Optimal. Leaf size=129 \[ \frac{\left (8 a^2+12 a b+5 b^2\right ) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{\left (8 a^2+12 a b+5 b^2\right ) \tan (e+f x) \sec (e+f x)}{16 f}+\frac{b (8 a+5 b) \tan (e+f x) \sec ^3(e+f x)}{24 f}+\frac{b \tan (e+f x) \sec ^5(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{6 f} \]
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Rubi [A] time = 0.134058, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4147, 413, 385, 199, 206} \[ \frac{\left (8 a^2+12 a b+5 b^2\right ) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{\left (8 a^2+12 a b+5 b^2\right ) \tan (e+f x) \sec (e+f x)}{16 f}+\frac{b (8 a+5 b) \tan (e+f x) \sec ^3(e+f x)}{24 f}+\frac{b \tan (e+f x) \sec ^5(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{6 f} \]
Antiderivative was successfully verified.
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Rule 4147
Rule 413
Rule 385
Rule 199
Rule 206
Rubi steps
\begin{align*} \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-a x^2\right )^2}{\left (1-x^2\right )^4} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f}-\frac{\operatorname{Subst}\left (\int \frac{-(a+b) (6 a+5 b)+3 a (2 a+b) x^2}{\left (1-x^2\right )^3} \, dx,x,\sin (e+f x)\right )}{6 f}\\ &=\frac{b (8 a+5 b) \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac{b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f}+\frac{\left (8 a^2+12 a b+5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{8 f}\\ &=\frac{\left (8 a^2+12 a b+5 b^2\right ) \sec (e+f x) \tan (e+f x)}{16 f}+\frac{b (8 a+5 b) \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac{b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f}+\frac{\left (8 a^2+12 a b+5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (e+f x)\right )}{16 f}\\ &=\frac{\left (8 a^2+12 a b+5 b^2\right ) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{\left (8 a^2+12 a b+5 b^2\right ) \sec (e+f x) \tan (e+f x)}{16 f}+\frac{b (8 a+5 b) \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac{b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f}\\ \end{align*}
Mathematica [A] time = 0.397695, size = 94, normalized size = 0.73 \[ \frac{3 \left (8 a^2+12 a b+5 b^2\right ) \tanh ^{-1}(\sin (e+f x))+\tan (e+f x) \sec (e+f x) \left (3 \left (8 a^2+12 a b+5 b^2\right )+2 b (12 a+5 b) \sec ^2(e+f x)+8 b^2 \sec ^4(e+f x)\right )}{48 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 191, normalized size = 1.5 \begin{align*}{\frac{{a}^{2}\tan \left ( fx+e \right ) \sec \left ( fx+e \right ) }{2\,f}}+{\frac{{a}^{2}\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{2\,f}}+{\frac{ab\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{3}}{2\,f}}+{\frac{3\,ab\tan \left ( fx+e \right ) \sec \left ( fx+e \right ) }{4\,f}}+{\frac{3\,ab\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{4\,f}}+{\frac{{b}^{2}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{5}}{6\,f}}+{\frac{5\,{b}^{2}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{3}}{24\,f}}+{\frac{5\,{b}^{2}\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{16\,f}}+{\frac{5\,{b}^{2}\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{16\,f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0066, size = 224, normalized size = 1.74 \begin{align*} \frac{3 \,{\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \,{\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (f x + e\right ) - 1\right ) - \frac{2 \,{\left (3 \,{\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \sin \left (f x + e\right )^{5} - 8 \,{\left (6 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \sin \left (f x + e\right )^{3} + 3 \,{\left (8 \, a^{2} + 20 \, a b + 11 \, b^{2}\right )} \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1}}{96 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.528086, size = 355, normalized size = 2.75 \begin{align*} \frac{3 \,{\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \,{\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \,{\left (3 \,{\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \,{\left (12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, b^{2}\right )} \sin \left (f x + e\right )}{96 \, f \cos \left (f x + e\right )^{6}} \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 \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \sec ^{3}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27548, size = 263, normalized size = 2.04 \begin{align*} \frac{3 \,{\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \,{\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - \frac{2 \,{\left (24 \, a^{2} \sin \left (f x + e\right )^{5} + 36 \, a b \sin \left (f x + e\right )^{5} + 15 \, b^{2} \sin \left (f x + e\right )^{5} - 48 \, a^{2} \sin \left (f x + e\right )^{3} - 96 \, a b \sin \left (f x + e\right )^{3} - 40 \, b^{2} \sin \left (f x + e\right )^{3} + 24 \, a^{2} \sin \left (f x + e\right ) + 60 \, a b \sin \left (f x + e\right ) + 33 \, b^{2} \sin \left (f x + e\right )\right )}}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{3}}}{96 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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