Optimal. Leaf size=99 \[ \frac {(b c-a d) \tan (e+f x)}{f \left (c^2-d^2\right ) (c+d \sec (e+f x))}+\frac {2 (a c-b d) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{f (c-d)^{3/2} (c+d)^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4003, 12, 3831, 2659, 208} \[ \frac {(b c-a d) \tan (e+f x)}{f \left (c^2-d^2\right ) (c+d \sec (e+f x))}+\frac {2 (a c-b d) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{f (c-d)^{3/2} (c+d)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 208
Rule 2659
Rule 3831
Rule 4003
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^2} \, dx &=\frac {(b c-a d) \tan (e+f x)}{\left (c^2-d^2\right ) f (c+d \sec (e+f x))}+\frac {\int \frac {(-a c+b d) \sec (e+f x)}{c+d \sec (e+f x)} \, dx}{-c^2+d^2}\\ &=\frac {(b c-a d) \tan (e+f x)}{\left (c^2-d^2\right ) f (c+d \sec (e+f x))}+\frac {(a c-b d) \int \frac {\sec (e+f x)}{c+d \sec (e+f x)} \, dx}{c^2-d^2}\\ &=\frac {(b c-a d) \tan (e+f x)}{\left (c^2-d^2\right ) f (c+d \sec (e+f x))}+\frac {(a c-b d) \int \frac {1}{1+\frac {c \cos (e+f x)}{d}} \, dx}{d \left (c^2-d^2\right )}\\ &=\frac {(b c-a d) \tan (e+f x)}{\left (c^2-d^2\right ) f (c+d \sec (e+f x))}+\frac {(2 (a c-b d)) \operatorname {Subst}\left (\int \frac {1}{1+\frac {c}{d}+\left (1-\frac {c}{d}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{d \left (c^2-d^2\right ) f}\\ &=\frac {2 (a c-b d) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{(c-d)^{3/2} (c+d)^{3/2} f}+\frac {(b c-a d) \tan (e+f x)}{\left (c^2-d^2\right ) f (c+d \sec (e+f x))}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 97, normalized size = 0.98 \[ \frac {\frac {(b c-a d) \sin (e+f x)}{(c-d) (c+d) (c \cos (e+f x)+d)}-\frac {2 (a c-b d) \tanh ^{-1}\left (\frac {(d-c) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{3/2}}}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 389, normalized size = 3.93 \[ \left [\frac {{\left (a c d - b d^{2} + {\left (a c^{2} - b c d\right )} \cos \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \log \left (\frac {2 \, c d \cos \left (f x + e\right ) - {\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {c^{2} - d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) + 2 \, {\left (b c^{3} - a c^{2} d - b c d^{2} + a d^{3}\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{5} - 2 \, c^{3} d^{2} + c d^{4}\right )} f \cos \left (f x + e\right ) + {\left (c^{4} d - 2 \, c^{2} d^{3} + d^{5}\right )} f\right )}}, \frac {{\left (a c d - b d^{2} + {\left (a c^{2} - b c d\right )} \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \arctan \left (-\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )}}{{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}\right ) + {\left (b c^{3} - a c^{2} d - b c d^{2} + a d^{3}\right )} \sin \left (f x + e\right )}{{\left (c^{5} - 2 \, c^{3} d^{2} + c d^{4}\right )} f \cos \left (f x + e\right ) + {\left (c^{4} d - 2 \, c^{2} d^{3} + d^{5}\right )} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 179, normalized size = 1.81 \[ -\frac {2 \, {\left (\frac {{\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, c - 2 \, d\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )} {\left (a c - b d\right )}}{{\left (c^{2} - d^{2}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {b c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )} {\left (c^{2} - d^{2}\right )}}\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 132, normalized size = 1.33 \[ \frac {\frac {2 \left (d a -c b \right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{\left (c^{2}-d^{2}\right ) \left (\left (\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c -\left (\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d -c -d \right )}+\frac {2 \left (c a -b d \right ) \arctanh \left (\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \left (c -d \right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \left (c -d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{f} \]
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 = 2.12, size = 106, normalized size = 1.07 \[ \frac {2\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {c-d}}{\sqrt {c+d}}\right )\,\left (a\,c-b\,d\right )}{f\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{3/2}}-\frac {2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a\,d-b\,c\right )}{f\,\left (c+d\right )\,\left (c-d\right )\,\left (\left (d-c\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+c+d\right )} \]
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 \[ \int \frac {\left (a + b \sec {\left (e + f x \right )}\right ) \sec {\left (e + f x \right )}}{\left (c + d \sec {\left (e + f x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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