Optimal. Leaf size=86 \[ b \sin (2 a) \text {Ci}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}+b \cos (2 a) \text {Si}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x} \]
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Rubi [A] time = 0.18, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6720, 3313, 12, 3303, 3299, 3302} \[ b \sin (2 a) \text {CosIntegral}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}+b \cos (2 a) \text {Si}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 3302
Rule 3303
Rule 3313
Rule 6720
Rubi steps
\begin {align*} \int \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x^2} \, 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^2} \, dx\\ &=-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x}+\left (2 b \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac {\sin (2 a+2 b x)}{2 x} \, dx\\ &=-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x}+\left (b \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac {\sin (2 a+2 b x)}{x} \, dx\\ &=-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x}+\left (b \cos (2 a) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac {\sin (2 b x)}{x} \, dx+\left (b \csc ^2(a+b x) \sin (2 a) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac {\cos (2 b x)}{x} \, dx\\ &=-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x}+b \text {Ci}(2 b x) \csc ^2(a+b x) \sin (2 a) \left (c \sin ^3(a+b x)\right )^{2/3}+b \cos (2 a) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \text {Si}(2 b x)\\ \end {align*}
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Mathematica [A] time = 0.15, size = 65, normalized size = 0.76 \[ \frac {\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} (2 b x \sin (2 a) \text {Ci}(2 b x)+2 b x \cos (2 a) \text {Si}(2 b x)+\cos (2 (a+b x))-1)}{2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 108, normalized size = 1.26 \[ -\frac {4^{\frac {2}{3}} {\left (2 \cdot 4^{\frac {1}{3}} b x \cos \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x\right ) + 2 \cdot 4^{\frac {1}{3}} \cos \left (b x + a\right )^{2} + {\left (4^{\frac {1}{3}} b x \operatorname {Ci}\left (2 \, b x\right ) + 4^{\frac {1}{3}} b x \operatorname {Ci}\left (-2 \, b x\right )\right )} \sin \left (2 \, a\right ) - 2 \cdot 4^{\frac {1}{3}}\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {2}{3}}}{8 \, {\left (x \cos \left (b x + a\right )^{2} - x\right )}} \]
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 \[ \int \frac {\left (c \sin \left (b x + a\right )^{3}\right )^{\frac {2}{3}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 211, normalized size = 2.45 \[ \frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} b \left (\frac {i}{b x}+2 \,{\mathrm e}^{2 i b x} \Ei \left (1, 2 i b x \right )\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {i b \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} \left (\frac {i {\mathrm e}^{4 i \left (b x +a \right )}}{x b}-2 \Ei \left (1, -2 i b x \right ) {\mathrm e}^{2 i \left (b x +2 a \right )}\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b x +a \right )}}{2 x \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.05, size = 280, normalized size = 3.26 \[ \frac {{\left (64 \, {\left ({\left (-i \, \sqrt {3} + 1\right )} E_{2}\left (2 i \, b x\right ) + {\left (i \, \sqrt {3} + 1\right )} E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{3} - {\left ({\left (64 \, \sqrt {3} + 64 i\right )} E_{2}\left (2 i \, b x\right ) + {\left (64 \, \sqrt {3} - 64 i\right )} E_{2}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )^{3} + 64 \, {\left ({\left ({\left (-i \, \sqrt {3} + 1\right )} E_{2}\left (2 i \, b x\right ) + {\left (i \, \sqrt {3} + 1\right )} E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) - 4\right )} \sin \left (2 \, a\right )^{2} + 64 \, {\left ({\left (i \, \sqrt {3} + 1\right )} E_{2}\left (2 i \, b x\right ) + {\left (-i \, \sqrt {3} + 1\right )} E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) - 256 \, \cos \left (2 \, a\right )^{2} - {\left ({\left ({\left (64 \, \sqrt {3} + 64 i\right )} E_{2}\left (2 i \, b x\right ) + {\left (64 \, \sqrt {3} - 64 i\right )} E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} - {\left (64 \, \sqrt {3} - 64 i\right )} E_{2}\left (2 i \, b x\right ) - {\left (64 \, \sqrt {3} + 64 i\right )} E_{2}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )\right )} b c^{\frac {2}{3}}}{1024 \, {\left (a \cos \left (2 \, a\right )^{2} + a \sin \left (2 \, a\right )^{2} - {\left (b x + a\right )} {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )}\right )}} \]
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 \[ \int \frac {{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{2/3}}{x^2} \,d x \]
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 \[ \int \frac {\left (c \sin ^{3}{\left (a + b x \right )}\right )^{\frac {2}{3}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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