Optimal. Leaf size=88 \[ -\frac{\sin (a+b x) \cos ^7(a+b x)}{8 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{48 b}+\frac{5 \sin (a+b x) \cos ^3(a+b x)}{192 b}+\frac{5 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{5 x}{128} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0658742, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ -\frac{\sin (a+b x) \cos ^7(a+b x)}{8 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{48 b}+\frac{5 \sin (a+b x) \cos ^3(a+b x)}{192 b}+\frac{5 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{5 x}{128} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx &=-\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{1}{8} \int \cos ^6(a+b x) \, dx\\ &=\frac{\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{5}{48} \int \cos ^4(a+b x) \, dx\\ &=\frac{5 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{5}{64} \int \cos ^2(a+b x) \, dx\\ &=\frac{5 \cos (a+b x) \sin (a+b x)}{128 b}+\frac{5 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{5 \int 1 \, dx}{128}\\ &=\frac{5 x}{128}+\frac{5 \cos (a+b x) \sin (a+b x)}{128 b}+\frac{5 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}\\ \end{align*}
Mathematica [A] time = 0.144025, size = 52, normalized size = 0.59 \[ \frac{48 \sin (2 (a+b x))-24 \sin (4 (a+b x))-16 \sin (6 (a+b x))-3 \sin (8 (a+b x))+120 b x}{3072 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 64, normalized size = 0.7 \begin{align*}{\frac{1}{b} \left ( -{\frac{\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{8}}+{\frac{\sin \left ( bx+a \right ) }{48} \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( bx+a \right ) }{8}} \right ) }+{\frac{5\,bx}{128}}+{\frac{5\,a}{128}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.997626, size = 65, normalized size = 0.74 \begin{align*} \frac{64 \, \sin \left (2 \, b x + 2 \, a\right )^{3} + 120 \, b x + 120 \, a - 3 \, \sin \left (8 \, b x + 8 \, a\right ) - 24 \, \sin \left (4 \, b x + 4 \, a\right )}{3072 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.62223, size = 149, normalized size = 1.69 \begin{align*} \frac{15 \, b x -{\left (48 \, \cos \left (b x + a\right )^{7} - 8 \, \cos \left (b x + a\right )^{5} - 10 \, \cos \left (b x + a\right )^{3} - 15 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{384 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 12.0965, size = 189, normalized size = 2.15 \begin{align*} \begin{cases} \frac{5 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac{5 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac{15 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac{5 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac{5 x \cos ^{8}{\left (a + b x \right )}}{128} + \frac{5 \sin ^{7}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{128 b} + \frac{55 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{384 b} + \frac{73 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{384 b} - \frac{5 \sin{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text{for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} \cos ^{6}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17535, size = 81, normalized size = 0.92 \begin{align*} \frac{5}{128} \, x - \frac{\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} - \frac{\sin \left (6 \, b x + 6 \, a\right )}{192 \, b} - \frac{\sin \left (4 \, b x + 4 \, a\right )}{128 \, b} + \frac{\sin \left (2 \, b x + 2 \, a\right )}{64 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]