Optimal. Leaf size=167 \[ \frac{11 \sec (e+f x)}{96 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{11 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{128 \sqrt{2} a^{5/2} f}-\frac{11 \cos (e+f x)}{128 a f (a \sin (e+f x)+a)^{3/2}}+\frac{17 \sec (e+f x)}{48 a f (a \sin (e+f x)+a)^{3/2}}-\frac{\sec (e+f x)}{6 f (a \sin (e+f x)+a)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30282, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2712, 2859, 2687, 2650, 2649, 206} \[ \frac{11 \sec (e+f x)}{96 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{11 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{128 \sqrt{2} a^{5/2} f}-\frac{11 \cos (e+f x)}{128 a f (a \sin (e+f x)+a)^{3/2}}+\frac{17 \sec (e+f x)}{48 a f (a \sin (e+f x)+a)^{3/2}}-\frac{\sec (e+f x)}{6 f (a \sin (e+f x)+a)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2712
Rule 2859
Rule 2687
Rule 2650
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\tan ^2(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac{\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}+\frac{\int \frac{\sec ^2(e+f x) \left (-\frac{5 a}{2}+6 a \sin (e+f x)\right )}{(a+a \sin (e+f x))^{3/2}} \, dx}{6 a^2}\\ &=-\frac{\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}+\frac{17 \sec (e+f x)}{48 a f (a+a \sin (e+f x))^{3/2}}+\frac{11 \int \frac{\sec ^2(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx}{96 a^2}\\ &=-\frac{\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}+\frac{17 \sec (e+f x)}{48 a f (a+a \sin (e+f x))^{3/2}}+\frac{11 \sec (e+f x)}{96 a^2 f \sqrt{a+a \sin (e+f x)}}+\frac{11 \int \frac{1}{(a+a \sin (e+f x))^{3/2}} \, dx}{64 a}\\ &=-\frac{\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}-\frac{11 \cos (e+f x)}{128 a f (a+a \sin (e+f x))^{3/2}}+\frac{17 \sec (e+f x)}{48 a f (a+a \sin (e+f x))^{3/2}}+\frac{11 \sec (e+f x)}{96 a^2 f \sqrt{a+a \sin (e+f x)}}+\frac{11 \int \frac{1}{\sqrt{a+a \sin (e+f x)}} \, dx}{256 a^2}\\ &=-\frac{\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}-\frac{11 \cos (e+f x)}{128 a f (a+a \sin (e+f x))^{3/2}}+\frac{17 \sec (e+f x)}{48 a f (a+a \sin (e+f x))^{3/2}}+\frac{11 \sec (e+f x)}{96 a^2 f \sqrt{a+a \sin (e+f x)}}-\frac{11 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{128 a^2 f}\\ &=-\frac{11 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a+a \sin (e+f x)}}\right )}{128 \sqrt{2} a^{5/2} f}-\frac{\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}-\frac{11 \cos (e+f x)}{128 a f (a+a \sin (e+f x))^{3/2}}+\frac{17 \sec (e+f x)}{48 a f (a+a \sin (e+f x))^{3/2}}+\frac{11 \sec (e+f x)}{96 a^2 f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.408413, size = 284, normalized size = 1.7 \[ \frac{\frac{48 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5}{\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )}+15 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4-30 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3+52 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2-104 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+\frac{64 \sin \left (\frac{1}{2} (e+f x)\right )}{\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )}+(33+33 i) (-1)^{3/4} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (e+f x)\right )-1\right )\right )-32}{384 f (a (\sin (e+f x)+1))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.648, size = 266, normalized size = 1.6 \begin{align*} -{\frac{1}{768\, \left ( 1+\sin \left ( fx+e \right ) \right ) ^{2}\cos \left ( fx+e \right ) f} \left ( \left ( 66\,{a}^{7/2}-33\,\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+ \left ( -448\,{a}^{7/2}+132\,\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ) \sin \left ( fx+e \right ) + \left ( 154\,{a}^{7/2}-99\,\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}-320\,{a}^{7/2}+132\,\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ){a}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
________________________________________________________________________________________
Fricas [A] time = 1.69739, size = 751, normalized size = 4.5 \begin{align*} \frac{33 \, \sqrt{2}{\left (3 \, \cos \left (f x + e\right )^{3} +{\left (\cos \left (f x + e\right )^{3} - 4 \, \cos \left (f x + e\right )\right )} \sin \left (f x + e\right ) - 4 \, \cos \left (f x + e\right )\right )} \sqrt{a} \log \left (-\frac{a \cos \left (f x + e\right )^{2} - 2 \, \sqrt{2} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{a}{\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) -{\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\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 (77 \, \cos \left (f x + e\right )^{2} +{\left (33 \, \cos \left (f x + e\right )^{2} - 224\right )} \sin \left (f x + e\right ) - 160\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{1536 \,{\left (3 \, a^{3} f \cos \left (f x + e\right )^{3} - 4 \, a^{3} f \cos \left (f x + e\right ) +{\left (a^{3} f \cos \left (f x + e\right )^{3} - 4 \, a^{3} f \cos \left (f x + e\right )\right )} \sin \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]