Optimal. Leaf size=115 \[ -\frac {1}{4} \cos (2 a) \text {Ci}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac {1}{2} \csc ^2\left (a+b x^2\right ) \log (x) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac {1}{4} \csc ^2\left (a+b x^2\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \text {Si}\left (2 b x^2\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6852, 3484,
3459, 3457, 3456} \begin {gather*} -\frac {1}{4} \cos (2 a) \text {CosIntegral}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac {1}{4} \sin (2 a) \text {Si}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac {1}{2} \log (x) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3456
Rule 3457
Rule 3459
Rule 3484
Rule 6852
Rubi steps
\begin {align*} \int \frac {\left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{x} \, dx &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \frac {\sin ^2\left (a+b x^2\right )}{x} \, dx\\ &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \left (\frac {1}{2 x}-\frac {\cos \left (2 a+2 b x^2\right )}{2 x}\right ) \, dx\\ &=\frac {1}{2} \csc ^2\left (a+b x^2\right ) \log (x) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac {1}{2} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \frac {\cos \left (2 a+2 b x^2\right )}{x} \, dx\\ &=\frac {1}{2} \csc ^2\left (a+b x^2\right ) \log (x) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac {1}{2} \left (\cos (2 a) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \frac {\cos \left (2 b x^2\right )}{x} \, dx+\frac {1}{2} \left (\csc ^2\left (a+b x^2\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \frac {\sin \left (2 b x^2\right )}{x} \, dx\\ &=-\frac {1}{4} \cos (2 a) \text {Ci}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac {1}{2} \csc ^2\left (a+b x^2\right ) \log (x) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac {1}{4} \csc ^2\left (a+b x^2\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \text {Si}\left (2 b x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 60, normalized size = 0.52 \begin {gather*} \frac {1}{4} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \left (-\cos (2 a) \text {Ci}\left (2 b x^2\right )+2 \log (x)+\sin (2 a) \text {Si}\left (2 b x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.17, size = 331, normalized size = 2.88
method | result | size |
risch | \(\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i x^{2} b} \pi \,\mathrm {csgn}\left (b \,x^{2}\right )}{8 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i x^{2} b} \sinIntegral \left (2 b \,x^{2}\right )}{4 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}-\frac {\left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i x^{2} b} \expIntegral \left (1, -2 i x^{2} b \right )}{8 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}-\frac {\expIntegral \left (1, -2 i x^{2} b \right ) \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b \,x^{2}+2 a \right )}}{8 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}-\frac {\ln \left (x \right ) \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b \,x^{2}+a \right )}}{2 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}\) | \(331\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.66, size = 55, normalized size = 0.48 \begin {gather*} \frac {1}{16} \, {\left ({\left ({\rm Ei}\left (2 i \, b x^{2}\right ) + {\rm Ei}\left (-2 i \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) - {\left (-i \, {\rm Ei}\left (2 i \, b x^{2}\right ) + i \, {\rm Ei}\left (-2 i \, b x^{2}\right )\right )} \sin \left (2 \, a\right ) - 4 \, \log \left (x\right )\right )} c^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 100, normalized size = 0.87 \begin {gather*} -\frac {4^{\frac {2}{3}} {\left (2 \cdot 4^{\frac {1}{3}} \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{2}\right ) - {\left (4^{\frac {1}{3}} \operatorname {Ci}\left (2 \, b x^{2}\right ) + 4^{\frac {1}{3}} \operatorname {Ci}\left (-2 \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) + 4 \cdot 4^{\frac {1}{3}} \log \left (x\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac {2}{3}}}{32 \, {\left (\cos \left (b x^{2} + a\right )^{2} - 1\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^{2} \right )}\right )^{\frac {2}{3}}}{x}\, 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 (b\,x^2+a\right )}^3\right )}^{2/3}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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