Optimal. Leaf size=79 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d \sqrt{a-b}}+\frac{\sin (c+d x)}{2 a d \left (a-(a-b) \sin ^2(c+d x)\right )} \]
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Rubi [A] time = 0.0795542, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3676, 199, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d \sqrt{a-b}}+\frac{\sin (c+d x)}{2 a d \left (a-(a-b) \sin ^2(c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 3676
Rule 199
Rule 208
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a-(a-b) x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\sin (c+d x)}{2 a d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\sin (c+d x)\right )}{2 a d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{a-b} d}+\frac{\sin (c+d x)}{2 a d \left (a-(a-b) \sin ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.237823, size = 75, normalized size = 0.95 \[ \frac{\frac{\sqrt{a} \sin (c+d x)}{(b-a) \sin ^2(c+d x)+a}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a-b}}}{2 a^{3/2} d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 80, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -{\frac{\sin \left ( dx+c \right ) }{2\,a \left ( a \left ( \sin \left ( dx+c \right ) \right ) ^{2}-b \left ( \sin \left ( dx+c \right ) \right ) ^{2}-a \right ) }}+{\frac{1}{2\,a}{\it Artanh} \left ({ \left ( a-b \right ) \sin \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \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 = 1.54069, size = 598, normalized size = 7.57 \begin{align*} \left [\frac{{\left ({\left (a - b\right )} \cos \left (d x + c\right )^{2} + b\right )} \sqrt{a^{2} - a b} \log \left (-\frac{{\left (a - b\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt{a^{2} - a b} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right ) + 2 \,{\left (a^{2} - a b\right )} \sin \left (d x + c\right )}{4 \,{\left ({\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} d \cos \left (d x + c\right )^{2} +{\left (a^{3} b - a^{2} b^{2}\right )} d\right )}}, -\frac{{\left ({\left (a - b\right )} \cos \left (d x + c\right )^{2} + b\right )} \sqrt{-a^{2} + a b} \arctan \left (\frac{\sqrt{-a^{2} + a b} \sin \left (d x + c\right )}{a}\right ) -{\left (a^{2} - a b\right )} \sin \left (d x + c\right )}{2 \,{\left ({\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} d \cos \left (d x + c\right )^{2} +{\left (a^{3} b - a^{2} b^{2}\right )} d\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{\sec ^{3}{\left (c + d x \right )}}{\left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.73869, size = 123, normalized size = 1.56 \begin{align*} \frac{\frac{\arctan \left (-\frac{a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt{-a^{2} + a b}}\right )}{\sqrt{-a^{2} + a b} a} - \frac{\sin \left (d x + c\right )}{{\left (a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right )^{2} - a\right )} a}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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