Optimal. Leaf size=77 \[ \frac{(a-b)^2 \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2} d}-\frac{(a-2 b) \tan (c+d x)}{b^2 d}+\frac{\tan ^3(c+d x)}{3 b d} \]
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Rubi [A] time = 0.091478, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3675, 390, 205} \[ \frac{(a-b)^2 \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2} d}-\frac{(a-2 b) \tan (c+d x)}{b^2 d}+\frac{\tan ^3(c+d x)}{3 b d} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 390
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec ^6(c+d x)}{a+b \tan ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{a+b x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a-2 b}{b^2}+\frac{x^2}{b}+\frac{a^2-2 a b+b^2}{b^2 \left (a+b x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{(a-2 b) \tan (c+d x)}{b^2 d}+\frac{\tan ^3(c+d x)}{3 b d}+\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tan (c+d x)\right )}{b^2 d}\\ &=\frac{(a-b)^2 \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2} d}-\frac{(a-2 b) \tan (c+d x)}{b^2 d}+\frac{\tan ^3(c+d x)}{3 b d}\\ \end{align*}
Mathematica [A] time = 0.330773, size = 74, normalized size = 0.96 \[ \frac{\frac{3 (a-b)^2 \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right )}{\sqrt{a}}+\sqrt{b} \tan (c+d x) \left (-3 a+b \sec ^2(c+d x)+5 b\right )}{3 b^{5/2} d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.07, size = 127, normalized size = 1.7 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,bd}}-{\frac{a\tan \left ( dx+c \right ) }{d{b}^{2}}}+2\,{\frac{\tan \left ( dx+c \right ) }{bd}}+{\frac{{a}^{2}}{d{b}^{2}}\arctan \left ({b\tan \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-2\,{\frac{a}{bd\sqrt{ab}}\arctan \left ({\frac{b\tan \left ( dx+c \right ) }{\sqrt{ab}}} \right ) }+{\frac{1}{d}\arctan \left ({b\tan \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.53481, size = 807, normalized size = 10.48 \begin{align*} \left [-\frac{3 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt{-a b} \cos \left (d x + c\right )^{3} \log \left (\frac{{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left ({\left (a + b\right )} \cos \left (d x + c\right )^{3} - b \cos \left (d x + c\right )\right )} \sqrt{-a b} \sin \left (d x + c\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (a b - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 4 \,{\left (a b^{2} -{\left (3 \, a^{2} b - 5 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, a b^{3} d \cos \left (d x + c\right )^{3}}, -\frac{3 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt{a b} \arctan \left (\frac{{\left ({\left (a + b\right )} \cos \left (d x + c\right )^{2} - b\right )} \sqrt{a b}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \cos \left (d x + c\right )^{3} - 2 \,{\left (a b^{2} -{\left (3 \, a^{2} b - 5 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, a b^{3} d \cos \left (d x + c\right )^{3}}\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 ^{6}{\left (c + d x \right )}}{a + b \tan ^{2}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.6116, size = 130, normalized size = 1.69 \begin{align*} \frac{\frac{3 \,{\left (\pi \left \lfloor \frac{d x + c}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (d x + c\right )}{\sqrt{a b}}\right )\right )}{\left (a^{2} - 2 \, a b + b^{2}\right )}}{\sqrt{a b} b^{2}} + \frac{b^{2} \tan \left (d x + c\right )^{3} - 3 \, a b \tan \left (d x + c\right ) + 6 \, b^{2} \tan \left (d x + c\right )}{b^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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