Optimal. Leaf size=86 \[ -\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x}+b \text {Ci}(2 b x) \csc ^2(a+b x) \sin (2 a) \left (c \sin ^3(a+b x)\right )^{2/3}+b \cos (2 a) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \text {Si}(2 b x) \]
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Rubi [A]
time = 0.13, 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 = {6852, 3394, 12,
3384, 3380, 3383} \begin {gather*} 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 3394
Rule 6852
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 = 0.10, size = 65, normalized size = 0.76 \begin {gather*} \frac {\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} (-1+\cos (2 (a+b x))+2 b x \text {Ci}(2 b x) \sin (2 a)+2 b x \cos (2 a) \text {Si}(2 b x))}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.12, size = 211, normalized size = 2.45
method | result | size |
risch | \(\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} b \left (\frac {i}{b x}+2 \,{\mathrm e}^{2 i b x} \expIntegral \left (1, 2 i b x \right )\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {i b \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} \left (\frac {i {\mathrm e}^{4 i \left (b x +a \right )}}{x b}-2 \expIntegral \left (1, -2 i b x \right ) {\mathrm e}^{2 i \left (b x +2 a \right )}\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b x +a \right )}}{2 x \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}\) | \(211\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.57, size = 265, normalized size = 3.08 \begin {gather*} \frac {{\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 )^{3} - {\left ({\left (\sqrt {3} + i\right )} E_{2}\left (2 i \, b x\right ) + {\left (\sqrt {3} - i\right )} E_{2}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )^{3} + {\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} + {\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 \, \cos \left (2 \, a\right )^{2} - {\left ({\left ({\left (\sqrt {3} + i\right )} E_{2}\left (2 i \, b x\right ) + {\left (\sqrt {3} - i\right )} E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} - {\left (\sqrt {3} - i\right )} E_{2}\left (2 i \, b x\right ) - {\left (\sqrt {3} + i\right )} E_{2}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )\right )} b c^{\frac {2}{3}}}{16 \, {\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 108, normalized size = 1.26 \begin {gather*} -\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 )}} \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 \frac {\left (c \sin ^{3}{\left (a + b x \right )}\right )^{\frac {2}{3}}}{x^{2}}\, 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 \frac {{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{2/3}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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