Optimal. Leaf size=123 \[ \frac {a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac {a^2 (12 A+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} a^2 x (12 A+7 C)+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}-\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3024, 2751, 2644} \[ \frac {a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac {a^2 (12 A+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} a^2 x (12 A+7 C)+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}-\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2644
Rule 2751
Rule 3024
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d}+\frac {\int (a+a \cos (c+d x))^2 (a (4 A+3 C)-a C \cos (c+d x)) \, dx}{4 a}\\ &=-\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d}+\frac {1}{12} (12 A+7 C) \int (a+a \cos (c+d x))^2 \, dx\\ &=\frac {1}{8} a^2 (12 A+7 C) x+\frac {a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac {a^2 (12 A+7 C) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.27, size = 73, normalized size = 0.59 \[ \frac {a^2 (48 (4 A+3 C) \sin (c+d x)+24 (A+2 C) \sin (2 (c+d x))+144 A d x+16 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+84 C d x)}{96 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 86, normalized size = 0.70 \[ \frac {3 \, {\left (12 \, A + 7 \, C\right )} a^{2} d x + {\left (6 \, C a^{2} \cos \left (d x + c\right )^{3} + 16 \, C a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right ) + 16 \, {\left (3 \, A + 2 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.39, size = 103, normalized size = 0.84 \[ \frac {C a^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {C a^{2} \sin \left (3 \, d x + 3 \, c\right )}{6 \, d} + \frac {1}{8} \, {\left (12 \, A a^{2} + 7 \, C a^{2}\right )} x + \frac {{\left (A a^{2} + 2 \, C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A a^{2} + 3 \, C a^{2}\right )} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.22, size = 142, normalized size = 1.15 \[ \frac {a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a^{2} A \sin \left (d x +c \right )+a^{2} A \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.71, size = 132, normalized size = 1.07 \[ \frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 96 \, {\left (d x + c\right )} A a^{2} - 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 192 \, A a^{2} \sin \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.92, size = 117, normalized size = 0.95 \[ \frac {3\,A\,a^2\,x}{2}+\frac {7\,C\,a^2\,x}{8}+\frac {2\,A\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {C\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {C\,a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.24, size = 309, normalized size = 2.51 \[ \begin {cases} \frac {A a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + A a^{2} x + \frac {A a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 A a^{2} \sin {\left (c + d x \right )}}{d} + \frac {3 C a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {C a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 C a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {C a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {4 C a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 C a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {2 C a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a \cos {\relax (c )} + a\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________