Optimal. Leaf size=31 \[ \frac {\sin ^5(a+b x)}{5 b}-\frac {\sin ^7(a+b x)}{7 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2564, 14} \[ \frac {\sin ^5(a+b x)}{5 b}-\frac {\sin ^7(a+b x)}{7 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2564
Rubi steps
\begin {align*} \int \cos ^3(a+b x) \sin ^4(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {\sin ^5(a+b x)}{5 b}-\frac {\sin ^7(a+b x)}{7 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 27, normalized size = 0.87 \[ \frac {\sin ^5(a+b x) (5 \cos (2 (a+b x))+9)}{70 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 41, normalized size = 1.32 \[ \frac {{\left (5 \, \cos \left (b x + a\right )^{6} - 8 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2} + 2\right )} \sin \left (b x + a\right )}{35 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 26, normalized size = 0.84 \[ -\frac {5 \, \sin \left (b x + a\right )^{7} - 7 \, \sin \left (b x + a\right )^{5}}{35 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 58, normalized size = 1.87 \[ \frac {-\frac {\left (\cos ^{4}\left (b x +a \right )\right ) \left (\sin ^{3}\left (b x +a \right )\right )}{7}-\frac {3 \sin \left (b x +a \right ) \left (\cos ^{4}\left (b x +a \right )\right )}{35}+\frac {\left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{35}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 26, normalized size = 0.84 \[ -\frac {5 \, \sin \left (b x + a\right )^{7} - 7 \, \sin \left (b x + a\right )^{5}}{35 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 26, normalized size = 0.84 \[ \frac {7\,{\sin \left (a+b\,x\right )}^5-5\,{\sin \left (a+b\,x\right )}^7}{35\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 7.36, size = 44, normalized size = 1.42 \[ \begin {cases} \frac {2 \sin ^{7}{\left (a + b x \right )}}{35 b} + \frac {\sin ^{5}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{5 b} & \text {for}\: b \neq 0 \\x \sin ^{4}{\relax (a )} \cos ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________