Optimal. Leaf size=94 \[ -\frac{\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left (a^2+b^2\right )}+\frac{a^2 b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{a x \left (a^2-b^2\right )}{2 \left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.163218, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3516, 1647, 801, 635, 203, 260} \[ -\frac{\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left (a^2+b^2\right )}+\frac{a^2 b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{a x \left (a^2-b^2\right )}{2 \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3516
Rule 1647
Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{\sin ^2(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{x^2}{(a+x) \left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{a^2 b^2}{a^2+b^2}+\frac{a b^2 x}{a^2+b^2}}{(a+x) \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=-\frac{\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac{\operatorname{Subst}\left (\int \left (-\frac{2 a^2 b^2}{\left (a^2+b^2\right )^2 (a+x)}-\frac{a b^2 \left (a^2-b^2-2 a x\right )}{\left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=\frac{a^2 b \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac{(a b) \operatorname{Subst}\left (\int \frac{a^2-b^2-2 a x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=\frac{a^2 b \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac{\left (a^2 b\right ) \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac{\left (a b \left (a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=\frac{a \left (a^2-b^2\right ) x}{2 \left (a^2+b^2\right )^2}+\frac{a^2 b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac{a^2 b \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 0.722595, size = 170, normalized size = 1.81 \[ -\frac{2 b^2 \left (a^2+b^2\right ) \cos ^2(c+d x)+2 a b \left (a^2+b^2\right ) \tan ^{-1}(\tan (c+d x))+a \left (b \left (a^2+b^2\right ) \sin (2 (c+d x))+2 a \left (-2 b^2 \log (a+b \tan (c+d x))+\left (a \sqrt{-b^2}+b^2\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )+\left (b^2-a \sqrt{-b^2}\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )\right )\right )}{4 b d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 238, normalized size = 2.5 \begin{align*} -{\frac{{a}^{3}\tan \left ( dx+c \right ) }{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{a\tan \left ( dx+c \right ){b}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{b{a}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{{b}^{3}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{2}b}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{b{a}^{2}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5721, size = 194, normalized size = 2.06 \begin{align*} \frac{\frac{2 \, a^{2} b \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{a^{2} b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{{\left (a^{3} - a b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{a \tan \left (d x + c\right ) + b}{{\left (a^{2} + b^{2}\right )} \tan \left (d x + c\right )^{2} + a^{2} + b^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12276, size = 278, normalized size = 2.96 \begin{align*} \frac{a^{2} 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 ) +{\left (a^{3} - a b^{2}\right )} d x -{\left (a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2} -{\left (a^{3} + a b^{2}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14829, size = 248, normalized size = 2.64 \begin{align*} \frac{\frac{2 \, a^{2} b^{2} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac{a^{2} b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{{\left (a^{3} - a b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{a^{2} b \tan \left (d x + c\right )^{2} - a^{3} \tan \left (d x + c\right ) - a b^{2} \tan \left (d x + c\right ) - b^{3}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}{\left (\tan \left (d x + c\right )^{2} + 1\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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