Optimal. Leaf size=69 \[ \frac {x}{a^2 c}+\frac {\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}-\frac {\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3989, 3967, 8}
\begin {gather*} -\frac {\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac {\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}+\frac {x}{a^2 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3967
Rule 3989
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))} \, dx &=\frac {\int \cot ^4(e+f x) (c-c \sec (e+f x)) \, dx}{a^2 c^2}\\ &=-\frac {\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac {\int \cot ^2(e+f x) (-3 c+2 c \sec (e+f x)) \, dx}{3 a^2 c^2}\\ &=\frac {\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}-\frac {\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac {\int 3 c \, dx}{3 a^2 c^2}\\ &=\frac {x}{a^2 c}+\frac {\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}-\frac {\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.58, size = 135, normalized size = 1.96 \begin {gather*} \frac {\csc \left (\frac {e}{2}\right ) \csc \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {e}{2}\right ) \sec ^3\left (\frac {1}{2} (e+f x)\right ) (6 f x \cos (f x)-6 f x \cos (2 e+f x)+3 f x \cos (e+2 f x)-3 f x \cos (3 e+2 f x)-10 \sin (f x)+10 \sin (e+f x)+5 \sin (2 (e+f x))-6 \sin (2 e+f x)-8 \sin (e+2 f x))}{96 a^2 c f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 60, normalized size = 0.87
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+8 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f \,a^{2} c}\) | \(60\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+8 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f \,a^{2} c}\) | \(60\) |
risch | \(\frac {x}{a^{2} c}-\frac {2 i \left (3 \,{\mathrm e}^{3 i \left (f x +e \right )}-5 \,{\mathrm e}^{i \left (f x +e \right )}-4\right )}{3 f \,a^{2} c \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}\) | \(72\) |
norman | \(\frac {\frac {x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c a}+\frac {1}{4 a c f}-\frac {\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}{a c f}+\frac {\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )}{12 a c f}}{a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 110, normalized size = 1.59 \begin {gather*} -\frac {\frac {\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}}}{a^{2} c} - \frac {24 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2} c} - \frac {3 \, {\left (\cos \left (f x + e\right ) + 1\right )}}{a^{2} c \sin \left (f x + e\right )}}{12 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.40, size = 76, normalized size = 1.10 \begin {gather*} \frac {4 \, \cos \left (f x + e\right )^{2} + 3 \, {\left (f x \cos \left (f x + e\right ) + f x\right )} \sin \left (f x + e\right ) + \cos \left (f x + e\right ) - 2}{3 \, {\left (a^{2} c f \cos \left (f x + e\right ) + a^{2} c 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 {1}{\sec ^{3}{\left (e + f x \right )} + \sec ^{2}{\left (e + f x \right )} - \sec {\left (e + f x \right )} - 1}\, dx}{a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.49, size = 81, normalized size = 1.17 \begin {gather*} \frac {\frac {12 \, {\left (f x + e\right )}}{a^{2} c} + \frac {3}{a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \frac {a^{4} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 12 \, a^{4} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6} c^{3}}}{12 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.42, size = 69, normalized size = 1.00 \begin {gather*} \frac {x}{a^2\,c}+\frac {\frac {4\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{3}-\frac {7\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{6}+\frac {1}{12}}{a^2\,c\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________