Optimal. Leaf size=112 \[ -\frac {2 \sqrt {2} a F_1\left (\frac {1}{2};-\frac {1}{2},\frac {4}{3};\frac {3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}{(c+d) f \sqrt {1+\sin (e+f x)} \sqrt [3]{c+d \sin (e+f x)}} \]
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Rubi [A]
time = 0.07, 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 = {2834, 144, 143}
\begin {gather*} -\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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 144
Rule 2834
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)) \text {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 ) \text {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] Leaf count is larger than twice the leaf count of optimal. \(942\) vs. \(2(112)=224\).
time = 6.49, size = 942, normalized size = 8.41 \begin {gather*} a \left (\frac {(1+\sin (e+f x)) (c+d \sin (e+f x))^{2/3} \left (-\frac {3 \csc (e) \sec (e)}{d (c+d) f}+\frac {3 \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d) f (c+d \sin (e+f x))}\right )}{\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) (1+\sin (e+f x)) \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 {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )},-\frac {\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (-1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )}\right ) \cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt {1+\cot ^2(e)} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}-c \csc (e)}} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}-d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}+c \csc (e)}} \sqrt [3]{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)}}-\frac {\frac {3 d \sin (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \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 {1+\cot ^2(e)}}}{\sqrt [3]{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)}}\right )}{(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 {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )},-\frac {\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (-1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )}\right ) \sec (e) \sec \left (f x+\tan ^{-1}(\tan (e))\right ) (1+\sin (e+f x)) \sqrt {\frac {d \sqrt {1+\tan ^2(e)}-d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}}{c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {\frac {d \sqrt {1+\tan ^2(e)}+d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}}{-c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}\right )^{2/3}}{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 {1+\tan ^2(e)}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.14, 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 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {a+a\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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