Optimal. Leaf size=42 \[ -\frac{a^2 \cot (c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d} \]
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Rubi [A] time = 0.0474087, antiderivative size = 42, 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, 43} \[ -\frac{a^2 \cot (c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3516
Rule 43
Rubi steps
\begin{align*} \int \csc ^2(c+d x) (a+b \tan (c+d x))^2 \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{(a+x)^2}{x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (1+\frac{a^2}{x^2}+\frac{2 a}{x}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [B] time = 0.544128, size = 91, normalized size = 2.17 \[ -\frac{\cos (c+d x) (a+b \tan (c+d x))^2 \left (a \cos (c+d x) (a \cot (c+d x)+2 b (\log (\cos (c+d x))-\log (\sin (c+d x))))-b^2 \sin (c+d x)\right )}{d (a \cos (c+d x)+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 43, normalized size = 1. \begin{align*} -{\frac{{a}^{2}\cot \left ( dx+c \right ) }{d}}+2\,{\frac{ab\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{2}\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10725, size = 53, normalized size = 1.26 \begin{align*} \frac{2 \, a b \log \left (\tan \left (d x + c\right )\right ) + b^{2} \tan \left (d x + c\right ) - \frac{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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Fricas [B] time = 2.28245, size = 246, normalized size = 5.86 \begin{align*} -\frac{a b \cos \left (d x + c\right ) \log \left (\cos \left (d x + c\right )^{2}\right ) \sin \left (d x + c\right ) - a b \cos \left (d x + c\right ) \log \left (-\frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4}\right ) \sin \left (d x + c\right ) +{\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} - b^{2}}{d \cos \left (d x + c\right ) \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 \left (a + b \tan{\left (c + d x \right )}\right )^{2} \csc ^{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 = 1.53881, size = 69, normalized size = 1.64 \begin{align*} \frac{2 \, a b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + b^{2} \tan \left (d x + c\right ) - \frac{2 \, a b \tan \left (d x + c\right ) + 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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