Optimal. Leaf size=50 \[ -\frac{b \log (\tan (c+d x))}{a^2 d}+\frac{b \log (a+b \tan (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d} \]
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Rubi [A] time = 0.0602185, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3516, 44} \[ -\frac{b \log (\tan (c+d x))}{a^2 d}+\frac{b \log (a+b \tan (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 3516
Rule 44
Rubi steps
\begin{align*} \int \frac{\csc ^2(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{1}{x^2 (a+x)} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{1}{a^2 x}+\frac{1}{a^2 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{\cot (c+d x)}{a d}-\frac{b \log (\tan (c+d x))}{a^2 d}+\frac{b \log (a+b \tan (c+d x))}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.130935, size = 47, normalized size = 0.94 \[ \frac{b (\log (a \cos (c+d x)+b \sin (c+d x))-\log (\sin (c+d x)))-a \cot (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 53, normalized size = 1.1 \begin{align*} -{\frac{1}{ad\tan \left ( dx+c \right ) }}-{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{2}d}}+{\frac{b\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{{a}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06775, size = 63, normalized size = 1.26 \begin{align*} \frac{\frac{b \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2}} - \frac{b \log \left (\tan \left (d x + c\right )\right )}{a^{2}} - \frac{1}{a \tan \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05537, size = 246, normalized size = 4.92 \begin{align*} \frac{b \log \left (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 ) \sin \left (d x + c\right ) - b \log \left (-\frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4}\right ) \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right )}{2 \, a^{2} d \sin \left (d x + c\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{\csc ^{2}{\left (c + d x \right )}}{a + 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 = 1.20827, size = 81, normalized size = 1.62 \begin{align*} \frac{\frac{b \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2}} - \frac{b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac{b \tan \left (d x + c\right ) - a}{a^{2} \tan \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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