Optimal. Leaf size=211 \[ \frac {d \left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 a^2 f (c-d)^3 (c+d) (c+d \sec (e+f x))}+\frac {2 d^2 (3 c+2 d) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a^2 f (c-d)^{7/2} (c+d)^{3/2}}+\frac {(c-6 d) \tan (e+f x)}{3 a^2 f (c-d)^2 (\sec (e+f x)+1) (c+d \sec (e+f x))}+\frac {\tan (e+f x)}{3 f (c-d) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))} \]
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Rubi [A] time = 0.37, antiderivative size = 260, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3987, 103, 152, 12, 93, 205} \[ \frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 f (c-d)^3 (c+d) \left (a^2 \sec (e+f x)+a^2\right )}-\frac {d \tan (e+f x)}{f \left (c^2-d^2\right ) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))}-\frac {2 d^2 (3 c+2 d) \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{a f (c-d)^{7/2} (c+d)^{3/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {(c+4 d) \tan (e+f x)}{3 f (c-d)^2 (c+d) (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 152
Rule 205
Rule 3987
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x} (a+a x)^{5/2} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {a^2 (c+2 d)-2 a^2 d x}{\sqrt {a-a x} (a+a x)^{5/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{\left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}+\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {-a^4 (c-6 d) (c+d)-a^4 d (c+4 d) x}{\sqrt {a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{3 a^3 (c-d) \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {3 a^6 d^2 (3 c+2 d)}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{3 a^6 (c-d)^2 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac {\left (d^2 (3 c+2 d) \tan (e+f x)\right ) \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)^2 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac {\left (2 d^2 (3 c+2 d) \tan (e+f x)\right ) \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)^2 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}-\frac {2 d^2 (3 c+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)}{a (c-d)^{7/2} (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}\\ \end {align*}
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Mathematica [C] time = 3.96, size = 376, normalized size = 1.78 \[ \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \sec ^4(e+f x) (c \cos (e+f x)+d) \left (\frac {12 d^2 (3 c+2 d) (\sin (e)+i \cos (e)) \cos ^3\left (\frac {1}{2} (e+f x)\right ) (c \cos (e+f x)+d) \tan ^{-1}\left (\frac {(\sin (e)+i \cos (e)) \left (\tan \left (\frac {f x}{2}\right ) (c \cos (e)-d)+c \sin (e)\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right )}{(c+d) \sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+\frac {6 d^3 \cos ^3\left (\frac {1}{2} (e+f x)\right ) (c \sin (f x)-d \sin (e))}{c (c+d) \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right )}+(c-d) \tan \left (\frac {e}{2}\right ) \cos \left (\frac {1}{2} (e+f x)\right ) (c \cos (e+f x)+d)-4 (c-4 d) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right ) (c \cos (e+f x)+d)+(c-d) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) (c \cos (e+f x)+d)\right )}{3 a^2 f (d-c)^3 (\sec (e+f x)+1)^2 (c+d \sec (e+f x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 1242, normalized size = 5.89 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 490, normalized size = 2.32 \[ \frac {\frac {12 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}} + \frac {12 \, {\left (3 \, c d^{2} + 2 \, d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right )} \sqrt {-c^{2} + d^{2}}} - \frac {a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 54 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 48 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6} c^{6} - 6 \, a^{6} c^{5} d + 15 \, a^{6} c^{4} d^{2} - 20 \, a^{6} c^{3} d^{3} + 15 \, a^{6} c^{2} d^{4} - 6 \, a^{6} c d^{5} + a^{6} d^{6}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.80, size = 203, normalized size = 0.96 \[ \frac {-\frac {\frac {\left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c}{3}-\frac {\left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d}{3}-\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c +5 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) d}{\left (c^{2}-2 c d +d^{2}\right ) \left (c -d \right )}-\frac {4 d^{2} \left (-\frac {d \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{\left (c +d \right ) \left (\left (\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c -\left (\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d -c -d \right )}-\frac {\left (3 c +2 d \right ) \arctanh \left (\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \left (c -d \right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{3}}}{2 f \,a^{2}} \]
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 = 2.18, size = 314, normalized size = 1.49 \[ \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3}{2\,a^2\,{\left (c-d\right )}^2}-\frac {c^2-d^2}{a^2\,{\left (c-d\right )}^4}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{6\,a^2\,f\,{\left (c-d\right )}^2}+\frac {2\,d^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (c+d\right )\,\left (a^2\,d^4-a^2\,c^4+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (a^2\,c^4-4\,a^2\,c^3\,d+6\,a^2\,c^2\,d^2-4\,a^2\,c\,d^3+a^2\,d^4\right )-2\,a^2\,c\,d^3+2\,a^2\,c^3\,d\right )}-\frac {d^2\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^4-4{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3\,d+6{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^2\,d^2-4{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c\,d^3+1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,d^4}{\sqrt {c+d}\,{\left (c-d\right )}^{7/2}}\right )\,\left (3\,c+2\,d\right )\,2{}\mathrm {i}}{a^2\,f\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{7/2}} \]
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 \[ \frac {\int \frac {\sec {\left (e + f x \right )}}{c^{2} \sec ^{2}{\left (e + f x \right )} + 2 c^{2} \sec {\left (e + f x \right )} + c^{2} + 2 c d \sec ^{3}{\left (e + f x \right )} + 4 c d \sec ^{2}{\left (e + f x \right )} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{4}{\left (e + f x \right )} + 2 d^{2} \sec ^{3}{\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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