Optimal. Leaf size=60 \[ \frac{\sin (c+d x)}{d (a-b)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a-b)^{3/2}} \]
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Rubi [A] time = 0.0797293, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3676, 388, 208} \[ \frac{\sin (c+d x)}{d (a-b)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a-b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3676
Rule 388
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{a+b \tan ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{a-(a-b) x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\sin (c+d x)}{(a-b) d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\sin (c+d x)\right )}{(a-b) d}\\ &=-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} (a-b)^{3/2} d}+\frac{\sin (c+d x)}{(a-b) d}\\ \end{align*}
Mathematica [A] time = 0.0937933, size = 60, normalized size = 1. \[ \frac{\sin (c+d x)}{d (a-b)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a-b)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 61, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{\sin \left ( dx+c \right ) }{a-b}}-{\frac{b}{a-b}{\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 = 1.53467, size = 414, normalized size = 6.9 \begin{align*} \left [-\frac{\sqrt{a^{2} - a b} 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 )}{2 \,{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} d}, \frac{\sqrt{-a^{2} + a b} 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 )}{{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} 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{\cos{\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 = 1.70302, size = 99, normalized size = 1.65 \begin{align*} -\frac{\frac{b \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}{\left (a - b\right )}} - \frac{\sin \left (d x + c\right )}{a - b}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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