Optimal. Leaf size=121 \[ \frac {7 c^4 \tan ^3(e+f x)}{3 a f}+\frac {28 c^4 \tan (e+f x)}{a f}-\frac {35 c^4 \tanh ^{-1}(\sin (e+f x))}{2 a f}-\frac {21 c^4 \tan (e+f x) \sec (e+f x)}{2 a f}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{f (a \sec (e+f x)+a)} \]
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Rubi [A] time = 0.16, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3957, 3791, 3770, 3767, 8, 3768} \[ \frac {7 c^4 \tan ^3(e+f x)}{3 a f}+\frac {28 c^4 \tan (e+f x)}{a f}-\frac {35 c^4 \tanh ^{-1}(\sin (e+f x))}{2 a f}-\frac {21 c^4 \tan (e+f x) \sec (e+f x)}{2 a f}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{f (a \sec (e+f x)+a)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3768
Rule 3770
Rule 3791
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx &=\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {(7 c) \int \sec (e+f x) (c-c \sec (e+f x))^3 \, dx}{a}\\ &=\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {(7 c) \int \left (c^3 \sec (e+f x)-3 c^3 \sec ^2(e+f x)+3 c^3 \sec ^3(e+f x)-c^3 \sec ^4(e+f x)\right ) \, dx}{a}\\ &=\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {\left (7 c^4\right ) \int \sec (e+f x) \, dx}{a}+\frac {\left (7 c^4\right ) \int \sec ^4(e+f x) \, dx}{a}+\frac {\left (21 c^4\right ) \int \sec ^2(e+f x) \, dx}{a}-\frac {\left (21 c^4\right ) \int \sec ^3(e+f x) \, dx}{a}\\ &=-\frac {7 c^4 \tanh ^{-1}(\sin (e+f x))}{a f}-\frac {21 c^4 \sec (e+f x) \tan (e+f x)}{2 a f}+\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {\left (21 c^4\right ) \int \sec (e+f x) \, dx}{2 a}-\frac {\left (7 c^4\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{a f}-\frac {\left (21 c^4\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{a f}\\ &=-\frac {35 c^4 \tanh ^{-1}(\sin (e+f x))}{2 a f}+\frac {28 c^4 \tan (e+f x)}{a f}-\frac {21 c^4 \sec (e+f x) \tan (e+f x)}{2 a f}+\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}+\frac {7 c^4 \tan ^3(e+f x)}{3 a f}\\ \end {align*}
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Mathematica [B] time = 6.42, size = 1036, normalized size = 8.56 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 153, normalized size = 1.26 \[ -\frac {105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + c^{4} \cos \left (f x + e\right )^{3}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + c^{4} \cos \left (f x + e\right )^{3}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (166 \, c^{4} \cos \left (f x + e\right )^{3} + 55 \, c^{4} \cos \left (f x + e\right )^{2} - 13 \, c^{4} \cos \left (f x + e\right ) + 2 \, c^{4}\right )} \sin \left (f x + e\right )}{12 \, {\left (a f \cos \left (f x + e\right )^{4} + a f \cos \left (f x + e\right )^{3}\right )}} \]
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 = 0.75, size = 212, normalized size = 1.75 \[ \frac {16 c^{4} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f a}-\frac {c^{4}}{3 f a \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )^{3}}-\frac {3 c^{4}}{f a \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )^{2}}-\frac {29 c^{4}}{2 f a \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}+\frac {35 c^{4} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{2 f a}-\frac {c^{4}}{3 f a \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )^{3}}+\frac {3 c^{4}}{f a \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )^{2}}-\frac {29 c^{4}}{2 f a \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}-\frac {35 c^{4} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{2 f a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 591, normalized size = 4.88 \[ \frac {c^{4} {\left (\frac {2 \, {\left (\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {16 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a - \frac {3 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {a \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}} - \frac {9 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} + \frac {9 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} + \frac {6 \, \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} + 12 \, c^{4} {\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 )} - 36 \, c^{4} {\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 )} - 24 \, c^{4} {\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 {6 \, c^{4} \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.85, size = 112, normalized size = 0.93 \[ \frac {16\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a\,f}-\frac {29\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-\frac {136\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}+19\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}^3}-\frac {35\,c^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {c^{4} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {4 \sec ^{2}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {6 \sec ^{3}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {4 \sec ^{4}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{5}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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