Optimal. Leaf size=119 \[ -\frac {5 c^3 \tan (e+f x)}{a^2 f}+\frac {5 c^3 \tanh ^{-1}(\sin (e+f x))}{a^2 f}-\frac {10 \tan (e+f x) \left (c^3-c^3 \sec (e+f x)\right )}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2} \]
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Rubi [A] time = 0.19, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3957, 3787, 3770, 3767, 8} \[ -\frac {5 c^3 \tan (e+f x)}{a^2 f}+\frac {5 c^3 \tanh ^{-1}(\sin (e+f x))}{a^2 f}-\frac {10 \tan (e+f x) \left (c^3-c^3 \sec (e+f x)\right )}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^2}{3 f (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3787
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^2} \, dx &=\frac {2 c (c-c \sec (e+f x))^2 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {(5 c) \int \frac {\sec (e+f x) (c-c \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx}{3 a}\\ &=\frac {2 c (c-c \sec (e+f x))^2 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {10 \left (c^3-c^3 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {\left (5 c^2\right ) \int \sec (e+f x) (c-c \sec (e+f x)) \, dx}{a^2}\\ &=\frac {2 c (c-c \sec (e+f x))^2 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {10 \left (c^3-c^3 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {\left (5 c^3\right ) \int \sec (e+f x) \, dx}{a^2}-\frac {\left (5 c^3\right ) \int \sec ^2(e+f x) \, dx}{a^2}\\ &=\frac {5 c^3 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac {2 c (c-c \sec (e+f x))^2 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {10 \left (c^3-c^3 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {\left (5 c^3\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{a^2 f}\\ &=\frac {5 c^3 \tanh ^{-1}(\sin (e+f x))}{a^2 f}-\frac {5 c^3 \tan (e+f x)}{a^2 f}+\frac {2 c (c-c \sec (e+f x))^2 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {10 \left (c^3-c^3 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}\\ \end {align*}
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Mathematica [B] time = 4.32, size = 485, normalized size = 4.08 \[ \frac {c^3 (\cos (e+f x)-1)^3 \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right ) \left (-\frac {1}{16} \sec ^3\left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) (-80 \cos (e+f x)-40 \cos (2 (e+f x))+66 \cos (2 e+f x)+23 \cos (e+2 f x)+17 \cos (3 e+2 f x)+40 \cos (e)+78 \cos (f x)-40) \csc ^5\left (\frac {1}{2} (e+f x)\right )-26 \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \cot ^4\left (\frac {1}{2} (e+f x)\right ) \csc \left (\frac {1}{2} (e+f x)\right )+20 \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \csc \left (\frac {1}{2} (e+f x)\right )+2 \left (\sin \left (\frac {3 e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \sec ^3\left (\frac {e}{2}\right ) \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )-15 \cos (e) \sec ^2\left (\frac {e}{2}\right ) \cot ^5\left (\frac {1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )-\left (\tan ^2\left (\frac {e}{2}\right )-1\right ) \cot ^3\left (\frac {1}{2} (e+f x)\right ) \left (15 \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )-4 \tan \left (\frac {e}{2}\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )}{6 a^2 f \left (\tan \left (\frac {e}{2}\right )-1\right ) \left (\tan \left (\frac {e}{2}\right )+1\right ) (\cos (e+f x)+1)^2 \left (\cot \left (\frac {1}{2} (e+f x)\right )-1\right ) \left (\cot \left (\frac {1}{2} (e+f x)\right )+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 178, normalized size = 1.50 \[ \frac {15 \, {\left (c^{3} \cos \left (f x + e\right )^{3} + 2 \, c^{3} \cos \left (f x + e\right )^{2} + c^{3} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, {\left (c^{3} \cos \left (f x + e\right )^{3} + 2 \, c^{3} \cos \left (f x + e\right )^{2} + c^{3} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (23 \, c^{3} \cos \left (f x + e\right )^{2} + 34 \, c^{3} \cos \left (f x + e\right ) + 3 \, c^{3}\right )} \sin \left (f x + e\right )}{6 \, {\left (a^{2} f \cos \left (f x + e\right )^{3} + 2 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f \cos \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [A] time = 0.68, size = 136, normalized size = 1.14 \[ -\frac {4 c^{3} \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 f \,a^{2}}-\frac {8 c^{3} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f \,a^{2}}+\frac {c^{3}}{f \,a^{2} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}-\frac {5 c^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{f \,a^{2}}+\frac {c^{3}}{f \,a^{2} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}+\frac {5 c^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{f \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 341, normalized size = 2.87 \[ -\frac {c^{3} {\left (\frac {\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (f x + e\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} + 3 \, c^{3} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} + \frac {3 \, c^{3} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} - \frac {c^{3} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.65, size = 104, normalized size = 0.87 \[ \frac {10\,c^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^2\,f}-\frac {4\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^2\,f}-\frac {8\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^2\,f}+\frac {2\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-a^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {c^{3} \left (\int \left (- \frac {\sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {3 \sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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