Optimal. Leaf size=145 \[ \frac{\left (a^2 (3 A+4 C)+2 A b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left (a^2 (3 A+4 C)+4 b^2 (A+2 C)\right )+\frac{2 a b (2 A+3 C) \sin (c+d x)}{3 d}+\frac{a A b \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.381924, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4095, 4074, 4047, 2637, 4045, 8} \[ \frac{\left (a^2 (3 A+4 C)+2 A b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left (a^2 (3 A+4 C)+4 b^2 (A+2 C)\right )+\frac{2 a b (2 A+3 C) \sin (c+d x)}{3 d}+\frac{a A b \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4095
Rule 4074
Rule 4047
Rule 2637
Rule 4045
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac{1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (2 A b+a (3 A+4 C) \sec (c+d x)+b (A+4 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a A b \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}-\frac{1}{12} \int \cos ^2(c+d x) \left (-3 \left (2 A b^2+a^2 (3 A+4 C)\right )-8 a b (2 A+3 C) \sec (c+d x)-3 b^2 (A+4 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a A b \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}-\frac{1}{12} \int \cos ^2(c+d x) \left (-3 \left (2 A b^2+a^2 (3 A+4 C)\right )-3 b^2 (A+4 C) \sec ^2(c+d x)\right ) \, dx+\frac{1}{3} (2 a b (2 A+3 C)) \int \cos (c+d x) \, dx\\ &=\frac{2 a b (2 A+3 C) \sin (c+d x)}{3 d}+\frac{\left (2 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a A b \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}-\frac{1}{8} \left (-4 b^2 (A+2 C)-a^2 (3 A+4 C)\right ) \int 1 \, dx\\ &=\frac{1}{8} \left (4 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) x+\frac{2 a b (2 A+3 C) \sin (c+d x)}{3 d}+\frac{\left (2 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a A b \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.391724, size = 104, normalized size = 0.72 \[ \frac{12 (c+d x) \left (a^2 (3 A+4 C)+4 b^2 (A+2 C)\right )+24 \left (a^2 (A+C)+A b^2\right ) \sin (2 (c+d x))+3 a^2 A \sin (4 (c+d x))+48 a b (3 A+4 C) \sin (c+d x)+16 a A b \sin (3 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.071, size = 140, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({a}^{2}A \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{2\,Aab \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+A{b}^{2} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{a}^{2}C \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +2\,abC\sin \left ( dx+c \right ) +{b}^{2}C \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.996142, size = 176, normalized size = 1.21 \begin{align*} \frac{3 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 64 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b + 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{2} + 96 \,{\left (d x + c\right )} C b^{2} + 192 \, C a b \sin \left (d x + c\right )}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.510309, size = 248, normalized size = 1.71 \begin{align*} \frac{3 \,{\left ({\left (3 \, A + 4 \, C\right )} a^{2} + 4 \,{\left (A + 2 \, C\right )} b^{2}\right )} d x +{\left (6 \, A a^{2} \cos \left (d x + c\right )^{3} + 16 \, A a b \cos \left (d x + c\right )^{2} + 16 \,{\left (2 \, A + 3 \, C\right )} a b + 3 \,{\left ({\left (3 \, A + 4 \, C\right )} a^{2} + 4 \, A b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \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 [B] time = 1.16697, size = 510, normalized size = 3.52 \begin{align*} \frac{3 \,{\left (3 \, A a^{2} + 4 \, C a^{2} + 4 \, A b^{2} + 8 \, C b^{2}\right )}{\left (d x + c\right )} - \frac{2 \,{\left (15 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 12 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 48 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 48 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 9 \, A a^{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 )^{5} - 80 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 144 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 80 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 144 \, C a b \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} - 15 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 12 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 48 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 48 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 12 \, A 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 )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]