Optimal. Leaf size=112 \[ -\frac {2 \sqrt {2} a \cos (e+f x) \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}} F_1\left (\frac {1}{2};-\frac {1}{2},\frac {4}{3};\frac {3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{f (c+d) \sqrt {\sin (e+f x)+1} \sqrt [3]{c+d \sin (e+f x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2755, 139, 138} \[ -\frac {2 \sqrt {2} a \cos (e+f x) \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}} F_1\left (\frac {1}{2};-\frac {1}{2},\frac {4}{3};\frac {3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{f (c+d) \sqrt {\sin (e+f x)+1} \sqrt [3]{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 2755
Rubi steps
\begin {align*} \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx &=\frac {(a \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{\sqrt {1-x} (c+d x)^{4/3}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}\\ &=\frac {\left (a \cos (e+f x) \sqrt [3]{-\frac {c+d \sin (e+f x)}{-c-d}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{\sqrt {1-x} \left (-\frac {c}{-c-d}-\frac {d x}{-c-d}\right )^{4/3}} \, dx,x,\sin (e+f x)\right )}{(c+d) f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)} \sqrt [3]{c+d \sin (e+f x)}}\\ &=-\frac {2 \sqrt {2} a F_1\left (\frac {1}{2};-\frac {1}{2},\frac {4}{3};\frac {3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}{(c+d) f \sqrt {1+\sin (e+f x)} \sqrt [3]{c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 6.42, size = 942, normalized size = 8.41 \[ a \left (\frac {(c+d \sin (e+f x))^{2/3} \left (\frac {3 \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d) f (c+d \sin (e+f x))}-\frac {3 \csc (e) \sec (e)}{d (c+d) f}\right ) (\sin (e+f x)+1)}{\left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}-\frac {2 \sec (e) \left (-\frac {F_1\left (-\frac {1}{3};-\frac {1}{2},-\frac {1}{2};\frac {2}{3};-\frac {\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {\cot ^2(e)+1} \sin (e)\right )}{d \sqrt {\cot ^2(e)+1} \left (1-\frac {c \csc (e)}{d \sqrt {\cot ^2(e)+1}}\right )},-\frac {\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {\cot ^2(e)+1} \sin (e)\right )}{d \sqrt {\cot ^2(e)+1} \left (-\frac {c \csc (e)}{d \sqrt {\cot ^2(e)+1}}-1\right )}\right ) \cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt {\cot ^2(e)+1} \sqrt {\frac {\cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {\cot ^2(e)+1} d+\sqrt {\cot ^2(e)+1} d}{d \sqrt {\cot ^2(e)+1}-c \csc (e)}} \sqrt {\frac {d \sqrt {\cot ^2(e)+1}-d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {\cot ^2(e)+1}}{\sqrt {\cot ^2(e)+1} d+c \csc (e)}} \sqrt [3]{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {\cot ^2(e)+1} \sin (e)}}-\frac {\frac {3 d \sin (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {\cot ^2(e)+1} \sin (e)\right )}{2 \left (d^2 \cos ^2(e)+d^2 \sin ^2(e)\right )}-\frac {\cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt {\cot ^2(e)+1}}}{\sqrt [3]{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {\cot ^2(e)+1} \sin (e)}}\right ) (\sin (e+f x)+1)}{(c+d) f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}+\frac {3 F_1\left (\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {5}{3};-\frac {\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {\tan ^2(e)+1}\right )}{d \sqrt {\tan ^2(e)+1} \left (1-\frac {c \sec (e)}{d \sqrt {\tan ^2(e)+1}}\right )},-\frac {\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {\tan ^2(e)+1}\right )}{d \sqrt {\tan ^2(e)+1} \left (-\frac {c \sec (e)}{d \sqrt {\tan ^2(e)+1}}-1\right )}\right ) \sec (e) \sec \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {\frac {d \sqrt {\tan ^2(e)+1}-d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {\tan ^2(e)+1}}{\sqrt {\tan ^2(e)+1} d+c \sec (e)}} \sqrt {\frac {\sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {\tan ^2(e)+1} d+\sqrt {\tan ^2(e)+1} d}{d \sqrt {\tan ^2(e)+1}-c \sec (e)}} \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {\tan ^2(e)+1}\right )^{2/3} (\sin (e+f x)+1)}{2 d (c+d) f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2 \sqrt {\tan ^2(e)+1}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {2}{3}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}, x\right ) \]
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 \[ \int \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {a +a \sin \left (f x +e \right )}{\left (c +d \sin \left (f x +e \right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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