Optimal. Leaf size=103 \[ \frac{d (2 b c-a d) \tanh ^{-1}(\sin (e+f x))}{b^2 f}+\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{a+b}}\right )}{b^2 f \sqrt{a-b} \sqrt{a+b}}+\frac{d^2 \tan (e+f x)}{b f} \]
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Rubi [A] time = 0.297448, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {3988, 2952, 2659, 208, 3770, 3767, 8} \[ \frac{d (2 b c-a d) \tanh ^{-1}(\sin (e+f x))}{b^2 f}+\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{a+b}}\right )}{b^2 f \sqrt{a-b} \sqrt{a+b}}+\frac{d^2 \tan (e+f x)}{b f} \]
Antiderivative was successfully verified.
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Rule 3988
Rule 2952
Rule 2659
Rule 208
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (c+d \sec (e+f x))^2}{a+b \sec (e+f x)} \, dx &=\int \frac{(d+c \cos (e+f x))^2 \sec ^2(e+f x)}{b+a \cos (e+f x)} \, dx\\ &=\int \left (\frac{(b c-a d)^2}{b^2 (b+a \cos (e+f x))}+\frac{d (2 b c-a d) \sec (e+f x)}{b^2}+\frac{d^2 \sec ^2(e+f x)}{b}\right ) \, dx\\ &=\frac{d^2 \int \sec ^2(e+f x) \, dx}{b}+\frac{(b c-a d)^2 \int \frac{1}{b+a \cos (e+f x)} \, dx}{b^2}+\frac{(d (2 b c-a d)) \int \sec (e+f x) \, dx}{b^2}\\ &=\frac{d (2 b c-a d) \tanh ^{-1}(\sin (e+f x))}{b^2 f}-\frac{d^2 \operatorname{Subst}(\int 1 \, dx,x,-\tan (e+f x))}{b f}+\frac{\left (2 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+(-a+b) x^2} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{b^2 f}\\ &=\frac{d (2 b c-a d) \tanh ^{-1}(\sin (e+f x))}{b^2 f}+\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b^2 \sqrt{a+b} f}+\frac{d^2 \tan (e+f x)}{b f}\\ \end{align*}
Mathematica [A] time = 0.804488, size = 135, normalized size = 1.31 \[ \frac{d \left (b d \tan (e+f x)-(2 b c-a d) \left (\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )\right )-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}}{b^2 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.065, size = 288, normalized size = 2.8 \begin{align*} 2\,{\frac{{a}^{2}{d}^{2}}{f{b}^{2}\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,fx+e/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-4\,{\frac{acd}{fb\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,fx+e/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+2\,{\frac{{c}^{2}}{f\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,fx+e/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-{\frac{{d}^{2}}{fb} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-1}}-{\frac{a{d}^{2}}{f{b}^{2}}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) }+2\,{\frac{d\ln \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) c}{fb}}-{\frac{{d}^{2}}{fb} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-1}}+{\frac{a{d}^{2}}{f{b}^{2}}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) }-2\,{\frac{d\ln \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) c}{fb}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 12.7316, size = 1139, normalized size = 11.06 \begin{align*} \left [\frac{2 \,{\left (a^{2} b - b^{3}\right )} d^{2} \sin \left (f x + e\right ) +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a^{2} - b^{2}} \cos \left (f x + e\right ) \log \left (\frac{2 \, a b \cos \left (f x + e\right ) -{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (f x + e\right )^{2} + 2 \, a b \cos \left (f x + e\right ) + b^{2}}\right ) +{\left (2 \,{\left (a^{2} b - b^{3}\right )} c d -{\left (a^{3} - a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right ) \log \left (\sin \left (f x + e\right ) + 1\right ) -{\left (2 \,{\left (a^{2} b - b^{3}\right )} c d -{\left (a^{3} - a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right ) \log \left (-\sin \left (f x + e\right ) + 1\right )}{2 \,{\left (a^{2} b^{2} - b^{4}\right )} f \cos \left (f x + e\right )}, \frac{2 \,{\left (a^{2} b - b^{3}\right )} d^{2} \sin \left (f x + e\right ) + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-a^{2} + b^{2}} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cos \left (f x + e\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right ) +{\left (2 \,{\left (a^{2} b - b^{3}\right )} c d -{\left (a^{3} - a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right ) \log \left (\sin \left (f x + e\right ) + 1\right ) -{\left (2 \,{\left (a^{2} b - b^{3}\right )} c d -{\left (a^{3} - a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right ) \log \left (-\sin \left (f x + e\right ) + 1\right )}{2 \,{\left (a^{2} b^{2} - b^{4}\right )} f \cos \left (f x + e\right )}\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{\left (c + d \sec{\left (e + f x \right )}\right )^{2} \sec{\left (e + f x \right )}}{a + b \sec{\left (e + f x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.41525, size = 273, normalized size = 2.65 \begin{align*} -\frac{\frac{2 \, d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )} b} - \frac{{\left (2 \, b c d - a d^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{b^{2}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{b^{2}} + \frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left (\pi \left \lfloor \frac{f x + e}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{\sqrt{-a^{2} + b^{2}}}\right )\right )}}{\sqrt{-a^{2} + b^{2}} b^{2}}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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