Optimal. Leaf size=517 \[ -\frac {3 \sqrt {\frac {3}{2} \left (3-i \sqrt {3}\right )} \sqrt [3]{c} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {1-\frac {(c \sin (a+b x))^{2/3}}{c^{2/3}}}}{\sqrt {3+i \sqrt {3}}}\right )|\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}\right ) \sec (a+b x) \sqrt {1-\frac {(c \sin (a+b x))^{2/3}}{c^{2/3}}} \sqrt {\frac {i+\sqrt {3}}{3 i+\sqrt {3}}+\frac {2 (c \sin (a+b x))^{2/3}}{\left (3-i \sqrt {3}\right ) c^{2/3}}} \sqrt {\frac {i-\sqrt {3}}{3 i-\sqrt {3}}+\frac {2 (c \sin (a+b x))^{2/3}}{\left (3+i \sqrt {3}\right ) c^{2/3}}}}{b}+\frac {3 \left (1-i \sqrt {3}\right ) \sqrt {3-i \sqrt {3}} \sqrt [3]{c} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {1-\frac {(c \sin (a+b x))^{2/3}}{c^{2/3}}}}{\sqrt {3-i \sqrt {3}}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right ) \sec (a+b x) \sqrt {1-\frac {(c \sin (a+b x))^{2/3}}{c^{2/3}}} \sqrt {\frac {i+\sqrt {3}}{3 i+\sqrt {3}}+\frac {2 (c \sin (a+b x))^{2/3}}{\left (3-i \sqrt {3}\right ) c^{2/3}}} \sqrt {\frac {i-\sqrt {3}}{3 i-\sqrt {3}}+\frac {2 (c \sin (a+b x))^{2/3}}{\left (3+i \sqrt {3}\right ) c^{2/3}}}}{2 \sqrt {2} b} \]
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Rubi [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 0.11, number of steps
used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722}
\begin {gather*} \frac {3 \cos (a+b x) (c \sin (a+b x))^{4/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\sin ^2(a+b x)\right )}{4 b c \sqrt {\cos ^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rubi steps
\begin {align*} \int \sqrt [3]{c \sin (a+b x)} \, dx &=\frac {3 \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\sin ^2(a+b x)\right ) (c \sin (a+b x))^{4/3}}{4 b c \sqrt {\cos ^2(a+b x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.03, size = 55, normalized size = 0.11 \begin {gather*} \frac {3 \sqrt {\cos ^2(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\sin ^2(a+b x)\right ) \sqrt [3]{c \sin (a+b x)} \tan (a+b x)}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (c \sin \left (b x +a \right )\right )^{\frac {1}{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{c \sin {\left (a + b x \right )}}\, 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 = -0.00 \begin {gather*} \int {\left (c\,\sin \left (a+b\,x\right )\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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