Optimal. Leaf size=156 \[ -\frac{5 \sec (e+f x)}{8 a c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{15 \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{32 \sqrt{2} a c^{5/2} f}+\frac{15 \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac{\sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.266892, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2736, 2681, 2687, 2650, 2649, 206} \[ -\frac{5 \sec (e+f x)}{8 a c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{15 \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{32 \sqrt{2} a c^{5/2} f}+\frac{15 \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac{\sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2736
Rule 2681
Rule 2687
Rule 2650
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx &=\frac{\int \frac{\sec ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx}{a c}\\ &=\frac{\sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}+\frac{5 \int \frac{\sec ^2(e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx}{8 a c^2}\\ &=\frac{\sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}-\frac{5 \sec (e+f x)}{8 a c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{15 \int \frac{1}{(c-c \sin (e+f x))^{3/2}} \, dx}{16 a c}\\ &=\frac{15 \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac{\sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}-\frac{5 \sec (e+f x)}{8 a c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{15 \int \frac{1}{\sqrt{c-c \sin (e+f x)}} \, dx}{64 a c^2}\\ &=\frac{15 \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac{\sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}-\frac{5 \sec (e+f x)}{8 a c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{15 \operatorname{Subst}\left (\int \frac{1}{2 c-x^2} \, dx,x,-\frac{c \cos (e+f x)}{\sqrt{c-c \sin (e+f x)}}\right )}{32 a c^2 f}\\ &=\frac{15 \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{32 \sqrt{2} a c^{5/2} f}+\frac{15 \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac{\sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}-\frac{5 \sec (e+f x)}{8 a c^2 f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.778693, size = 162, normalized size = 1.04 \[ \frac{\left (\frac{1}{128}+\frac{i}{128}\right ) \cos (e+f x) \left ((1-i) (40 \sin (e+f x)+15 \cos (2 (e+f x))-9)-60 \sqrt [4]{-1} \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac{1}{4} (e+f x)\right )+1\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{a c^2 f (\sin (e+f x)-1)^2 (\sin (e+f x)+1) \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.703, size = 210, normalized size = 1.4 \begin{align*} -{\frac{1}{64\,a \left ( -1+\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) f} \left ( 15\,\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}{c}^{2}-30\,{c}^{5/2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}-30\,\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ) \sin \left ( fx+e \right ){c}^{2}+40\,{c}^{5/2}\sin \left ( fx+e \right ) +15\,\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{2}+6\,{c}^{5/2} \right ){c}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{c-c\sin \left ( fx+e \right ) }}}} \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{1}{{\left (a \sin \left (f x + e\right ) + a\right )}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.13242, size = 663, normalized size = 4.25 \begin{align*} \frac{15 \, \sqrt{2}{\left (\cos \left (f x + e\right )^{3} + 2 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right )\right )} \sqrt{c} \log \left (-\frac{c \cos \left (f x + e\right )^{2} + 2 \, \sqrt{2} \sqrt{-c \sin \left (f x + e\right ) + c} \sqrt{c}{\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) +{\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \,{\left (15 \, \cos \left (f x + e\right )^{2} + 20 \, \sin \left (f x + e\right ) - 12\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{128 \,{\left (a c^{3} f \cos \left (f x + e\right )^{3} + 2 \, a c^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a c^{3} f \cos \left (f x + e\right )\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*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]