Optimal. Leaf size=126 \[ \frac{2 \cot ^5(e+f x)}{5 a^3 c f}-\frac{\cot ^3(e+f x)}{3 a^3 c f}+\frac{\cot (e+f x)}{a^3 c f}-\frac{2 \csc ^5(e+f x)}{5 a^3 c f}+\frac{4 \csc ^3(e+f x)}{3 a^3 c f}-\frac{2 \csc (e+f x)}{a^3 c f}+\frac{x}{a^3 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.193262, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3904, 3886, 3473, 8, 2606, 194, 2607, 30} \[ \frac{2 \cot ^5(e+f x)}{5 a^3 c f}-\frac{\cot ^3(e+f x)}{3 a^3 c f}+\frac{\cot (e+f x)}{a^3 c f}-\frac{2 \csc ^5(e+f x)}{5 a^3 c f}+\frac{4 \csc ^3(e+f x)}{3 a^3 c f}-\frac{2 \csc (e+f x)}{a^3 c f}+\frac{x}{a^3 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3904
Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 194
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))} \, dx &=-\frac{\int \cot ^6(e+f x) (c-c \sec (e+f x))^2 \, dx}{a^3 c^3}\\ &=-\frac{\int \left (c^2 \cot ^6(e+f x)-2 c^2 \cot ^5(e+f x) \csc (e+f x)+c^2 \cot ^4(e+f x) \csc ^2(e+f x)\right ) \, dx}{a^3 c^3}\\ &=-\frac{\int \cot ^6(e+f x) \, dx}{a^3 c}-\frac{\int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a^3 c}+\frac{2 \int \cot ^5(e+f x) \csc (e+f x) \, dx}{a^3 c}\\ &=\frac{\cot ^5(e+f x)}{5 a^3 c f}+\frac{\int \cot ^4(e+f x) \, dx}{a^3 c}-\frac{\operatorname{Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a^3 c f}-\frac{2 \operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a^3 c f}\\ &=-\frac{\cot ^3(e+f x)}{3 a^3 c f}+\frac{2 \cot ^5(e+f x)}{5 a^3 c f}-\frac{\int \cot ^2(e+f x) \, dx}{a^3 c}-\frac{2 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c f}\\ &=\frac{\cot (e+f x)}{a^3 c f}-\frac{\cot ^3(e+f x)}{3 a^3 c f}+\frac{2 \cot ^5(e+f x)}{5 a^3 c f}-\frac{2 \csc (e+f x)}{a^3 c f}+\frac{4 \csc ^3(e+f x)}{3 a^3 c f}-\frac{2 \csc ^5(e+f x)}{5 a^3 c f}+\frac{\int 1 \, dx}{a^3 c}\\ &=\frac{x}{a^3 c}+\frac{\cot (e+f x)}{a^3 c f}-\frac{\cot ^3(e+f x)}{3 a^3 c f}+\frac{2 \cot ^5(e+f x)}{5 a^3 c f}-\frac{2 \csc (e+f x)}{a^3 c f}+\frac{4 \csc ^3(e+f x)}{3 a^3 c f}-\frac{2 \csc ^5(e+f x)}{5 a^3 c f}\\ \end{align*}
Mathematica [A] time = 0.929026, size = 197, normalized size = 1.56 \[ -\frac{\csc \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}\right ) \csc \left (\frac{1}{2} (e+f x)\right ) \sec ^5\left (\frac{1}{2} (e+f x)\right ) (-445 \sin (e+f x)-356 \sin (2 (e+f x))-89 \sin (3 (e+f x))+240 \sin (2 e+f x)+296 \sin (e+2 f x)+120 \sin (3 e+2 f x)+104 \sin (2 e+3 f x)+150 f x \cos (2 e+f x)-120 f x \cos (e+2 f x)+120 f x \cos (3 e+2 f x)-30 f x \cos (2 e+3 f x)+30 f x \cos (4 e+3 f x)+80 \sin (e)+280 \sin (f x)-150 f x \cos (f x))}{3840 a^3 c f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.061, size = 109, normalized size = 0.9 \begin{align*} -{\frac{1}{40\,f{a}^{3}c} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}+{\frac{5}{24\,f{a}^{3}c} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{11}{8\,f{a}^{3}c}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{3}c}}+{\frac{1}{8\,f{a}^{3}c} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.55291, size = 165, normalized size = 1.31 \begin{align*} -\frac{\frac{\frac{165 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{25 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3} c} - \frac{240 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3} c} - \frac{15 \,{\left (\cos \left (f x + e\right ) + 1\right )}}{a^{3} c \sin \left (f x + e\right )}}{120 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.03937, size = 284, normalized size = 2.25 \begin{align*} \frac{26 \, \cos \left (f x + e\right )^{3} + 22 \, \cos \left (f x + e\right )^{2} + 15 \,{\left (f x \cos \left (f x + e\right )^{2} + 2 \, f x \cos \left (f x + e\right ) + f x\right )} \sin \left (f x + e\right ) - 17 \, \cos \left (f x + e\right ) - 16}{15 \,{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{1}{\sec ^{4}{\left (e + f x \right )} + 2 \sec ^{3}{\left (e + f x \right )} - 2 \sec{\left (e + f x \right )} - 1}\, dx}{a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.33161, size = 144, normalized size = 1.14 \begin{align*} \frac{\frac{120 \,{\left (f x + e\right )}}{a^{3} c} + \frac{15}{a^{3} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )} - \frac{3 \, a^{12} c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 25 \, a^{12} c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 165 \, a^{12} c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a^{15} c^{5}}}{120 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]