Optimal. Leaf size=131 \[ -\frac{b \left (2 a^2 A-3 a b B-A b^2\right ) \sin (c+d x)}{d}+\frac{1}{2} b x \left (6 a^2 B+6 a A b+b^2 B\right )+\frac{a^2 (a B+3 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (2 a A-b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a A \tan (c+d x) (a+b \cos (c+d x))^2}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.331931, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {2989, 3033, 3023, 2735, 3770} \[ -\frac{b \left (2 a^2 A-3 a b B-A b^2\right ) \sin (c+d x)}{d}+\frac{1}{2} b x \left (6 a^2 B+6 a A b+b^2 B\right )+\frac{a^2 (a B+3 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (2 a A-b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a A \tan (c+d x) (a+b \cos (c+d x))^2}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2989
Rule 3033
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx &=\frac{a A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\int (a+b \cos (c+d x)) \left (a (3 A b+a B)+b (A b+2 a B) \cos (c+d x)-b (2 a A-b B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b^2 (2 a A-b B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\frac{1}{2} \int \left (2 a^2 (3 A b+a B)+b \left (6 a A b+6 a^2 B+b^2 B\right ) \cos (c+d x)-2 b \left (2 a^2 A-A b^2-3 a b B\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b \left (2 a^2 A-A b^2-3 a b B\right ) \sin (c+d x)}{d}-\frac{b^2 (2 a A-b B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\frac{1}{2} \int \left (2 a^2 (3 A b+a B)+b \left (6 a A b+6 a^2 B+b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{2} b \left (6 a A b+6 a^2 B+b^2 B\right ) x-\frac{b \left (2 a^2 A-A b^2-3 a b B\right ) \sin (c+d x)}{d}-\frac{b^2 (2 a A-b B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\left (a^2 (3 A b+a B)\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} b \left (6 a A b+6 a^2 B+b^2 B\right ) x+\frac{a^2 (3 A b+a B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b \left (2 a^2 A-A b^2-3 a b B\right ) \sin (c+d x)}{d}-\frac{b^2 (2 a A-b B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.650147, size = 217, normalized size = 1.66 \[ \frac{2 b (c+d x) \left (6 a^2 B+6 a A b+b^2 B\right )-4 a^2 (a B+3 A b) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 a^2 (a B+3 A b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{4 a^3 A \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{4 a^3 A \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+4 b^2 (3 a B+A b) \sin (c+d x)+b^3 B \sin (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.075, size = 168, normalized size = 1.3 \begin{align*}{\frac{A{a}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{A{a}^{2}b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+3\,{a}^{2}bBx+3\,{\frac{B{a}^{2}bc}{d}}+3\,Aa{b}^{2}x+3\,{\frac{Aa{b}^{2}c}{d}}+3\,{\frac{Ba{b}^{2}\sin \left ( dx+c \right ) }{d}}+{\frac{A{b}^{3}\sin \left ( dx+c \right ) }{d}}+{\frac{B{b}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{{b}^{3}Bx}{2}}+{\frac{B{b}^{3}c}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.16678, size = 194, normalized size = 1.48 \begin{align*} \frac{12 \,{\left (d x + c\right )} B a^{2} b + 12 \,{\left (d x + c\right )} A a b^{2} +{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{3} + 2 \, B a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, A a^{2} b{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, B a b^{2} \sin \left (d x + c\right ) + 4 \, A b^{3} \sin \left (d x + c\right ) + 4 \, A a^{3} \tan \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.56437, size = 369, normalized size = 2.82 \begin{align*} \frac{{\left (6 \, B a^{2} b + 6 \, A a b^{2} + B b^{3}\right )} d x \cos \left (d x + c\right ) +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (B b^{3} \cos \left (d x + c\right )^{2} + 2 \, A a^{3} + 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.54625, size = 316, normalized size = 2.41 \begin{align*} -\frac{\frac{4 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1} -{\left (6 \, B a^{2} b + 6 \, A a b^{2} + B b^{3}\right )}{\left (d x + c\right )} - 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (6 \, B a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, A b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, B a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, A b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + B b^{3} \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.
[In]
[Out]