Optimal. Leaf size=140 \[ -\frac{\tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{4 a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 a f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.424006, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3960, 3959, 3770} \[ -\frac{\tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{4 a^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 a f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2} \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3960
Rule 3959
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)}} \, dx &=\frac{\tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)}}+\frac{\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)}} \, dx}{2 a}\\ &=\frac{\tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 a f (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)}}+\frac{\int \frac{\sec (e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}} \, dx}{4 a^2}\\ &=\frac{\tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 a f (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \int \csc (e+f x) \, dx}{4 a^2 \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=\frac{\tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 a f (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)}}-\frac{\tanh ^{-1}(\cos (e+f x)) \tan (e+f x)}{4 a^2 f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.05336, size = 91, normalized size = 0.65 \[ -\frac{\tan (e+f x) \left (3 \cos (e+f x)+8 \cos ^4\left (\frac{1}{2} (e+f x)\right ) \tanh ^{-1}\left (e^{i (e+f x)}\right )+2\right )}{4 a^2 f (\cos (e+f x)+1)^2 \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.3, size = 164, normalized size = 1.2 \begin{align*} -{\frac{ \left ( -1+\cos \left ( fx+e \right ) \right ) ^{3}}{16\,f{a}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{5}}\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ( 4\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+5\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+8\,\cos \left ( fx+e \right ) \ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -2\,\cos \left ( fx+e \right ) +4\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -3 \right ){\frac{1}{\sqrt{{\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 2.14867, size = 1608, normalized size = 11.49 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.693415, size = 1138, normalized size = 8.13 \begin{align*} \left [-\frac{\sqrt{-a c}{\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \log \left (-\frac{4 \,{\left (2 \, \sqrt{-a c} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} +{\left (a c \cos \left (f x + e\right )^{2} + a c\right )} \sin \left (f x + e\right )\right )}}{{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \,{\left (3 \, \cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{8 \,{\left (a^{3} c f \cos \left (f x + e\right )^{2} + 2 \, a^{3} c f \cos \left (f x + e\right ) + a^{3} c f\right )} \sin \left (f x + e\right )}, \frac{\sqrt{a c}{\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \arctan \left (\frac{\sqrt{a c} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{a c \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) +{\left (3 \, \cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{4 \,{\left (a^{3} c f \cos \left (f x + e\right )^{2} + 2 \, a^{3} c f \cos \left (f x + e\right ) + a^{3} c f\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(-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]