Optimal. Leaf size=67 \[ \frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.0936005, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3090, 3770, 2606, 8, 2611} \[ \frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3090
Rule 3770
Rule 2606
Rule 8
Rule 2611
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx &=\int \left (a^2 \sec (c+d x)+2 a b \sec (c+d x) \tan (c+d x)+b^2 \sec (c+d x) \tan ^2(c+d x)\right ) \, dx\\ &=a^2 \int \sec (c+d x) \, dx+(2 a b) \int \sec (c+d x) \tan (c+d x) \, dx+b^2 \int \sec (c+d x) \tan ^2(c+d x) \, dx\\ &=\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac{1}{2} b^2 \int \sec (c+d x) \, dx+\frac{(2 a b) \operatorname{Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0469392, size = 67, normalized size = 1. \[ \frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.102, size = 98, normalized size = 1.5 \begin{align*}{\frac{{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{ab}{d\cos \left ( dx+c \right ) }}+{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{2}\sin \left ( dx+c \right ) }{2\,d}}-{\frac{{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19866, size = 120, normalized size = 1.79 \begin{align*} -\frac{b^{2}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2 \, a^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac{8 \, a b}{\cos \left (d x + c\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.498405, size = 234, normalized size = 3.49 \begin{align*} \frac{{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8 \, a b \cos \left (d x + c\right ) + 2 \, b^{2} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18976, size = 165, normalized size = 2.46 \begin{align*} \frac{{\left (2 \, a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (2 \, a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 4 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4 \, a b\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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