Optimal. Leaf size=59 \[ \frac{\tanh ^{-1}(\sin (c+d x))}{b d}-\frac{\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b d} \]
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Rubi [A] time = 0.0811648, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3676, 391, 206, 208} \[ \frac{\tanh ^{-1}(\sin (c+d x))}{b d}-\frac{\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b d} \]
Antiderivative was successfully verified.
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Rule 3676
Rule 391
Rule 206
Rule 208
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{a+b \tan ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a-(a-b) x^2\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{b d}-\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\sin (c+d x)\right )}{b d}\\ &=\frac{\tanh ^{-1}(\sin (c+d x))}{b d}-\frac{\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b d}\\ \end{align*}
Mathematica [A] time = 0.103528, size = 53, normalized size = 0.9 \[ \frac{\tanh ^{-1}(\sin (c+d x))-\frac{\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a}}}{b d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.077, size = 111, normalized size = 1.9 \begin{align*} -{\frac{a}{db}{\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 ) }}}}+{\frac{1}{d}{\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 ) }}}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) +1 \right ) }{2\,db}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{2\,db}} \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 = 1.69346, size = 413, normalized size = 7. \begin{align*} \left [\frac{\sqrt{\frac{a - b}{a}} \log \left (-\frac{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + 2 \, a \sqrt{\frac{a - b}{a}} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right ) + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, b d}, \frac{2 \, \sqrt{-\frac{a - b}{a}} \arctan \left (\sqrt{-\frac{a - b}{a}} \sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, b d}\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 )}}{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 = 2.22181, size = 119, normalized size = 2.02 \begin{align*} -\frac{\frac{2 \,{\left (a - b\right )} \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} b} - \frac{\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{b} + \frac{\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{b}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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