Optimal. Leaf size=79 \[ \frac{2 \tan (e+f x)}{3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{3 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}} \]
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Rubi [A] time = 0.0928341, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {4146, 378, 191} \[ \frac{2 \tan (e+f x)}{3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{3 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4146
Rule 378
Rule 191
Rubi steps
\begin{align*} \int \frac{\sec ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{\left (a+b+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 (a+b) f}\\ &=\frac{\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac{2 \tan (e+f x)}{3 (a+b)^2 f \sqrt{a+b+b \tan ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 4.55023, size = 74, normalized size = 0.94 \[ \frac{\tan (e+f x) \sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) (a \cos (2 (e+f x))+2 a+3 b)}{6 f (a+b)^2 \left (a+b \sec ^2(e+f x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.304, size = 76, normalized size = 1. \begin{align*}{\frac{\sin \left ( fx+e \right ) \left ( b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \left ( 2\,a \left ( \cos \left ( fx+e \right ) \right ) ^{2}+a+3\,b \right ) }{3\,f \left ( a+b \right ) ^{2} \left ( \cos \left ( fx+e \right ) \right ) ^{5}} \left ({\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18301, size = 158, normalized size = 2. \begin{align*} \frac{\frac{2 \, \tan \left (f x + e\right )}{\sqrt{b \tan \left (f x + e\right )^{2} + a + b}{\left (a + b\right )}^{2}} + \frac{\tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac{3}{2}}{\left (a + b\right )}} - \frac{\tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac{3}{2}} b} + \frac{\tan \left (f x + e\right )}{\sqrt{b \tan \left (f x + e\right )^{2} + a + b}{\left (a + b\right )} b}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.842701, size = 313, normalized size = 3.96 \begin{align*} \frac{{\left (2 \, a \cos \left (f x + e\right )^{3} +{\left (a + 3 \, b\right )} \cos \left (f x + e\right )\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )}{3 \,{\left ({\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} f \cos \left (f x + e\right )^{4} + 2 \,{\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} f \cos \left (f x + e\right )^{2} +{\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} 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*} \int \frac{\sec ^{4}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.4051, size = 560, normalized size = 7.09 \begin{align*} -\frac{2 \,{\left ({\left (\frac{3 \,{\left (a^{6} b^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) + 2 \, a^{5} b^{5} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) + a^{4} b^{6} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )\right )} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2}}{a^{7} b^{4} + 3 \, a^{6} b^{5} + 3 \, a^{5} b^{6} + a^{4} b^{7}} - \frac{2 \,{\left (a^{6} b^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) - 2 \, a^{5} b^{5} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) - 3 \, a^{4} b^{6} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )\right )}}{a^{7} b^{4} + 3 \, a^{6} b^{5} + 3 \, a^{5} b^{6} + a^{4} b^{7}}\right )} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + \frac{3 \,{\left (a^{6} b^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) + 2 \, a^{5} b^{5} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) + a^{4} b^{6} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )\right )}}{a^{7} b^{4} + 3 \, a^{6} b^{5} + 3 \, a^{5} b^{6} + a^{4} b^{7}}\right )} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{3 \,{\left (a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 2 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a + b\right )}^{\frac{3}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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