Optimal. Leaf size=83 \[ \frac {\tan (e+f x)}{f (c-d) (a \sec (e+f x)+a)}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a f (c-d)^{3/2} \sqrt {c+d}} \]
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Rubi [A] time = 0.16, antiderivative size = 134, normalized size of antiderivative = 1.61, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3987, 96, 93, 205} \[ \frac {2 d \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{f (c-d)^{3/2} \sqrt {c+d} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {\tan (e+f x)}{f (c-d) (a \sec (e+f x)+a)} \]
Antiderivative was successfully verified.
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Rule 93
Rule 96
Rule 205
Rule 3987
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x)) (c+d \sec (e+f x))} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {\tan (e+f x)}{(c-d) f (a+a \sec (e+f x))}+\frac {(a d \tan (e+f x)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{(c-d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {\tan (e+f x)}{(c-d) f (a+a \sec (e+f x))}+\frac {(2 a d \tan (e+f x)) \operatorname {Subst}\left (\int \frac {1}{a c-a d-(-a c-a d) x^2} \, dx,x,\frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {a-a \sec (e+f x)}}\right )}{(c-d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {\tan (e+f x)}{(c-d) f (a+a \sec (e+f x))}+\frac {2 d \tan ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+a \sec (e+f x)}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right ) \tan (e+f x)}{(c-d)^{3/2} \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.67, size = 160, normalized size = 1.93 \[ \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (\sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )+\frac {2 d (\sin (e)+i \cos (e)) \cos \left (\frac {1}{2} (e+f x)\right ) \tan ^{-1}\left (\frac {(\sin (e)+i \cos (e)) \left (\tan \left (\frac {f x}{2}\right ) (c \cos (e)-d)+c \sin (e)\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right )}{a f (c-d) (\cos (e+f x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 353, normalized size = 4.25 \[ \left [-\frac {\sqrt {c^{2} - d^{2}} {\left (d \cos \left (f x + e\right ) + d\right )} \log \left (\frac {2 \, c d \cos \left (f x + e\right ) - {\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {c^{2} - d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) - 2 \, {\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f \cos \left (f x + e\right ) + {\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f\right )}}, -\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cos \left (f x + e\right ) + d\right )} \arctan \left (-\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )}}{{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}\right ) - {\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}{{\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f \cos \left (f x + e\right ) + {\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 114, normalized size = 1.37 \[ \frac {\frac {2 \, {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, c - 2 \, d\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )} d}{{\left (a c - a d\right )} \sqrt {-c^{2} + d^{2}}} + \frac {\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a c - a d}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 74, normalized size = 0.89 \[ \frac {\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{c -d}-\frac {2 d \arctanh \left (\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \left (c -d \right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{f a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.95, size = 110, normalized size = 1.33 \[ \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a\,f\,\left (c-d\right )}-\frac {2\,d\,\mathrm {atanh}\left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^2-2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c\,d+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,d^2}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {c+d}\,{\left (c-d\right )}^{3/2}}\right )}{a\,f\,\sqrt {c+d}\,{\left (c-d\right )}^{3/2}} \]
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 \[ \frac {\int \frac {\sec {\left (e + f x \right )}}{c \sec {\left (e + f x \right )} + c + d \sec ^{2}{\left (e + f x \right )} + d \sec {\left (e + f x \right )}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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