Optimal. Leaf size=91 \[ -\frac {\sin (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac {3 \sin (a+x (b-d)-c)}{8 (b-d)}-\frac {3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac {\sin (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4569, 2637} \[ -\frac {\sin (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac {3 \sin (a+x (b-d)-c)}{8 (b-d)}-\frac {3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac {\sin (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2637
Rule 4569
Rubi steps
\begin {align*} \int \sin (a+b x) \sin ^3(c+d x) \, dx &=\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*}
________________________________________________________________________________________
Mathematica [A] time = 0.54, size = 86, normalized size = 0.95 \[ \frac {1}{8} \left (-\frac {\sin (a+b x-3 c-3 d x)}{b-3 d}+\frac {3 \sin (a+b x-c-d x)}{b-d}+\frac {\sin (a+b x+3 c+3 d x)}{b+3 d}-\frac {3 \sin (a+x (b+d)+c)}{b+d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 122, normalized size = 1.34 \[ -\frac {3 \, {\left ({\left (b^{2} d - d^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (b^{2} d - 3 \, d^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (b x + a\right ) - {\left ({\left (b^{3} - b d^{2}\right )} \cos \left (b x + a\right ) \cos \left (d x + c\right )^{2} - {\left (b^{3} - 7 \, b d^{2}\right )} \cos \left (b x + a\right )\right )} \sin \left (d x + c\right )}{b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.75, size = 84, normalized size = 0.92 \[ \frac {\sin \left (b x + 3 \, d x + a + 3 \, c\right )}{8 \, {\left (b + 3 \, d\right )}} - \frac {3 \, \sin \left (b x + d x + a + c\right )}{8 \, {\left (b + d\right )}} + \frac {3 \, \sin \left (b x - d x + a - c\right )}{8 \, {\left (b - d\right )}} - \frac {\sin \left (b x - 3 \, d x + a - 3 \, c\right )}{8 \, {\left (b - 3 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.86, size = 84, normalized size = 0.92 \[ -\frac {\sin \left (a -3 c +\left (b -3 d \right ) x \right )}{8 \left (b -3 d \right )}+\frac {3 \sin \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {3 \sin \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sin \left (a +3 c +\left (b +3 d \right ) x \right )}{8 b +24 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.41, size = 916, normalized size = 10.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.57, size = 311, normalized size = 3.42 \[ -{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,3{}\mathrm {i}+b\,x\,1{}\mathrm {i}-d\,x\,3{}\mathrm {i}}\,\left (\frac {b+3\,d}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (b-3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}\right )+{\mathrm {e}}^{a\,1{}\mathrm {i}+c\,3{}\mathrm {i}+b\,x\,1{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,\left (\frac {b-3\,d}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (b+3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}\right )+{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {3\,b+3\,d}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (3\,b-3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}\right )-{\mathrm {e}}^{a\,1{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {3\,b-3\,d}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (3\,b+3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 32.89, size = 921, normalized size = 10.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________