Optimal. Leaf size=103 \[ \frac{\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \tan (c+d x)}{3 d}+a^2 A x+\frac{a b (2 A+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b C \tan (c+d x) \sec (c+d x)}{3 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d} \]
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Rubi [A] time = 0.139301, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4057, 4048, 3770, 3767, 8} \[ \frac{\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \tan (c+d x)}{3 d}+a^2 A x+\frac{a b (2 A+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b C \tan (c+d x) \sec (c+d x)}{3 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 4057
Rule 4048
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+b \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac{1}{3} \int (a+b \sec (c+d x)) \left (3 a A+b (3 A+2 C) \sec (c+d x)+2 a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a b C \sec (c+d x) \tan (c+d x)}{3 d}+\frac{C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac{1}{6} \int \left (6 a^2 A+6 a b (2 A+C) \sec (c+d x)+2 \left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \sec ^2(c+d x)\right ) \, dx\\ &=a^2 A x+\frac{a b C \sec (c+d x) \tan (c+d x)}{3 d}+\frac{C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+(a b (2 A+C)) \int \sec (c+d x) \, dx+\frac{1}{3} \left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \int \sec ^2(c+d x) \, dx\\ &=a^2 A x+\frac{a b (2 A+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b C \sec (c+d x) \tan (c+d x)}{3 d}+\frac{C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}-\frac{\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=a^2 A x+\frac{a b (2 A+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \tan (c+d x)}{3 d}+\frac{a b C \sec (c+d x) \tan (c+d x)}{3 d}+\frac{C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [B] time = 1.24115, size = 242, normalized size = 2.35 \[ \frac{\sec ^3(c+d x) \left (2 \sin (c+d x) \left (\left (3 a^2 C+3 A b^2+2 b^2 C\right ) \cos (2 (c+d x))+3 a^2 C+6 a b C \cos (c+d x)+3 A b^2+4 b^2 C\right )+9 a \cos (c+d x) \left (a A (c+d x)-b (2 A+C) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+b (2 A+C) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+3 a \cos (3 (c+d x)) \left (a A (c+d x)-b (2 A+C) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+b (2 A+C) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 145, normalized size = 1.4 \begin{align*}{a}^{2}Ax+{\frac{A{a}^{2}c}{d}}+{\frac{{a}^{2}C\tan \left ( dx+c \right ) }{d}}+2\,{\frac{Aab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{abC\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+{\frac{abC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{A{b}^{2}\tan \left ( dx+c \right ) }{d}}+{\frac{2\,{b}^{2}C\tan \left ( dx+c \right ) }{3\,d}}+{\frac{{b}^{2}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988857, size = 174, normalized size = 1.69 \begin{align*} \frac{6 \,{\left (d x + c\right )} A a^{2} + 2 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C b^{2} - 3 \, C 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 )} + 12 \, A a b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 6 \, C a^{2} \tan \left (d x + c\right ) + 6 \, A b^{2} \tan \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.531963, size = 347, normalized size = 3.37 \begin{align*} \frac{6 \, A a^{2} d x \cos \left (d x + c\right )^{3} + 3 \,{\left (2 \, A + C\right )} a b \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (2 \, A + C\right )} a b \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (3 \, C a b \cos \left (d x + c\right ) + C b^{2} +{\left (3 \, C a^{2} +{\left (3 \, A + 2 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{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 + C \sec ^{2}{\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 = 1.24665, size = 354, normalized size = 3.44 \begin{align*} \frac{3 \,{\left (d x + c\right )} A a^{2} + 3 \,{\left (2 \, A a b + C a b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (2 \, A a b + C a b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (3 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 6 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, C 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 )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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