Optimal. Leaf size=80 \[ \frac {\cot ^5(e+f x)}{5 a^2 c^3 f}+\frac {\csc (e+f x)}{a^2 c^3 f}-\frac {2 \csc ^3(e+f x)}{3 a^2 c^3 f}+\frac {\csc ^5(e+f x)}{5 a^2 c^3 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {4043, 2686,
200, 2687, 30} \begin {gather*} \frac {\cot ^5(e+f x)}{5 a^2 c^3 f}+\frac {\csc ^5(e+f x)}{5 a^2 c^3 f}-\frac {2 \csc ^3(e+f x)}{3 a^2 c^3 f}+\frac {\csc (e+f x)}{a^2 c^3 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 200
Rule 2686
Rule 2687
Rule 4043
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx &=-\frac {\int \left (a \cot ^5(e+f x) \csc (e+f x)+a \cot ^4(e+f x) \csc ^2(e+f x)\right ) \, dx}{a^3 c^3}\\ &=-\frac {\int \cot ^5(e+f x) \csc (e+f x) \, dx}{a^2 c^3}-\frac {\int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a^2 c^3}\\ &=-\frac {\text {Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a^2 c^3 f}+\frac {\text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a^2 c^3 f}\\ &=\frac {\cot ^5(e+f x)}{5 a^2 c^3 f}+\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^2 c^3 f}\\ &=\frac {\cot ^5(e+f x)}{5 a^2 c^3 f}+\frac {\csc (e+f x)}{a^2 c^3 f}-\frac {2 \csc ^3(e+f x)}{3 a^2 c^3 f}+\frac {\csc ^5(e+f x)}{5 a^2 c^3 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.99, size = 147, normalized size = 1.84 \begin {gather*} -\frac {\csc (e) \csc ^2\left (\frac {1}{2} (e+f x)\right ) \csc ^3(e+f x) (-200 \sin (e)+104 \sin (f x)+534 \sin (e+f x)-178 \sin (2 (e+f x))-178 \sin (3 (e+f x))+89 \sin (4 (e+f x))+40 \sin (2 e+f x)-168 \sin (e+2 f x)+120 \sin (3 e+2 f x)+72 \sin (2 e+3 f x)-120 \sin (4 e+3 f x)+24 \sin (3 e+4 f x))}{1920 a^2 c^3 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.17, size = 76, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {6}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}-\frac {4}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}}{16 f \,c^{3} a^{2}}\) | \(76\) |
default | \(\frac {-\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {6}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}-\frac {4}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}}{16 f \,c^{3} a^{2}}\) | \(76\) |
risch | \(\frac {2 i \left (15 \,{\mathrm e}^{7 i \left (f x +e \right )}-15 \,{\mathrm e}^{6 i \left (f x +e \right )}-5 \,{\mathrm e}^{5 i \left (f x +e \right )}+25 \,{\mathrm e}^{4 i \left (f x +e \right )}+13 \,{\mathrm e}^{3 i \left (f x +e \right )}-21 \,{\mathrm e}^{2 i \left (f x +e \right )}+9 \,{\mathrm e}^{i \left (f x +e \right )}+3\right )}{15 f \,c^{3} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5}}\) | \(118\) |
norman | \(\frac {\frac {1}{80 a c f}-\frac {\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}{12 a c f}+\frac {3 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 a c f}+\frac {\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )}{4 a c f}-\frac {\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )}{48 a c f}}{a \,c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 131, normalized size = 1.64 \begin {gather*} \frac {\frac {5 \, {\left (\frac {12 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} c^{3}} - \frac {{\left (\frac {20 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {90 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 3\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{a^{2} c^{3} \sin \left (f x + e\right )^{5}}}{240 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.85, size = 117, normalized size = 1.46 \begin {gather*} \frac {3 \, \cos \left (f x + e\right )^{4} + 12 \, \cos \left (f x + e\right )^{3} - 12 \, \cos \left (f x + e\right )^{2} - 8 \, \cos \left (f x + e\right ) + 8}{15 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} - a^{2} c^{3} f \cos \left (f x + e\right )^{2} - a^{2} c^{3} f \cos \left (f x + e\right ) + a^{2} c^{3} f\right )} \sin \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - \sec ^{4}{\left (e + f x \right )} - 2 \sec ^{3}{\left (e + f x \right )} + 2 \sec ^{2}{\left (e + f x \right )} + \sec {\left (e + f x \right )} - 1}\, dx}{a^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.50, size = 96, normalized size = 1.20 \begin {gather*} \frac {\frac {90 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 20 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3}{a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}} - \frac {5 \, {\left (a^{4} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 12 \, a^{4} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} c^{9}}}{240 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.01, size = 76, normalized size = 0.95 \begin {gather*} \frac {-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+60\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+90\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+3}{240\,a^2\,c^3\,f\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________