Optimal. Leaf size=78 \[ -\frac {2 \cot ^5(e+f x)}{5 a^3 c f}+\frac {2 \csc ^5(e+f x)}{5 a^3 c f}-\frac {\csc ^3(e+f x)}{a^3 c f}+\frac {\csc (e+f x)}{a^3 c f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3958, 2606, 194, 2607, 30, 14} \[ -\frac {2 \cot ^5(e+f x)}{5 a^3 c f}+\frac {2 \csc ^5(e+f x)}{5 a^3 c f}-\frac {\csc ^3(e+f x)}{a^3 c f}+\frac {\csc (e+f x)}{a^3 c f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 30
Rule 194
Rule 2606
Rule 2607
Rule 3958
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))} \, dx &=-\frac {\int \left (c^2 \cot ^5(e+f x) \csc (e+f x)-2 c^2 \cot ^4(e+f x) \csc ^2(e+f x)+c^2 \cot ^3(e+f x) \csc ^3(e+f x)\right ) \, dx}{a^3 c^3}\\ &=-\frac {\int \cot ^5(e+f x) \csc (e+f x) \, dx}{a^3 c}-\frac {\int \cot ^3(e+f x) \csc ^3(e+f x) \, dx}{a^3 c}+\frac {2 \int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a^3 c}\\ &=\frac {\operatorname {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c f}+\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a^3 c f}+\frac {2 \operatorname {Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a^3 c f}\\ &=-\frac {2 \cot ^5(e+f x)}{5 a^3 c f}+\frac {\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c f}+\frac {\operatorname {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c f}\\ &=-\frac {2 \cot ^5(e+f x)}{5 a^3 c f}+\frac {\csc (e+f x)}{a^3 c f}-\frac {\csc ^3(e+f x)}{a^3 c f}+\frac {2 \csc ^5(e+f x)}{5 a^3 c f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.82, size = 109, normalized size = 1.40 \[ -\frac {\csc (e) \sin ^4\left (\frac {1}{2} (e+f x)\right ) (65 \sin (e+f x)+52 \sin (2 (e+f x))+13 \sin (3 (e+f x))-40 \sin (2 e+f x)-12 \sin (e+2 f x)-20 \sin (3 e+2 f x)-8 \sin (2 e+3 f x)-40 \sin (e)) \csc ^5(e+f x)}{20 a^3 c f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 76, normalized size = 0.97 \[ -\frac {2 \, \cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )^{2} - 4 \, \cos \left (f x + e\right ) - 2}{5 \, {\left (a^{3} c f \cos \left (f x + e\right )^{2} + 2 \, a^{3} c f \cos \left (f x + e\right ) + a^{3} c f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.37, size = 91, normalized size = 1.17 \[ \frac {\frac {5}{a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \frac {a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 5 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15} c^{5}}}{40 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.70, size = 61, normalized size = 0.78 \[ \frac {\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{5}-\left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )+3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )+\frac {1}{\tan \left (\frac {e}{2}+\frac {f x}{2}\right )}}{8 f \,a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 95, normalized size = 1.22 \[ \frac {\frac {\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3} c} + \frac {5 \, {\left (\cos \left (f x + e\right ) + 1\right )}}{a^{3} c \sin \left (f x + e\right )}}{40 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.62, size = 74, normalized size = 0.95 \[ -\frac {16\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-28\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+8\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1}{40\,a^3\,c\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} + 2 \sec ^{3}{\left (e + f x \right )} - 2 \sec {\left (e + f x \right )} - 1}\, dx}{a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________