Optimal. Leaf size=100 \[ \frac {10 c^3 \tan (e+f x)}{a f}-\frac {15 c^3 \tanh ^{-1}(\sin (e+f x))}{2 a f}-\frac {5 c^3 \tan (e+f x) \sec (e+f x)}{2 a f}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3957, 3788, 3767, 8, 4046, 3770} \[ \frac {10 c^3 \tan (e+f x)}{a f}-\frac {15 c^3 \tanh ^{-1}(\sin (e+f x))}{2 a f}-\frac {5 c^3 \tan (e+f x) \sec (e+f x)}{2 a f}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{f (a \sec (e+f x)+a)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3767
Rule 3770
Rule 3788
Rule 3957
Rule 4046
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))^3}{a+a \sec (e+f x)} \, dx &=\frac {2 c (c-c \sec (e+f x))^2 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {(5 c) \int \sec (e+f x) (c-c \sec (e+f x))^2 \, dx}{a}\\ &=\frac {2 c (c-c \sec (e+f x))^2 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {(5 c) \int \sec (e+f x) \left (c^2+c^2 \sec ^2(e+f x)\right ) \, dx}{a}+\frac {\left (10 c^3\right ) \int \sec ^2(e+f x) \, dx}{a}\\ &=-\frac {5 c^3 \sec (e+f x) \tan (e+f x)}{2 a f}+\frac {2 c (c-c \sec (e+f x))^2 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {\left (15 c^3\right ) \int \sec (e+f x) \, dx}{2 a}-\frac {\left (10 c^3\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{a f}\\ &=-\frac {15 c^3 \tanh ^{-1}(\sin (e+f x))}{2 a f}+\frac {10 c^3 \tan (e+f x)}{a f}-\frac {5 c^3 \sec (e+f x) \tan (e+f x)}{2 a f}+\frac {2 c (c-c \sec (e+f x))^2 \tan (e+f x)}{f (a+a \sec (e+f x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 2.65, size = 287, normalized size = 2.87 \[ \frac {\cos ^2(e+f x) \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^4\left (\frac {1}{2} (e+f x)\right ) (c-c \sec (e+f x))^3 \left (\cot \left (\frac {1}{2} (e+f x)\right ) \left (-\frac {16 \sin (f x)}{\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {1}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}-\frac {1}{\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2}-30 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+30 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )-32 \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \csc \left (\frac {1}{2} (e+f x)\right )\right )}{16 a f (\sec (e+f x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 140, normalized size = 1.40 \[ -\frac {15 \, {\left (c^{3} \cos \left (f x + e\right )^{3} + c^{3} \cos \left (f x + e\right )^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, {\left (c^{3} \cos \left (f x + e\right )^{3} + c^{3} \cos \left (f x + e\right )^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (24 \, c^{3} \cos \left (f x + e\right )^{2} + 7 \, c^{3} \cos \left (f x + e\right ) - c^{3}\right )} \sin \left (f x + e\right )}{4 \, {\left (a f \cos \left (f x + e\right )^{3} + a f \cos \left (f x + e\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.70, size = 164, normalized size = 1.64 \[ \frac {8 c^{3} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f a}-\frac {c^{3}}{2 f a \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )^{2}}-\frac {9 c^{3}}{2 f a \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}+\frac {15 c^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{2 f a}+\frac {c^{3}}{2 f a \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )^{2}}-\frac {9 c^{3}}{2 f a \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}-\frac {15 c^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{2 f a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.35, size = 386, normalized size = 3.86 \[ \frac {c^{3} {\left (\frac {2 \, {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a - \frac {2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} - \frac {3 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} + \frac {3 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} + \frac {2 \, \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} - 6 \, c^{3} {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} - \frac {2 \, \sin \left (f x + e\right )}{{\left (a - \frac {a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}} - \frac {\sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} - 6 \, c^{3} {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} - \frac {\sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} + \frac {2 \, c^{3} \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.66, size = 96, normalized size = 0.96 \[ \frac {8\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a\,f}-\frac {9\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-7\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}^2}-\frac {15\,c^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {c^{3} \left (\int \left (- \frac {\sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {3 \sec ^{3}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________