Optimal. Leaf size=112 \[ -\frac{2 \sqrt{2} a \cos (e+f x) \sqrt [3]{\frac{c+d \sin (e+f x)}{c+d}} F_1\left (\frac{1}{2};-\frac{1}{2},\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{f (c+d) \sqrt{\sin (e+f x)+1} \sqrt [3]{c+d \sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0963662, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, 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.
[In]
[Out]
Rule 2755
Rule 139
Rule 138
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*}
Mathematica [B] time = 6.40232, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.151, size = 0, normalized size = 0. \begin{align*} \int{(a+a\sin \left ( fx+e \right ) ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]