Optimal. Leaf size=61 \[ -\frac {16 \sin ^{11}(a+b x)}{11 b}+\frac {16 \sin ^9(a+b x)}{3 b}-\frac {48 \sin ^7(a+b x)}{7 b}+\frac {16 \sin ^5(a+b x)}{5 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4287, 2564, 270} \[ -\frac {16 \sin ^{11}(a+b x)}{11 b}+\frac {16 \sin ^9(a+b x)}{3 b}-\frac {48 \sin ^7(a+b x)}{7 b}+\frac {16 \sin ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 2564
Rule 4287
Rubi steps
\begin {align*} \int \cos ^3(a+b x) \sin ^4(2 a+2 b x) \, dx &=16 \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx\\ &=\frac {16 \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^3 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {16 \operatorname {Subst}\left (\int \left (x^4-3 x^6+3 x^8-x^{10}\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {16 \sin ^5(a+b x)}{5 b}-\frac {48 \sin ^7(a+b x)}{7 b}+\frac {16 \sin ^9(a+b x)}{3 b}-\frac {16 \sin ^{11}(a+b x)}{11 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.21, size = 47, normalized size = 0.77 \[ \frac {\sin ^5(a+b x) (3335 \cos (2 (a+b x))+910 \cos (4 (a+b x))+105 \cos (6 (a+b x))+3042)}{2310 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 63, normalized size = 1.03 \[ \frac {16 \, {\left (105 \, \cos \left (b x + a\right )^{10} - 140 \, \cos \left (b x + a\right )^{8} + 5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{1155 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.37, size = 82, normalized size = 1.34 \[ \frac {\sin \left (11 \, b x + 11 \, a\right )}{704 \, b} + \frac {\sin \left (9 \, b x + 9 \, a\right )}{192 \, b} - \frac {\sin \left (7 \, b x + 7 \, a\right )}{448 \, b} - \frac {11 \, \sin \left (5 \, b x + 5 \, a\right )}{320 \, b} - \frac {\sin \left (3 \, b x + 3 \, a\right )}{32 \, b} + \frac {7 \, \sin \left (b x + a\right )}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.25, size = 83, normalized size = 1.36 \[ \frac {7 \sin \left (b x +a \right )}{32 b}-\frac {\sin \left (3 b x +3 a \right )}{32 b}-\frac {11 \sin \left (5 b x +5 a \right )}{320 b}-\frac {\sin \left (7 b x +7 a \right )}{448 b}+\frac {\sin \left (9 b x +9 a \right )}{192 b}+\frac {\sin \left (11 b x +11 a \right )}{704 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 69, normalized size = 1.13 \[ \frac {105 \, \sin \left (11 \, b x + 11 \, a\right ) + 385 \, \sin \left (9 \, b x + 9 \, a\right ) - 165 \, \sin \left (7 \, b x + 7 \, a\right ) - 2541 \, \sin \left (5 \, b x + 5 \, a\right ) - 2310 \, \sin \left (3 \, b x + 3 \, a\right ) + 16170 \, \sin \left (b x + a\right )}{73920 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 45, normalized size = 0.74 \[ \frac {-\frac {16\,{\sin \left (a+b\,x\right )}^{11}}{11}+\frac {16\,{\sin \left (a+b\,x\right )}^9}{3}-\frac {48\,{\sin \left (a+b\,x\right )}^7}{7}+\frac {16\,{\sin \left (a+b\,x\right )}^5}{5}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________