Optimal. Leaf size=33 \[ \frac{(a-b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a+b) \]
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Rubi [A] time = 0.0385714, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3675, 385, 203} \[ \frac{(a-b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a+b) \]
Antiderivative was successfully verified.
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Rule 3675
Rule 385
Rule 203
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b x^2}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{(a-b) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{(a+b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{1}{2} (a+b) x+\frac{(a-b) \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0564466, size = 32, normalized size = 0.97 \[ \frac{2 (a+b) (c+d x)+(a-b) \sin (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 54, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ( b \left ( -{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +a \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.63994, size = 53, normalized size = 1.61 \begin{align*} \frac{{\left (d x + c\right )}{\left (a + b\right )} + \frac{{\left (a - b\right )} \tan \left (d x + c\right )}{\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 = 1.72183, size = 77, normalized size = 2.33 \begin{align*} \frac{{\left (a + b\right )} d x +{\left (a - b\right )} \cos \left (d x + c\right ) \sin \left (d x + c\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 ^{2}{\left (c + d x \right )}\right ) \cos ^{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 = 1.40934, size = 228, normalized size = 6.91 \begin{align*} \frac{a d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + b d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + a d x \tan \left (d x\right )^{2} + b d x \tan \left (d x\right )^{2} + a d x \tan \left (c\right )^{2} + b d x \tan \left (c\right )^{2} - a \tan \left (d x\right )^{2} \tan \left (c\right ) + b \tan \left (d x\right )^{2} \tan \left (c\right ) - a \tan \left (d x\right ) \tan \left (c\right )^{2} + b \tan \left (d x\right ) \tan \left (c\right )^{2} + a d x + b d x + a \tan \left (d x\right ) - b \tan \left (d x\right ) + a \tan \left (c\right ) - b \tan \left (c\right )}{2 \,{\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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