Optimal. Leaf size=80 \[ -\frac{b^2 \log (a \cos (c+d x)+b)}{a d \left (a^2-b^2\right )}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)} \]
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Rubi [A] time = 0.236899, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4397, 2837, 12, 1629} \[ -\frac{b^2 \log (a \cos (c+d x)+b)}{a d \left (a^2-b^2\right )}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 2837
Rule 12
Rule 1629
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx &=\int \frac{\cos (c+d x) \cot (c+d x)}{b+a \cos (c+d x)} \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{x^2}{a^2 (b+x) \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{(b+x) \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{a d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a}{2 (a+b) (a-x)}-\frac{a}{2 (a-b) (a+x)}+\frac{b^2}{(a-b) (a+b) (b+x)}\right ) \, dx,x,a \cos (c+d x)\right )}{a d}\\ &=\frac{\log (1-\cos (c+d x))}{2 (a+b) d}+\frac{\log (1+\cos (c+d x))}{2 (a-b) d}-\frac{b^2 \log (b+a \cos (c+d x))}{a \left (a^2-b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 0.100196, size = 70, normalized size = 0.88 \[ \frac{b^2 (-\log (a \cos (c+d x)+b))+a (a-b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+a (a+b) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a d (a-b) (a+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.101, size = 80, normalized size = 1. \begin{align*}{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{d \left ( 2\,a-2\,b \right ) }}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{d \left ( 2\,a+2\,b \right ) }}-{\frac{{b}^{2}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) \left ( a-b \right ) a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56268, size = 138, normalized size = 1.72 \begin{align*} -\frac{\frac{b^{2} \log \left (a + b - \frac{{\left (a - b\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{3} - a b^{2}} - \frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a + b} + \frac{\log \left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.564225, size = 190, normalized size = 2.38 \begin{align*} -\frac{2 \, b^{2} \log \left (a \cos \left (d x + c\right ) + b\right ) -{\left (a^{2} + a b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (a^{2} - a b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \,{\left (a^{3} - a b^{2}\right )} d} \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 \sin{\left (c + d x \right )} + b \tan{\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 = 1.21627, size = 180, normalized size = 2.25 \begin{align*} -\frac{\frac{2 \, b^{2} \log \left ({\left | -a - b - \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{3} - a b^{2}} - \frac{\log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a + b} + \frac{2 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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