Integrand size = 15, antiderivative size = 91 \[ \int \sin (a+b x) \sin ^3(c+d x) \, dx=-\frac {\sin (a-3 c+(b-3 d) x)}{8 (b-3 d)}+\frac {3 \sin (a-c+(b-d) x)}{8 (b-d)}-\frac {3 \sin (a+c+(b+d) x)}{8 (b+d)}+\frac {\sin (a+3 c+(b+3 d) x)}{8 (b+3 d)} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4665, 2717} \[ \int \sin (a+b x) \sin ^3(c+d x) \, dx=-\frac {\sin (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac {3 \sin (a+x (b-d)-c)}{8 (b-d)}-\frac {3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac {\sin (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]
[In]
[Out]
Rule 2717
Rule 4665
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{8} \cos (a-3 c+(b-3 d) x)+\frac {3}{8} \cos (a-c+(b-d) x)-\frac {3}{8} \cos (a+c+(b+d) x)+\frac {1}{8} \cos (a+3 c+(b+3 d) x)\right ) \, dx \\ & = -\left (\frac {1}{8} \int \cos (a-3 c+(b-3 d) x) \, dx\right )+\frac {1}{8} \int \cos (a+3 c+(b+3 d) x) \, dx+\frac {3}{8} \int \cos (a-c+(b-d) x) \, dx-\frac {3}{8} \int \cos (a+c+(b+d) x) \, dx \\ & = -\frac {\sin (a-3 c+(b-3 d) x)}{8 (b-3 d)}+\frac {3 \sin (a-c+(b-d) x)}{8 (b-d)}-\frac {3 \sin (a+c+(b+d) x)}{8 (b+d)}+\frac {\sin (a+3 c+(b+3 d) x)}{8 (b+3 d)} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.95 \[ \int \sin (a+b x) \sin ^3(c+d x) \, dx=\frac {1}{8} \left (-\frac {\sin (a-3 c+b x-3 d x)}{b-3 d}+\frac {3 \sin (a-c+b x-d x)}{b-d}+\frac {\sin (a+3 c+b x+3 d x)}{b+3 d}-\frac {3 \sin (a+c+(b+d) x)}{b+d}\right ) \]
[In]
[Out]
Time = 0.80 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {\sin \left (a -3 c +\left (b -3 d \right ) x \right )}{8 \left (b -3 d \right )}+\frac {3 \sin \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {3 \sin \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sin \left (a +3 c +\left (b +3 d \right ) x \right )}{8 b +24 d}\) | \(84\) |
risch | \(-\frac {\sin \left (x b -3 d x +a -3 c \right )}{8 \left (b -3 d \right )}+\frac {3 \sin \left (x b -d x +a -c \right )}{8 \left (b -d \right )}-\frac {3 \sin \left (x b +d x +a +c \right )}{8 \left (b +d \right )}+\frac {\sin \left (x b +3 d x +a +3 c \right )}{8 b +24 d}\) | \(85\) |
parallelrisch | \(\frac {12 d^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+12 b \,d^{2} \left (1-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+12 \left (-2 b^{2} d +3 d^{3}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+8 b \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )-1\right ) \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1\right ) \left (b -2 d \right ) \left (b +2 d \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+12 \left (2 b^{2} d -3 d^{3}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+12 b \,d^{2} \left (1-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-12 d^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{\left (b -d \right ) \left (b +3 d \right ) \left (b -3 d \right ) \left (b +d \right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )}\) | \(275\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.34 \[ \int \sin (a+b x) \sin ^3(c+d x) \, dx=-\frac {3 \, {\left ({\left (b^{2} d - d^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (b^{2} d - 3 \, d^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (b x + a\right ) - {\left ({\left (b^{3} - b d^{2}\right )} \cos \left (b x + a\right ) \cos \left (d x + c\right )^{2} - {\left (b^{3} - 7 \, b d^{2}\right )} \cos \left (b x + a\right )\right )} \sin \left (d x + c\right )}{b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 921 vs. \(2 (76) = 152\).
Time = 1.85 (sec) , antiderivative size = 921, normalized size of antiderivative = 10.12 \[ \int \sin (a+b x) \sin ^3(c+d x) \, dx=\text {Too large to display} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 916 vs. \(2 (83) = 166\).
Time = 0.29 (sec) , antiderivative size = 916, normalized size of antiderivative = 10.07 \[ \int \sin (a+b x) \sin ^3(c+d x) \, dx=\text {Too large to display} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.92 \[ \int \sin (a+b x) \sin ^3(c+d x) \, dx=\frac {\sin \left (b x + 3 \, d x + a + 3 \, c\right )}{8 \, {\left (b + 3 \, d\right )}} - \frac {3 \, \sin \left (b x + d x + a + c\right )}{8 \, {\left (b + d\right )}} + \frac {3 \, \sin \left (b x - d x + a - c\right )}{8 \, {\left (b - d\right )}} - \frac {\sin \left (b x - 3 \, d x + a - 3 \, c\right )}{8 \, {\left (b - 3 \, d\right )}} \]
[In]
[Out]
Time = 21.33 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.42 \[ \int \sin (a+b x) \sin ^3(c+d x) \, dx=-{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,3{}\mathrm {i}+b\,x\,1{}\mathrm {i}-d\,x\,3{}\mathrm {i}}\,\left (\frac {b+3\,d}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (b-3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}\right )+{\mathrm {e}}^{a\,1{}\mathrm {i}+c\,3{}\mathrm {i}+b\,x\,1{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,\left (\frac {b-3\,d}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (b+3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}\right )+{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {3\,b+3\,d}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (3\,b-3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}\right )-{\mathrm {e}}^{a\,1{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {3\,b-3\,d}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (3\,b+3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}\right ) \]
[In]
[Out]