Optimal. Leaf size=86 \[ \frac{\left (2 a^2 B+4 a A b+b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 A x+\frac{b (3 a B+2 A b) \tan (c+d x)}{2 d}+\frac{b B \tan (c+d x) (a+b \sec (c+d x))}{2 d} \]
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Rubi [A] time = 0.0807489, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3918, 3770, 3767, 8} \[ \frac{\left (2 a^2 B+4 a A b+b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 A x+\frac{b (3 a B+2 A b) \tan (c+d x)}{2 d}+\frac{b B \tan (c+d x) (a+b \sec (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 3918
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\frac{b B (a+b \sec (c+d x)) \tan (c+d x)}{2 d}+\frac{1}{2} \int \left (2 a^2 A+\left (4 a A b+2 a^2 B+b^2 B\right ) \sec (c+d x)+b (2 A b+3 a B) \sec ^2(c+d x)\right ) \, dx\\ &=a^2 A x+\frac{b B (a+b \sec (c+d x)) \tan (c+d x)}{2 d}+\frac{1}{2} (b (2 A b+3 a B)) \int \sec ^2(c+d x) \, dx+\frac{1}{2} \left (4 a A b+2 a^2 B+b^2 B\right ) \int \sec (c+d x) \, dx\\ &=a^2 A x+\frac{\left (4 a A b+2 a^2 B+b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b B (a+b \sec (c+d x)) \tan (c+d x)}{2 d}-\frac{(b (2 A b+3 a B)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{2 d}\\ &=a^2 A x+\frac{\left (4 a A b+2 a^2 B+b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b (2 A b+3 a B) \tan (c+d x)}{2 d}+\frac{b B (a+b \sec (c+d x)) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.262533, size = 67, normalized size = 0.78 \[ \frac{\left (2 a^2 B+4 a A b+b^2 B\right ) \tanh ^{-1}(\sin (c+d x))+2 a^2 A d x+b \tan (c+d x) (4 a B+2 A b+b B \sec (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 133, normalized size = 1.6 \begin{align*}{a}^{2}Ax+{\frac{A{a}^{2}c}{d}}+{\frac{B{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{Aab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{Bab\tan \left ( dx+c \right ) }{d}}+{\frac{A{b}^{2}\tan \left ( dx+c \right ) }{d}}+{\frac{B{b}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{B{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 = 0.965132, size = 170, normalized size = 1.98 \begin{align*} \frac{4 \,{\left (d x + c\right )} A a^{2} - B 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 )} + 4 \, B a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 8 \, A a b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 8 \, B a b \tan \left (d x + c\right ) + 4 \, A b^{2} \tan \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 = 0.506255, size = 335, normalized size = 3.9 \begin{align*} \frac{4 \, A a^{2} d x \cos \left (d x + c\right )^{2} +{\left (2 \, B a^{2} + 4 \, A a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, B a^{2} + 4 \, A a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (B b^{2} + 2 \,{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )\right )} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B \sec{\left (c + d x \right )}\right ) \left (a + b \sec{\left (c + d x \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20235, size = 259, normalized size = 3.01 \begin{align*} \frac{2 \,{\left (d x + c\right )} A a^{2} +{\left (2 \, B a^{2} + 4 \, A a b + B b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (2 \, B a^{2} + 4 \, A a b + B b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (4 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 4 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - B b^{2} \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 )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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