Optimal. Leaf size=271 \[ -\frac{a \left (3 a^2 C+A b^2-2 b^2 C\right ) \tan (c+d x)}{b^3 d \left (a^2-b^2\right )}+\frac{\left (C \left (6 a^2+b^2\right )+2 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{2 a \left (a^2 A b^2-4 a^2 b^2 C+3 a^4 C-2 A b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac{\left (3 a^2 C+2 A b^2-b^2 C\right ) \tan (c+d x) \sec (c+d x)}{2 b^2 d \left (a^2-b^2\right )} \]
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Rubi [A] time = 0.86113, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {4099, 4092, 4082, 3998, 3770, 3831, 2659, 208} \[ -\frac{a \left (3 a^2 C+A b^2-2 b^2 C\right ) \tan (c+d x)}{b^3 d \left (a^2-b^2\right )}+\frac{\left (C \left (6 a^2+b^2\right )+2 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{2 a \left (a^2 A b^2-4 a^2 b^2 C+3 a^4 C-2 A b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac{\left (3 a^2 C+2 A b^2-b^2 C\right ) \tan (c+d x) \sec (c+d x)}{2 b^2 d \left (a^2-b^2\right )} \]
Antiderivative was successfully verified.
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Rule 4099
Rule 4092
Rule 4082
Rule 3998
Rule 3770
Rule 3831
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx &=-\frac{\left (A b^2+a^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\int \frac{\sec ^2(c+d x) \left (2 \left (A b^2+a^2 C\right )-a b (A+C) \sec (c+d x)-\left (2 A b^2+3 a^2 C-b^2 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac{\left (2 A b^2+3 a^2 C-b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\int \frac{\sec (c+d x) \left (-a \left (2 A b^2+\left (3 a^2-b^2\right ) C\right )+b \left (2 A b^2+\left (a^2+b^2\right ) C\right ) \sec (c+d x)+2 a \left (A b^2+3 a^2 C-2 b^2 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=-\frac{a \left (A b^2+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (2 A b^2+3 a^2 C-b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\int \frac{\sec (c+d x) \left (-a b \left (b^2 (2 A-C)+3 a^2 C\right )-\left (a^2-b^2\right ) \left (2 A b^2+6 a^2 C+b^2 C\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )}\\ &=-\frac{a \left (A b^2+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (2 A b^2+3 a^2 C-b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac{\left (a \left (2 A b^4-a^2 b^2 (A-4 C)-3 a^4 C\right )\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{b^4 \left (a^2-b^2\right )}+\frac{\left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) \int \sec (c+d x) \, dx}{2 b^4}\\ &=\frac{\left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{a \left (A b^2+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (2 A b^2+3 a^2 C-b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac{\left (a \left (2 A b^4-a^2 b^2 (A-4 C)-3 a^4 C\right )\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{b^5 \left (a^2-b^2\right )}\\ &=\frac{\left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{a \left (A b^2+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (2 A b^2+3 a^2 C-b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac{\left (2 a \left (2 A b^4-a^2 b^2 (A-4 C)-3 a^4 C\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}+\left (1-\frac{a}{b}\right ) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^5 \left (a^2-b^2\right ) d}\\ &=\frac{\left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{2 a \left (a^2 A b^2-2 A b^4+3 a^4 C-4 a^2 b^2 C\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{3/2} b^4 (a+b)^{3/2} d}-\frac{a \left (A b^2+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (2 A b^2+3 a^2 C-b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 3.79693, size = 461, normalized size = 1.7 \[ \frac{(a \cos (c+d x)+b) \left (A+C \sec ^2(c+d x)\right ) \left (\frac{4 a^2 b \left (a^2 C+A b^2\right ) \sin (c+d x)}{(b-a) (a+b)}-2 \left (C \left (6 a^2+b^2\right )+2 A b^2\right ) (a \cos (c+d x)+b) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+2 \left (C \left (6 a^2+b^2\right )+2 A b^2\right ) (a \cos (c+d x)+b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{8 a \left (a^2 b^2 (A-4 C)+3 a^4 C-2 A b^4\right ) (a \cos (c+d x)+b) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac{b^2 C (a \cos (c+d x)+b)}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{b^2 C (a \cos (c+d x)+b)}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{8 a b C \sin \left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}-\frac{8 a b C \sin \left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}\right )}{2 b^4 d (a+b \sec (c+d x))^2 (A \cos (2 (c+d x))+A+2 C)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.104, size = 646, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 29.9701, size = 2553, normalized size = 9.42 \begin{align*} \text{result too large to display} \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 \frac{\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \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 [A] time = 1.39989, size = 483, normalized size = 1.78 \begin{align*} -\frac{\frac{4 \,{\left (3 \, C a^{5} + A a^{3} b^{2} - 4 \, C a^{3} b^{2} - 2 \, A a b^{4}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} \sqrt{-a^{2} + b^{2}}} - \frac{4 \,{\left (C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + A a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (a^{2} b^{3} - b^{5}\right )}{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a - b\right )}} - \frac{{\left (6 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} + \frac{{\left (6 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} - \frac{2 \,{\left (4 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 4 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + C 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 )}^{2} b^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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