Optimal. Leaf size=126 \[ \frac{\left (2 a^2-3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{a \tan (c+d x)}{b^2 d}-\frac{2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a b^3 d}+\frac{x}{a}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d} \]
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Rubi [A] time = 0.342726, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3898, 2893, 3057, 2659, 208, 3770} \[ \frac{\left (2 a^2-3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{a \tan (c+d x)}{b^2 d}-\frac{2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a b^3 d}+\frac{x}{a}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d} \]
Antiderivative was successfully verified.
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Rule 3898
Rule 2893
Rule 3057
Rule 2659
Rule 208
Rule 3770
Rubi steps
\begin{align*} \int \frac{\tan ^4(c+d x)}{a+b \sec (c+d x)} \, dx &=\int \frac{\sin (c+d x) \tan ^3(c+d x)}{b+a \cos (c+d x)} \, dx\\ &=-\frac{a \tan (c+d x)}{b^2 d}+\frac{\sec (c+d x) \tan (c+d x)}{2 b d}-\frac{\int \frac{\left (-2 a^2+3 b^2-a b \cos (c+d x)-2 b^2 \cos ^2(c+d x)\right ) \sec (c+d x)}{b+a \cos (c+d x)} \, dx}{2 b^2}\\ &=\frac{x}{a}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\sec (c+d x) \tan (c+d x)}{2 b d}-\frac{\left (a^2-b^2\right )^2 \int \frac{1}{b+a \cos (c+d x)} \, dx}{a b^3}-\frac{\left (-2 a^2+3 b^2\right ) \int \sec (c+d x) \, dx}{2 b^3}\\ &=\frac{x}{a}+\frac{\left (2 a^2-3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\sec (c+d x) \tan (c+d x)}{2 b d}-\frac{\left (2 \left (a^2-b^2\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+(-a+b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a b^3 d}\\ &=\frac{x}{a}+\frac{\left (2 a^2-3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{2 (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a b^3 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\sec (c+d x) \tan (c+d x)}{2 b d}\\ \end{align*}
Mathematica [B] time = 2.14776, size = 287, normalized size = 2.28 \[ \frac{\sec (c+d x) (a \cos (c+d x)+b) \left (\frac{8 \left (a^2-b^2\right )^{3/2} \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a b^3}-\frac{4 a^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{b^3}+\frac{4 a^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{b^3}-\frac{4 a \tan (c+d x)}{b^2}+\frac{4 c}{a}+\frac{4 d x}{a}+\frac{1}{b \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{1}{b \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{6 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{b}-\frac{6 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{b}\right )}{4 d (a+b \sec (c+d x))} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.072, size = 374, normalized size = 3. \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{ad}}-2\,{\frac{{a}^{3}}{d{b}^{3}\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+4\,{\frac{a}{db\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-2\,{\frac{b}{ad\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-{\frac{1}{2\,db} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{1}{2\,db} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{a}{d{b}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{{a}^{2}}{d{b}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{3}{2\,db}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{1}{2\,db} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{1}{2\,db} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{a}{d{b}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{{a}^{2}}{d{b}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+{\frac{3}{2\,db}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \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 [A] time = 1.68004, size = 1048, normalized size = 8.32 \begin{align*} \left [\frac{4 \, b^{3} d x \cos \left (d x + c\right )^{2} - 2 \,{\left (a^{2} - b^{2}\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{2} \log \left (\frac{2 \, a b \cos \left (d x + c\right ) -{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) +{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (2 \, a^{2} b \cos \left (d x + c\right ) - a b^{2}\right )} \sin \left (d x + c\right )}{4 \, a b^{3} d \cos \left (d x + c\right )^{2}}, \frac{4 \, b^{3} d x \cos \left (d x + c\right )^{2} - 4 \,{\left (a^{2} - b^{2}\right )} \sqrt{-a^{2} + b^{2}} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) \cos \left (d x + c\right )^{2} +{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (2 \, a^{2} b \cos \left (d x + c\right ) - a b^{2}\right )} \sin \left (d x + c\right )}{4 \, a b^{3} d \cos \left (d x + c\right )^{2}}\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{\tan ^{4}{\left (c + d x \right )}}{a + b \sec{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.50431, size = 324, normalized size = 2.57 \begin{align*} \frac{\frac{2 \,{\left (d x + c\right )}}{a} + \frac{{\left (2 \, a^{2} - 3 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{b^{3}} - \frac{{\left (2 \, a^{2} - 3 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{b^{3}} - \frac{4 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{-a^{2} + b^{2}}}\right )\right )}}{\sqrt{-a^{2} + b^{2}} a b^{3}} + \frac{2 \,{\left (2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2} b^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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