Optimal. Leaf size=73 \[ \frac {\sin ^6(c+d x)}{6 a d}-\frac {\sin ^5(c+d x)}{5 a d}-\frac {\sin ^4(c+d x)}{4 a d}+\frac {\sin ^3(c+d x)}{3 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2835, 2564, 14} \[ \frac {\sin ^6(c+d x)}{6 a d}-\frac {\sin ^5(c+d x)}{5 a d}-\frac {\sin ^4(c+d x)}{4 a d}+\frac {\sin ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2564
Rule 2835
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \cos ^3(c+d x) \sin ^2(c+d x) \, dx}{a}-\frac {\int \cos ^3(c+d x) \sin ^3(c+d x) \, dx}{a}\\ &=\frac {\operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int x^3 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int \left (x^3-x^5\right ) \, dx,x,\sin (c+d x)\right )}{a d}\\ &=\frac {\sin ^3(c+d x)}{3 a d}-\frac {\sin ^4(c+d x)}{4 a d}-\frac {\sin ^5(c+d x)}{5 a d}+\frac {\sin ^6(c+d x)}{6 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.21, size = 48, normalized size = 0.66 \[ \frac {\sin ^3(c+d x) \left (10 \sin ^3(c+d x)-12 \sin ^2(c+d x)-15 \sin (c+d x)+20\right )}{60 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 59, normalized size = 0.81 \[ -\frac {10 \, \cos \left (d x + c\right )^{6} - 15 \, \cos \left (d x + c\right )^{4} + 4 \, {\left (3 \, \cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right )}{60 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 49, normalized size = 0.67 \[ \frac {10 \, \sin \left (d x + c\right )^{6} - 12 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} + 20 \, \sin \left (d x + c\right )^{3}}{60 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.24, size = 49, normalized size = 0.67 \[ \frac {\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.58, size = 49, normalized size = 0.67 \[ \frac {10 \, \sin \left (d x + c\right )^{6} - 12 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} + 20 \, \sin \left (d x + c\right )^{3}}{60 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 57, normalized size = 0.78 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^3}{3\,a}-\frac {{\sin \left (c+d\,x\right )}^4}{4\,a}-\frac {{\sin \left (c+d\,x\right )}^5}{5\,a}+\frac {{\sin \left (c+d\,x\right )}^6}{6\,a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 51.02, size = 862, normalized size = 11.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________