Optimal. Leaf size=106 \[ -\frac {\sqrt {2} \cos (c+d x) \sqrt [3]{a+b \sin (c+d x)} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {1}{3};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right )}{d \sqrt {\sin (c+d x)+1} \sqrt [3]{\frac {a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.07, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2665, 139, 138} \[ -\frac {\sqrt {2} \cos (c+d x) \sqrt [3]{a+b \sin (c+d x)} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {1}{3};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right )}{d \sqrt {\sin (c+d x)+1} \sqrt [3]{\frac {a+b \sin (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 2665
Rubi steps
\begin {align*} \int \sqrt [3]{a+b \sin (c+d x)} \, dx &=\frac {\cos (c+d x) \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (c+d x)\right )}{d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}\\ &=\frac {\left (\cos (c+d x) \sqrt [3]{a+b \sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {a}{-a-b}-\frac {b x}{-a-b}}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (c+d x)\right )}{d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)} \sqrt [3]{-\frac {a+b \sin (c+d x)}{-a-b}}}\\ &=-\frac {\sqrt {2} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {1}{3};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right ) \cos (c+d x) \sqrt [3]{a+b \sin (c+d x)}}{d \sqrt {1+\sin (c+d x)} \sqrt [3]{\frac {a+b \sin (c+d x)}{a+b}}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 118, normalized size = 1.11 \[ \frac {3 \sec (c+d x) \sqrt {-\frac {b (\sin (c+d x)-1)}{a+b}} \sqrt {\frac {b (\sin (c+d x)+1)}{b-a}} (a+b \sin (c+d x))^{4/3} F_1\left (\frac {4}{3};\frac {1}{2},\frac {1}{2};\frac {7}{3};\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right )}{4 b d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}, x\right ) \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \left (a +b \sin \left (d x +c \right )\right )^{\frac {1}{3}}\, dx \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\,{d x} \]
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 \[ \int {\left (a+b\,\sin \left (c+d\,x\right )\right )}^{1/3} \,d x \]
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 \[ \int \sqrt [3]{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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