Optimal. Leaf size=186 \[ \frac{\left (3 a^2 C+b^2 (3 A+2 C)\right ) \tan (c+d x)}{3 b^3 d}-\frac{a \left (C \left (2 a^2+b^2\right )+2 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{2 a^2 \left (a^2 C+A b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{a C \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{C \tan (c+d x) \sec ^2(c+d x)}{3 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.645232, antiderivative size = 186, 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 = {4103, 4092, 4082, 3998, 3770, 3831, 2659, 208} \[ \frac{\left (3 a^2 C+b^2 (3 A+2 C)\right ) \tan (c+d x)}{3 b^3 d}-\frac{a \left (C \left (2 a^2+b^2\right )+2 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{2 a^2 \left (a^2 C+A b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{a C \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{C \tan (c+d x) \sec ^2(c+d x)}{3 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4103
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)} \, dx &=\frac{C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac{\int \frac{\sec ^2(c+d x) \left (2 a C+b (3 A+2 C) \sec (c+d x)-3 a C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 b}\\ &=-\frac{a C \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac{C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac{\int \frac{\sec (c+d x) \left (-3 a^2 C+a b C \sec (c+d x)+2 \left (3 a^2 C+b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^2}\\ &=\frac{\left (3 a^2 C+b^2 (3 A+2 C)\right ) \tan (c+d x)}{3 b^3 d}-\frac{a C \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac{C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac{\int \frac{\sec (c+d x) \left (-3 a^2 b C-3 a \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^3}\\ &=\frac{\left (3 a^2 C+b^2 (3 A+2 C)\right ) \tan (c+d x)}{3 b^3 d}-\frac{a C \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac{C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac{\left (a^2 \left (A b^2+a^2 C\right )\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{b^4}-\frac{\left (a \left (2 A b^2+\left (2 a^2+b^2\right ) C\right )\right ) \int \sec (c+d x) \, dx}{2 b^4}\\ &=-\frac{a \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{\left (3 a^2 C+b^2 (3 A+2 C)\right ) \tan (c+d x)}{3 b^3 d}-\frac{a C \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac{C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac{\left (a^2 \left (A b^2+a^2 C\right )\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{b^5}\\ &=-\frac{a \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{\left (3 a^2 C+b^2 (3 A+2 C)\right ) \tan (c+d x)}{3 b^3 d}-\frac{a C \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac{C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac{\left (2 a^2 \left (A b^2+a^2 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 d}\\ &=-\frac{a \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{2 a^2 \left (A b^2+a^2 C\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b^4 \sqrt{a+b} d}+\frac{\left (3 a^2 C+b^2 (3 A+2 C)\right ) \tan (c+d x)}{3 b^3 d}-\frac{a C \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac{C \sec ^2(c+d x) \tan (c+d x)}{3 b d}\\ \end{align*}
Mathematica [C] time = 3.7097, size = 657, normalized size = 3.53 \[ \frac{\cos (c+d x) (a \cos (c+d x)+b) \left (A+C \sec ^2(c+d x)\right ) \left (\frac{4 b \sin \left (\frac{d x}{2}\right ) \left (3 a^2 C+3 A b^2+2 b^2 C\right )}{\left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{4 b \sin \left (\frac{d x}{2}\right ) \left (3 a^2 C+3 A b^2+2 b^2 C\right )}{\left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+6 a \left (C \left (2 a^2+b^2\right )+2 A b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-6 a \left (C \left (2 a^2+b^2\right )+2 A b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-\frac{24 i a^2 (\cos (c)-i \sin (c)) \left (a^2 C+A b^2\right ) \tan ^{-1}\left (\frac{(\sin (c)+i \cos (c)) \left (\tan \left (\frac{d x}{2}\right ) (a \cos (c)-b)+a \sin (c)\right )}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right )}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}+\frac{b^2 C \left ((3 a+b) \sin \left (\frac{c}{2}\right )+(b-3 a) \cos \left (\frac{c}{2}\right )\right )}{\left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \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 \left ((3 a+b) \sin \left (\frac{c}{2}\right )+(3 a-b) \cos \left (\frac{c}{2}\right )\right )}{\left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{2 b^3 C \sin \left (\frac{d x}{2}\right )}{\left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{2 b^3 C \sin \left (\frac{d x}{2}\right )}{\left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}\right )}{6 b^4 d (a+b \sec (c+d x)) (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.087, size = 554, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 7.08132, size = 1486, normalized size = 7.99 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}}{a + b \sec{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.45972, size = 502, normalized size = 2.7 \begin{align*} -\frac{\frac{3 \,{\left (2 \, C a^{3} + 2 \, A a b^{2} + C a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} - \frac{3 \,{\left (2 \, C a^{3} + 2 \, A a b^{2} + C a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} - \frac{12 \,{\left (C a^{4} + A a^{2} b^{2}\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 )}}{\sqrt{-a^{2} + b^{2}} b^{4}} + \frac{2 \,{\left (6 \, 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} + 6 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 4 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, 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 ) + 6 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, 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} b^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]