Optimal. Leaf size=49 \[ -\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))}{2 d}+\frac{a x}{2}-\frac{b \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0844788, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {819, 635, 203, 260} \[ -\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))}{2 d}+\frac{a x}{2}-\frac{b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 819
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \sin ^2(c+d x) (a+b \tan (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 (a+b x)}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))}{2 d}+\frac{\operatorname{Subst}\left (\int \frac{a+2 b x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac{\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))}{2 d}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{a x}{2}-\frac{b \log (\cos (c+d x))}{d}-\frac{\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0480753, size = 56, normalized size = 1.14 \[ \frac{a (c+d x)}{2 d}-\frac{a \sin (2 (c+d x))}{4 d}-\frac{b \left (\log (\cos (c+d x))-\frac{1}{2} \cos ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 58, normalized size = 1.2 \begin{align*} -{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{b\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-{\frac{a\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{2\,d}}+{\frac{ax}{2}}+{\frac{ac}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.51148, size = 70, normalized size = 1.43 \begin{align*} \frac{{\left (d x + c\right )} a + b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - \frac{a \tan \left (d x + c\right ) - b}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25526, size = 120, normalized size = 2.45 \begin{align*} \frac{a d x + b \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, b \log \left (-\cos \left (d x + c\right )\right )}{2 \, 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 \left (a + b \tan{\left (c + d x \right )}\right ) \sin ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.43679, size = 558, normalized size = 11.39 \begin{align*} \frac{2 \, a d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, b \log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 2 \, a d x \tan \left (d x\right )^{2} + 2 \, a d x \tan \left (c\right )^{2} + b \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, b \log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) \tan \left (d x\right )^{2} + 2 \, a \tan \left (d x\right )^{2} \tan \left (c\right ) - 2 \, b \log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) \tan \left (c\right )^{2} + 2 \, a \tan \left (d x\right ) \tan \left (c\right )^{2} + 2 \, a d x - b \tan \left (d x\right )^{2} - 4 \, b \tan \left (d x\right ) \tan \left (c\right ) - b \tan \left (c\right )^{2} - 2 \, b \log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) - 2 \, a \tan \left (d x\right ) - 2 \, a \tan \left (c\right ) + b}{4 \,{\left (d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + d \tan \left (d x\right )^{2} + d \tan \left (c\right )^{2} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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