Optimal. Leaf size=83 \[ -\frac{b}{d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))}-\frac{a \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{3/2}} \]
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Rubi [A] time = 0.0665358, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3155, 3074, 206} \[ -\frac{b}{d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))}-\frac{a \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3155
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx &=-\frac{b}{\left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))}+\frac{a \int \frac{1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{a^2+b^2}\\ &=-\frac{b}{\left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{a \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac{b}{\left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.213311, size = 79, normalized size = 0.95 \[ \frac{\frac{2 a \tanh ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )-b}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac{b}{\left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))}}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.167, size = 118, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( -2\,{\frac{1}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\,\tan \left ( 1/2\,dx+c/2 \right ) b-a} \left ( -{\frac{{b}^{2}\tan \left ( 1/2\,dx+c/2 \right ) }{ \left ({a}^{2}+{b}^{2} \right ) a}}-{\frac{b}{{a}^{2}+{b}^{2}}} \right ) }+2\,{\frac{a}{ \left ({a}^{2}+{b}^{2} \right ) ^{3/2}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \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 [B] time = 0.507247, size = 502, normalized size = 6.05 \begin{align*} -\frac{2 \, a^{2} b + 2 \, b^{3} -{\left (a^{2} \cos \left (d x + c\right ) + a b \sin \left (d x + c\right )\right )} \sqrt{a^{2} + b^{2}} \log \left (-\frac{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right )}{2 \,{\left ({\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cos \left (d x + c\right ) +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \sin \left (d x + c\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 = 1.26548, size = 186, normalized size = 2.24 \begin{align*} -\frac{\frac{a \log \left (\frac{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac{3}{2}}} - \frac{2 \,{\left (b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a b\right )}}{{\left (a^{3} + a b^{2}\right )}{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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