (x) \left (c \sin ^3(a+b x)\right )^{2/3}-\left (2 b^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \left (\frac {1}{2 x}-\frac {\cos (2 a+2 b x)}{2 x}\right ) \, dx\\ &=-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac {b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x}+\left (b^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac {\cos (2 a+2 b x)}{x} \, dx\\ &=-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac {b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x}+\left (b^2 \cos (2 a) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac {\cos (2 b x)}{x} \, dx-\left (b^2 \csc ^2(a+b x) \sin (2 a) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac {\sin (2 b x)}{x} \, dx\\ &=-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac {b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x}+b^2 \cos (2 a) \text {Ci}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-b^2 \csc ^2(a+b x) \sin (2 a) \left (c \sin ^3(a+b x)\right )^{2/3} \text {Si}(2 b x)\\ \end {align*}
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Mathematica [A] time = 0.21, size = 85, normalized size = 0.71 \[ \frac {\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (4 b^2 x^2 \cos (2 a) \text {Ci}(2 b x)-4 b^2 x^2 \sin (2 a) \text {Si}(2 b x)-2 b x \sin (2 (a+b x))+\cos (2 (a+b x))-1\right )}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 142, normalized size = 1.19 \[ \frac {4^{\frac {2}{3}} {\left (2 \cdot 4^{\frac {1}{3}} b^{2} x^{2} \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x\right ) + 2 \cdot 4^{\frac {1}{3}} b x \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 4^{\frac {1}{3}} \cos \left (b x + a\right )^{2} - {\left (4^{\frac {1}{3}} b^{2} x^{2} \operatorname {Ci}\left (2 \, b x\right ) + 4^{\frac {1}{3}} b^{2} x^{2} \operatorname {Ci}\left (-2 \, b x\right )\right )} \cos \left (2 \, a\right ) + 4^{\frac {1}{3}}\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {2}{3}}}{8 \, {\left (x^{2} \cos \left (b x + a\right )^{2} - x^{2}\right )}} \]
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 \[ \int \frac {\left (c \sin \left (b x + a\right )^{3}\right )^{\frac {2}{3}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 238, normalized size = 2.00 \[ -\frac {\left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} b^{2} \left (\frac {1}{2 x^{2} b^{2}}-\frac {i}{b x}-2 \,{\mathrm e}^{2 i b x} \Ei \left (1, 2 i b x \right )\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {b^{2} \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} \left (\frac {{\mathrm e}^{4 i \left (b x +a \right )}}{2 x^{2} b^{2}}+\frac {i {\mathrm e}^{4 i \left (b x +a \right )}}{x b}-2 \Ei \left (1, -2 i b x \right ) {\mathrm e}^{2 i \left (b x +2 a \right )}\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b x +a \right )}}{4 x^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.68, size = 311, normalized size = 2.61 \[ -\frac {{\left (128 \, {\left ({\left (-i \, \sqrt {3} + 1\right )} E_{3}\left (2 i \, b x\right ) + {\left (i \, \sqrt {3} + 1\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{3} - {\left ({\left (128 \, \sqrt {3} + 128 i\right )} E_{3}\left (2 i \, b x\right ) + {\left (128 \, \sqrt {3} - 128 i\right )} E_{3}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )^{3} + 128 \, {\left ({\left ({\left (-i \, \sqrt {3} + 1\right )} E_{3}\left (2 i \, b x\right ) + {\left (i \, \sqrt {3} + 1\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) - 2\right )} \sin \left (2 \, a\right )^{2} + 128 \, {\left ({\left (i \, \sqrt {3} + 1\right )} E_{3}\left (2 i \, b x\right ) + {\left (-i \, \sqrt {3} + 1\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) - 256 \, \cos \left (2 \, a\right )^{2} - {\left ({\left ({\left (128 \, \sqrt {3} + 128 i\right )} E_{3}\left (2 i \, b x\right ) + {\left (128 \, \sqrt {3} - 128 i\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} - {\left (128 \, \sqrt {3} - 128 i\right )} E_{3}\left (2 i \, b x\right ) - {\left (128 \, \sqrt {3} + 128 i\right )} E_{3}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )\right )} b^{2} c^{\frac {2}{3}}}{2048 \, {\left (a^{2} \cos \left (2 \, a\right )^{2} + a^{2} \sin \left (2 \, a\right )^{2} + {\left (b x + a\right )}^{2} {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} - 2 \, {\left (a \cos \left (2 \, a\right )^{2} + a \sin \left (2 \, a\right )^{2}\right )} {\left (b x + a\right )}\right )}} \]
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 \[ \int \frac {{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{2/3}}{x^3} \,d x \]
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 \[ \int \frac {\left (c \sin ^{3}{\left (a + b x \right )}\right )^{\frac {2}{3}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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