Optimal. Leaf size=96 \[ -\frac {\left (a^2 C-b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 b d}+\frac {1}{2} a x (2 A+C)+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 b d}-\frac {a C \sin (c+d x) \cos (c+d x)}{6 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3024, 2734} \[ -\frac {\left (a^2 C-b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 b d}+\frac {1}{2} a x (2 A+C)+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 b d}-\frac {a C \sin (c+d x) \cos (c+d x)}{6 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2734
Rule 3024
Rubi steps
\begin {align*} \int (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 b d}+\frac {\int (a+b \cos (c+d x)) (b (3 A+2 C)-a C \cos (c+d x)) \, dx}{3 b}\\ &=\frac {1}{2} a (2 A+C) x-\frac {\left (a^2 C-b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 b d}-\frac {a C \cos (c+d x) \sin (c+d x)}{6 d}+\frac {C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 b d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 64, normalized size = 0.67 \[ \frac {12 a A d x+3 a C \sin (2 (c+d x))+6 a c C+6 a C d x+3 b (4 A+3 C) \sin (c+d x)+b C \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 56, normalized size = 0.58 \[ \frac {3 \, {\left (2 \, A + C\right )} a d x + {\left (2 \, C b \cos \left (d x + c\right )^{2} + 3 \, C a \cos \left (d x + c\right ) + 2 \, {\left (3 \, A + 2 \, C\right )} b\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 64, normalized size = 0.67 \[ \frac {1}{2} \, {\left (2 \, A a + C a\right )} x + \frac {C b \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {C a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A b + 3 \, C b\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.18, size = 68, normalized size = 0.71 \[ \frac {\frac {C b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A b \sin \left (d x +c \right )+a A \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 67, normalized size = 0.70 \[ \frac {12 \, {\left (d x + c\right )} A a + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C b + 12 \, A b \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.28, size = 67, normalized size = 0.70 \[ A\,a\,x+\frac {C\,a\,x}{2}+\frac {A\,b\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,b\,\sin \left (c+d\,x\right )}{4\,d}+\frac {C\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.52, size = 121, normalized size = 1.26 \[ \begin {cases} A a x + \frac {A b \sin {\left (c + d x \right )}}{d} + \frac {C a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {C a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {C a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 C b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a + b \cos {\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________