Optimal. Leaf size=99 \[ \frac{\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+a^2 (-x)-\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{\tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b)^2}{3 d} \]
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Rubi [A] time = 0.466797, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4397, 2889, 3048, 3031, 3021, 2735, 3770} \[ \frac{\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+a^2 (-x)-\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{\tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 2889
Rule 3048
Rule 3031
Rule 3021
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx &=\int (b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan ^2(c+d x) \, dx\\ &=\int (b+a \cos (c+d x))^2 \left (1-\cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{3} \int (b+a \cos (c+d x)) \left (2 a-b \cos (c+d x)-3 a \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{a b \sec (c+d x) \tan (c+d x)}{3 d}+\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac{1}{6} \int \left (-2 \left (2 a^2-b^2\right )+6 a b \cos (c+d x)+6 a^2 \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+\frac{a b \sec (c+d x) \tan (c+d x)}{3 d}+\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac{1}{6} \int \left (6 a b+6 a^2 \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=-a^2 x+\frac{\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+\frac{a b \sec (c+d x) \tan (c+d x)}{3 d}+\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}-(a b) \int \sec (c+d x) \, dx\\ &=-a^2 x-\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+\frac{a b \sec (c+d x) \tan (c+d x)}{3 d}+\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [B] time = 1.14578, size = 201, normalized size = 2.03 \[ \frac{\sec ^3(c+d x) \left (2 \sin (c+d x) \left (\left (3 a^2-b^2\right ) \cos (2 (c+d x))+3 a^2+6 a b \cos (c+d x)+b^2\right )-9 a \cos (c+d x) \left (a (c+d x)-b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-3 a \cos (3 (c+d x)) \left (a (c+d x)-b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 109, normalized size = 1.1 \begin{align*} -{a}^{2}x+{\frac{{a}^{2}\tan \left ( dx+c \right ) }{d}}-{\frac{{a}^{2}c}{d}}+{\frac{ab \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{ab\sin \left ( dx+c \right ) }{d}}-{\frac{ab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.69924, size = 111, normalized size = 1.12 \begin{align*} \frac{2 \, b^{2} \tan \left (d x + c\right )^{3} - 6 \,{\left (d x + c - \tan \left (d x + c\right )\right )} a^{2} - 3 \, a b{\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 )}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.523383, size = 294, normalized size = 2.97 \begin{align*} -\frac{6 \, a^{2} d x \cos \left (d x + c\right )^{3} + 3 \, a b \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a b \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (3 \, a b \cos \left (d x + c\right ) +{\left (3 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \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 \sin{\left (c + d x \right )} + b \tan{\left (c + d x \right )}\right )^{2} \sec ^{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 = 2.90522, size = 213, normalized size = 2.15 \begin{align*} -\frac{3 \,{\left (d x + c\right )} a^{2} + 3 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 6 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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