Optimal. Leaf size=54 \[ -\frac {\sin ^7(a+b x)}{7 b}+\frac {3 \sin ^5(a+b x)}{5 b}-\frac {\sin ^3(a+b x)}{b}+\frac {\sin (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2633} \[ -\frac {\sin ^7(a+b x)}{7 b}+\frac {3 \sin ^5(a+b x)}{5 b}-\frac {\sin ^3(a+b x)}{b}+\frac {\sin (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2633
Rubi steps
\begin {align*} \int \cos ^7(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (a+b x)\right )}{b}\\ &=\frac {\sin (a+b x)}{b}-\frac {\sin ^3(a+b x)}{b}+\frac {3 \sin ^5(a+b x)}{5 b}-\frac {\sin ^7(a+b x)}{7 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 54, normalized size = 1.00 \[ -\frac {\sin ^7(a+b x)}{7 b}+\frac {3 \sin ^5(a+b x)}{5 b}-\frac {\sin ^3(a+b x)}{b}+\frac {\sin (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 43, normalized size = 0.80 \[ \frac {{\left (5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{35 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 44, normalized size = 0.81 \[ -\frac {5 \, \sin \left (b x + a\right )^{7} - 21 \, \sin \left (b x + a\right )^{5} + 35 \, \sin \left (b x + a\right )^{3} - 35 \, \sin \left (b x + a\right )}{35 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 42, normalized size = 0.78 \[ \frac {\left (\frac {16}{5}+\cos ^{6}\left (b x +a \right )+\frac {6 \left (\cos ^{4}\left (b x +a \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (b x +a \right )\right )}{5}\right ) \sin \left (b x +a \right )}{7 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.69, size = 44, normalized size = 0.81 \[ -\frac {5 \, \sin \left (b x + a\right )^{7} - 21 \, \sin \left (b x + a\right )^{5} + 35 \, \sin \left (b x + a\right )^{3} - 35 \, \sin \left (b x + a\right )}{35 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 43, normalized size = 0.80 \[ -\frac {\sin \left (a+b\,x\right )\,\left (5\,{\sin \left (a+b\,x\right )}^6-21\,{\sin \left (a+b\,x\right )}^4+35\,{\sin \left (a+b\,x\right )}^2-35\right )}{35\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.15, size = 78, normalized size = 1.44 \[ \begin {cases} \frac {16 \sin ^{7}{\left (a + b x \right )}}{35 b} + \frac {8 \sin ^{5}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{5 b} + \frac {2 \sin ^{3}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{b} + \frac {\sin {\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cos ^{7}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________