Optimal. Leaf size=115 \[ \frac {\left (2 c^2+6 c d+7 d^2\right ) \tan (e+f x)}{15 f \left (a^3 \sec (e+f x)+a^3\right )}+\frac {(c-d)^2 \tan (e+f x)}{5 f (a \sec (e+f x)+a)^3}+\frac {2 (c+4 d) (c-d) \tan (e+f x)}{15 a f (a \sec (e+f x)+a)^2} \]
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Rubi [A] time = 0.16, antiderivative size = 115, normalized size of antiderivative = 1.00, 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, 89, 78, 37} \[ \frac {\left (2 c^2+6 c d+7 d^2\right ) \tan (e+f x)}{15 f \left (a^3 \sec (e+f x)+a^3\right )}+\frac {(c-d)^2 \tan (e+f x)}{5 f (a \sec (e+f x)+a)^3}+\frac {2 (c+4 d) (c-d) \tan (e+f x)}{15 a f (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 89
Rule 3987
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^3} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^2}{\sqrt {a-a x} (a+a x)^{7/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d)^2 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {a^3 \left (2 c^2+6 c d-3 d^2\right )+5 a^3 d^2 x}{\sqrt {a-a x} (a+a x)^{5/2}} \, dx,x,\sec (e+f x)\right )}{5 a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d)^2 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {2 (c-d) (c+4 d) \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}-\frac {\left (\left (2 c^2+6 c d+7 d^2\right ) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x} (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{15 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d)^2 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {2 (c-d) (c+4 d) \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {\left (2 c^2+6 c d+7 d^2\right ) \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 180, normalized size = 1.57 \[ \frac {\sec \left (\frac {e}{2}\right ) \cos \left (\frac {1}{2} (e+f x)\right ) \left (10 \left (4 c^2+3 c d+2 d^2\right ) \sin \left (\frac {f x}{2}\right )+20 c^2 \sin \left (e+\frac {3 f x}{2}\right )-15 c^2 \sin \left (2 e+\frac {3 f x}{2}\right )+7 c^2 \sin \left (2 e+\frac {5 f x}{2}\right )-30 c (c+d) \sin \left (e+\frac {f x}{2}\right )+30 c d \sin \left (e+\frac {3 f x}{2}\right )+6 c d \sin \left (2 e+\frac {5 f x}{2}\right )+10 d^2 \sin \left (e+\frac {3 f x}{2}\right )+2 d^2 \sin \left (2 e+\frac {5 f x}{2}\right )\right )}{30 a^3 f (\cos (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 113, normalized size = 0.98 \[ \frac {{\left ({\left (7 \, c^{2} + 6 \, c d + 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, c^{2} + 6 \, c d + 7 \, d^{2} + 6 \, {\left (c^{2} + 3 \, c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 137, normalized size = 1.19 \[ \frac {3 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 10 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 30 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{60 \, a^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.83, size = 128, normalized size = 1.11 \[ \frac {\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c^{2}}{5}-\frac {2 \left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c d}{5}+\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d^{2}}{5}-\frac {2 \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c^{2}}{3}+\frac {2 \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d^{2}}{3}+\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c^{2}+2 c d \tan \left (\frac {e}{2}+\frac {f x}{2}\right )+\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) d^{2}}{4 f \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 184, normalized size = 1.60 \[ \frac {\frac {d^{2} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac {c^{2} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac {6 \, c d {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.83, size = 79, normalized size = 0.69 \[ \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\left (c+d\right )}^2}{4\,a^3\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,c^2-2\,d^2\right )}{12\,a^3\,f}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\left (c-d\right )}^2}{20\,a^3\,f} \]
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 \[ \frac {\int \frac {c^{2} \sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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