Optimal. Leaf size=190 \[ -\frac{a^2 c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}+\frac{a^2 c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}+\frac{a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.363996, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3909, 3906, 3905, 3475} \[ -\frac{a^2 c^2 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}+\frac{a^2 c^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 c \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}+\frac{a^2 \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3909
Rule 3906
Rule 3905
Rule 3475
Rubi steps
\begin{align*} \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{5/2} \, dx &=\frac{a^2 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}+a \int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2} \, dx\\ &=-\frac{a^2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{a+a \sec (e+f x)}}+\frac{a^2 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}+(a c) \int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx\\ &=-\frac{a^2 c^2 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}-\frac{a^2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{a+a \sec (e+f x)}}+\frac{a^2 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}+\left (a c^2\right ) \int \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)} \, dx\\ &=-\frac{a^2 c^2 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}-\frac{a^2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{a+a \sec (e+f x)}}+\frac{a^2 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}-\frac{\left (a^2 c^3 \tan (e+f x)\right ) \int \tan (e+f x) \, dx}{\sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=\frac{a^2 c^3 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}-\frac{a^2 c^2 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}-\frac{a^2 c (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{a+a \sec (e+f x)}}+\frac{a^2 (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.25331, size = 157, normalized size = 0.83 \[ \frac{i a c^2 \csc \left (\frac{1}{2} (e+f x)\right ) \sec \left (\frac{1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left (6 i \cos (2 (e+f x))+3 f x \cos (3 (e+f x))+\left (9 i \log \left (1+e^{2 i (e+f x)}\right )+9 f x+6 i\right ) \cos (e+f x)+3 i \log \left (1+e^{2 i (e+f x)}\right ) \cos (3 (e+f x))+2 i\right )}{24 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.29, size = 187, normalized size = 1. \begin{align*} -{\frac{a}{6\,f\sin \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2}} \left ( 6\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\ln \left ({\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +6\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -6\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\ln \left ( 2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \right ) + \left ( \cos \left ( fx+e \right ) \right ) ^{3}+6\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+3\,\cos \left ( fx+e \right ) -2 \right ) \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{5}{2}}}\sqrt{{\frac{a \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 = 2.17331, size = 1831, normalized size = 9.64 \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 = 1.69682, size = 1154, normalized size = 6.07 \begin{align*} \left [-\frac{{\left (7 \, a c^{2} \cos \left (f x + e\right )^{2} + a c^{2} \cos \left (f x + e\right ) - 2 \, a c^{2}\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 )}} \sin \left (f x + e\right ) - 3 \,{\left (a c^{2} \cos \left (f x + e\right )^{3} + a c^{2} \cos \left (f x + e\right )^{2}\right )} \sqrt{-a c} \log \left (\frac{a c \cos \left (f x + e\right )^{4} -{\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )\right )} \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 )}} \sin \left (f x + e\right ) + a c}{2 \, \cos \left (f x + e\right )^{2}}\right )}{6 \,{\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}, -\frac{{\left (7 \, a c^{2} \cos \left (f x + e\right )^{2} + a c^{2} \cos \left (f x + e\right ) - 2 \, a c^{2}\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 )}} \sin \left (f x + e\right ) - 6 \,{\left (a c^{2} \cos \left (f x + e\right )^{3} + a c^{2} \cos \left (f x + e\right )^{2}\right )} \sqrt{a c} \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 )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{a c \cos \left (f x + e\right )^{2} + a c}\right )}{6 \,{\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\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]