Optimal. Leaf size=88 \[ \frac {a^3 \tanh ^{-1}(\sin (e+f x))}{c^2 f}+\frac {4 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))}-\frac {8 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))^2}+\frac {a^3 x}{c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3903, 3777, 3919, 3794, 3796, 3797, 3799, 3998, 3770} \[ \frac {a^3 \tanh ^{-1}(\sin (e+f x))}{c^2 f}+\frac {4 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))}-\frac {8 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))^2}+\frac {a^3 x}{c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3770
Rule 3777
Rule 3794
Rule 3796
Rule 3797
Rule 3799
Rule 3903
Rule 3919
Rule 3998
Rubi steps
\begin {align*} \int \frac {(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^2} \, dx &=\frac {\int \left (\frac {a^3}{(1-\sec (e+f x))^2}+\frac {3 a^3 \sec (e+f x)}{(1-\sec (e+f x))^2}+\frac {3 a^3 \sec ^2(e+f x)}{(1-\sec (e+f x))^2}+\frac {a^3 \sec ^3(e+f x)}{(1-\sec (e+f x))^2}\right ) \, dx}{c^2}\\ &=\frac {a^3 \int \frac {1}{(1-\sec (e+f x))^2} \, dx}{c^2}+\frac {a^3 \int \frac {\sec ^3(e+f x)}{(1-\sec (e+f x))^2} \, dx}{c^2}+\frac {\left (3 a^3\right ) \int \frac {\sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{c^2}+\frac {\left (3 a^3\right ) \int \frac {\sec ^2(e+f x)}{(1-\sec (e+f x))^2} \, dx}{c^2}\\ &=-\frac {8 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))^2}-\frac {a^3 \int \frac {-3-\sec (e+f x)}{1-\sec (e+f x)} \, dx}{3 c^2}+\frac {a^3 \int \frac {(-2-3 \sec (e+f x)) \sec (e+f x)}{1-\sec (e+f x)} \, dx}{3 c^2}+\frac {a^3 \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{c^2}-\frac {\left (2 a^3\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{c^2}\\ &=\frac {a^3 x}{c^2}-\frac {8 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))^2}+\frac {a^3 \tan (e+f x)}{c^2 f (1-\sec (e+f x))}+\frac {a^3 \int \sec (e+f x) \, dx}{c^2}+\frac {\left (4 a^3\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{3 c^2}-\frac {\left (5 a^3\right ) \int \frac {\sec (e+f x)}{1-\sec (e+f x)} \, dx}{3 c^2}\\ &=\frac {a^3 x}{c^2}+\frac {a^3 \tanh ^{-1}(\sin (e+f x))}{c^2 f}-\frac {8 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))^2}+\frac {4 a^3 \tan (e+f x)}{3 c^2 f (1-\sec (e+f x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 1.18, size = 177, normalized size = 2.01 \[ \frac {a^3 (\cos (e+f x)+1)^3 \tan \left (\frac {1}{2} (e+f x)\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (-4 \cot \left (\frac {e}{2}\right ) \tan \left (\frac {1}{2} (e+f x)\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )+4 \csc \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \sec \left (\frac {1}{2} (e+f x)\right )+3 \tan ^3\left (\frac {1}{2} (e+f x)\right ) \left (-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )+f x\right )\right )}{6 c^2 f (\cos (e+f x)-1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 156, normalized size = 1.77 \[ \frac {8 \, a^{3} \cos \left (f x + e\right )^{2} + 16 \, a^{3} \cos \left (f x + e\right ) + 8 \, a^{3} + 3 \, {\left (a^{3} \cos \left (f x + e\right ) - a^{3}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - 3 \, {\left (a^{3} \cos \left (f x + e\right ) - a^{3}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) + 6 \, {\left (a^{3} f x \cos \left (f x + e\right ) - a^{3} f x\right )} \sin \left (f x + e\right )}{6 \, {\left (c^{2} f \cos \left (f x + e\right ) - c^{2} f\right )} \sin \left (f x + e\right )} \]
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 = 0.83, size = 90, normalized size = 1.02 \[ -\frac {4 a^{3}}{3 f \,c^{2} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{3}}-\frac {a^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{f \,c^{2}}+\frac {a^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{f \,c^{2}}+\frac {2 a^{3} \arctan \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.44, size = 274, normalized size = 3.11 \[ \frac {a^{3} {\left (\frac {12 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{2}} + \frac {{\left (\frac {9 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{3}}{c^{2} \sin \left (f x + e\right )^{3}}\right )} + a^{3} {\left (\frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c^{2}} - \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c^{2}} - \frac {{\left (\frac {9 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{3}}{c^{2} \sin \left (f x + e\right )^{3}}\right )} - \frac {3 \, a^{3} {\left (\frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{3}}{c^{2} \sin \left (f x + e\right )^{3}} + \frac {3 \, a^{3} {\left (\frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{3}}{c^{2} \sin \left (f x + e\right )^{3}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.44, size = 45, normalized size = 0.51 \[ \frac {a^3\,x}{c^2}+\frac {a^3\,\left (2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {4\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}\right )}{c^2\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{3} \left (\int \frac {3 \sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} - 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} - 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} - 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {1}{\sec ^{2}{\left (e + f x \right )} - 2 \sec {\left (e + f x \right )} + 1}\, dx\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________