Optimal. Leaf size=104 \[ -\frac{\left (3 a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^{5/2} d}+\frac{(a-b)^2 \tan (c+d x)}{2 a b^2 d \left (a+b \tan ^2(c+d x)\right )}+\frac{\tan (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.135615, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3675, 390, 385, 205} \[ -\frac{\left (3 a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^{5/2} d}+\frac{(a-b)^2 \tan (c+d x)}{2 a b^2 d \left (a+b \tan ^2(c+d x)\right )}+\frac{\tan (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 390
Rule 385
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec ^6(c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{\left (a+b x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{b^2}-\frac{a^2-b^2+2 (a-b) b x^2}{b^2 \left (a+b x^2\right )^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\tan (c+d x)}{b^2 d}-\frac{\operatorname{Subst}\left (\int \frac{a^2-b^2+2 (a-b) b x^2}{\left (a+b x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{b^2 d}\\ &=\frac{\tan (c+d x)}{b^2 d}+\frac{(a-b)^2 \tan (c+d x)}{2 a b^2 d \left (a+b \tan ^2(c+d x)\right )}-\frac{((a-b) (3 a+b)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tan (c+d x)\right )}{2 a b^2 d}\\ &=-\frac{(a-b) (3 a+b) \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^{5/2} d}+\frac{\tan (c+d x)}{b^2 d}+\frac{(a-b)^2 \tan (c+d x)}{2 a b^2 d \left (a+b \tan ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.620238, size = 104, normalized size = 1. \[ \frac{-\frac{(3 a+b) (a-b) \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\sqrt{b} (a-b)^2 \sin (2 (c+d x))}{a ((a-b) \cos (2 (c+d x))+a+b)}+2 \sqrt{b} \tan (c+d x)}{2 b^{5/2} d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.085, size = 181, normalized size = 1.7 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{{b}^{2}d}}+{\frac{a\tan \left ( dx+c \right ) }{2\,{b}^{2}d \left ( a+b \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{\tan \left ( dx+c \right ) }{db \left ( a+b \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}+{\frac{\tan \left ( dx+c \right ) }{2\,ad \left ( a+b \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{3\,a}{2\,{b}^{2}d}\arctan \left ({b\tan \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{1}{db}\arctan \left ({b\tan \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{1}{2\,ad}\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 [B] time = 1.72883, size = 1089, normalized size = 10.47 \begin{align*} \left [\frac{{\left ({\left (3 \, a^{3} - 5 \, a^{2} b + a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (3 \, a^{2} b - 2 \, a b^{2} - b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt{-a b} \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 (2 \, a^{2} b^{2} +{\left (3 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{8 \,{\left (a^{2} b^{4} d \cos \left (d x + c\right ) +{\left (a^{3} b^{3} - a^{2} b^{4}\right )} d \cos \left (d x + c\right )^{3}\right )}}, \frac{{\left ({\left (3 \, a^{3} - 5 \, a^{2} b + a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (3 \, a^{2} b - 2 \, a b^{2} - b^{3}\right )} \cos \left (d x + c\right )\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 ) + 2 \,{\left (2 \, a^{2} b^{2} +{\left (3 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \,{\left (a^{2} b^{4} d \cos \left (d x + c\right ) +{\left (a^{3} b^{3} - a^{2} b^{4}\right )} d \cos \left (d x + c\right )^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.73124, size = 173, normalized size = 1.66 \begin{align*} \frac{\frac{2 \, \tan \left (d x + c\right )}{b^{2}} - \frac{{\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 (3 \, a^{2} - 2 \, a b - b^{2}\right )}}{\sqrt{a b} a b^{2}} + \frac{a^{2} \tan \left (d x + c\right ) - 2 \, a b \tan \left (d x + c\right ) + b^{2} \tan \left (d x + c\right )}{{\left (b \tan \left (d x + c\right )^{2} + a\right )} a b^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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