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 = 0.162603, antiderivative size = 115, normalized size of antiderivative = 1., 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 3987
Rule 89
Rule 78
Rule 37
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*}
Mathematica [A] time = 0.470205, 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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Maple [A] time = 0.064, size = 128, normalized size = 1.1 \begin{align*}{\frac{1}{4\,f{a}^{3}} \left ({\frac{{c}^{2}}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}-{\frac{2\,cd}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}+{\frac{{d}^{2}}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}-{\frac{2\,{c}^{2}}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+{\frac{2\,{d}^{2}}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ){c}^{2}+2\,cd\tan \left ( 1/2\,fx+e/2 \right ) +\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ){d}^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0211, size = 248, normalized size = 2.16 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.45978, size = 270, normalized size = 2.35 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24547, size = 185, normalized size = 1.61 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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