Optimal. Leaf size=84 \[ -\frac {a (4 A+B) \cos ^3(c+d x)}{12 d}+\frac {a (4 A+B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x (4 A+B)-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)}{4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2860, 2669, 2635, 8} \[ -\frac {a (4 A+B) \cos ^3(c+d x)}{12 d}+\frac {a (4 A+B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x (4 A+B)-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rule 2669
Rule 2860
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))}{4 d}+\frac {1}{4} (4 A+B) \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {a (4 A+B) \cos ^3(c+d x)}{12 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))}{4 d}+\frac {1}{4} (a (4 A+B)) \int \cos ^2(c+d x) \, dx\\ &=-\frac {a (4 A+B) \cos ^3(c+d x)}{12 d}+\frac {a (4 A+B) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))}{4 d}+\frac {1}{8} (a (4 A+B)) \int 1 \, dx\\ &=\frac {1}{8} a (4 A+B) x-\frac {a (4 A+B) \cos ^3(c+d x)}{12 d}+\frac {a (4 A+B) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))}{4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.67, size = 64, normalized size = 0.76 \[ -\frac {a (24 (A+B) \cos (c+d x)+8 (A+B) \cos (3 (c+d x))-12 d x (4 A+B)-24 A \sin (2 (c+d x))+3 B \sin (4 (c+d x)))}{96 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 65, normalized size = 0.77 \[ -\frac {8 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{3} - 3 \, {\left (4 \, A + B\right )} a d x + 3 \, {\left (2 \, B a \cos \left (d x + c\right )^{3} - {\left (4 \, A + B\right )} a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 83, normalized size = 0.99 \[ \frac {1}{8} \, {\left (4 \, A a + B a\right )} x - \frac {B a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {A a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} - \frac {{\left (A a + B a\right )} \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {{\left (A a + B a\right )} \cos \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.36, size = 96, normalized size = 1.14 \[ \frac {a B \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {a A \left (\cos ^{3}\left (d x +c \right )\right )}{3}-\frac {a B \left (\cos ^{3}\left (d x +c \right )\right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.38, size = 74, normalized size = 0.88 \[ -\frac {32 \, A a \cos \left (d x + c\right )^{3} + 32 \, B a \cos \left (d x + c\right )^{3} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 3 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} B a}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 10.55, size = 276, normalized size = 3.29 \[ \frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,A+B\right )}{4\,\left (A\,a+\frac {B\,a}{4}\right )}\right )\,\left (4\,A+B\right )}{4\,d}-\frac {a\,\left (4\,A+B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d}-\frac {\left (A\,a-\frac {B\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (2\,A\,a+2\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (A\,a+\frac {7\,B\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (2\,A\,a+2\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-A\,a-\frac {7\,B\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {2\,A\,a}{3}+\frac {2\,B\,a}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (\frac {B\,a}{4}-A\,a\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {2\,A\,a}{3}+\frac {2\,B\,a}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.01, size = 199, normalized size = 2.37 \[ \begin {cases} \frac {A a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} - \frac {A a \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {B a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {B a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {B a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {B a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {B a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {B a \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a \sin {\relax (c )} + a\right ) \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________