Integrand size = 20, antiderivative size = 61 \[ \int \sin ^3(a+b x) \sin ^4(2 a+2 b x) \, dx=-\frac {16 \cos ^5(a+b x)}{5 b}+\frac {48 \cos ^7(a+b x)}{7 b}-\frac {16 \cos ^9(a+b x)}{3 b}+\frac {16 \cos ^{11}(a+b x)}{11 b} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4373, 2645, 276} \[ \int \sin ^3(a+b x) \sin ^4(2 a+2 b x) \, dx=\frac {16 \cos ^{11}(a+b x)}{11 b}-\frac {16 \cos ^9(a+b x)}{3 b}+\frac {48 \cos ^7(a+b x)}{7 b}-\frac {16 \cos ^5(a+b x)}{5 b} \]
[In]
[Out]
Rule 276
Rule 2645
Rule 4373
Rubi steps \begin{align*} \text {integral}& = 16 \int \cos ^4(a+b x) \sin ^7(a+b x) \, dx \\ & = -\frac {16 \text {Subst}\left (\int x^4 \left (1-x^2\right )^3 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {16 \text {Subst}\left (\int \left (x^4-3 x^6+3 x^8-x^{10}\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {16 \cos ^5(a+b x)}{5 b}+\frac {48 \cos ^7(a+b x)}{7 b}-\frac {16 \cos ^9(a+b x)}{3 b}+\frac {16 \cos ^{11}(a+b x)}{11 b} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int \sin ^3(a+b x) \sin ^4(2 a+2 b x) \, dx=\frac {\cos ^5(a+b x) (-3042+3335 \cos (2 (a+b x))-910 \cos (4 (a+b x))+105 \cos (6 (a+b x)))}{2310 b} \]
[In]
[Out]
Time = 5.66 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.16
method | result | size |
parallelrisch | \(\frac {-32768-16170 \cos \left (x b +a \right )-165 \cos \left (7 x b +7 a \right )+2541 \cos \left (5 x b +5 a \right )+105 \cos \left (11 x b +11 a \right )-385 \cos \left (9 x b +9 a \right )-2310 \cos \left (3 x b +3 a \right )}{73920 b}\) | \(71\) |
default | \(-\frac {7 \cos \left (x b +a \right )}{32 b}-\frac {\cos \left (3 x b +3 a \right )}{32 b}+\frac {11 \cos \left (5 x b +5 a \right )}{320 b}-\frac {\cos \left (7 x b +7 a \right )}{448 b}-\frac {\cos \left (9 x b +9 a \right )}{192 b}+\frac {\cos \left (11 x b +11 a \right )}{704 b}\) | \(83\) |
risch | \(-\frac {7 \cos \left (x b +a \right )}{32 b}-\frac {\cos \left (3 x b +3 a \right )}{32 b}+\frac {11 \cos \left (5 x b +5 a \right )}{320 b}-\frac {\cos \left (7 x b +7 a \right )}{448 b}-\frac {\cos \left (9 x b +9 a \right )}{192 b}+\frac {\cos \left (11 x b +11 a \right )}{704 b}\) | \(83\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \sin ^3(a+b x) \sin ^4(2 a+2 b x) \, dx=\frac {16 \, {\left (105 \, \cos \left (b x + a\right )^{11} - 385 \, \cos \left (b x + a\right )^{9} + 495 \, \cos \left (b x + a\right )^{7} - 231 \, \cos \left (b x + a\right )^{5}\right )}}{1155 \, b} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (53) = 106\).
Time = 11.11 (sec) , antiderivative size = 366, normalized size of antiderivative = 6.00 \[ \int \sin ^3(a+b x) \sin ^4(2 a+2 b x) \, dx=\begin {cases} - \frac {472 \sin ^{3}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{1155 b} - \frac {64 \sin ^{3}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{231 b} - \frac {211 \sin ^{2}{\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{1155 b} - \frac {304 \sin ^{2}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{385 b} - \frac {128 \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{231 b} + \frac {272 \sin {\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{1155 b} + \frac {256 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{1155 b} - \frac {46 \sin ^{4}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )}}{165 b} - \frac {192 \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{385 b} - \frac {256 \cos ^{3}{\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{1155 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} \sin ^{4}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.13 \[ \int \sin ^3(a+b x) \sin ^4(2 a+2 b x) \, dx=\frac {105 \, \cos \left (11 \, b x + 11 \, a\right ) - 385 \, \cos \left (9 \, b x + 9 \, a\right ) - 165 \, \cos \left (7 \, b x + 7 \, a\right ) + 2541 \, \cos \left (5 \, b x + 5 \, a\right ) - 2310 \, \cos \left (3 \, b x + 3 \, a\right ) - 16170 \, \cos \left (b x + a\right )}{73920 \, b} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.13 \[ \int \sin ^3(a+b x) \sin ^4(2 a+2 b x) \, dx=\frac {105 \, \cos \left (11 \, b x + 11 \, a\right ) - 385 \, \cos \left (9 \, b x + 9 \, a\right ) - 165 \, \cos \left (7 \, b x + 7 \, a\right ) + 2541 \, \cos \left (5 \, b x + 5 \, a\right ) - 2310 \, \cos \left (3 \, b x + 3 \, a\right ) - 16170 \, \cos \left (b x + a\right )}{73920 \, b} \]
[In]
[Out]
Time = 19.42 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \sin ^3(a+b x) \sin ^4(2 a+2 b x) \, dx=-\frac {-\frac {16\,{\cos \left (a+b\,x\right )}^{11}}{11}+\frac {16\,{\cos \left (a+b\,x\right )}^9}{3}-\frac {48\,{\cos \left (a+b\,x\right )}^7}{7}+\frac {16\,{\cos \left (a+b\,x\right )}^5}{5}}{b} \]
[In]
[Out]