Optimal. Leaf size=68 \[ -\frac {(a-a \sin (c+d x))^6}{6 a^7 d}+\frac {4 (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(a-a \sin (c+d x))^4}{a^5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ -\frac {(a-a \sin (c+d x))^6}{6 a^7 d}+\frac {4 (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(a-a \sin (c+d x))^4}{a^5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (a-x)^3-4 a (a-x)^4+(a-x)^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac {(a-a \sin (c+d x))^4}{a^5 d}+\frac {4 (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(a-a \sin (c+d x))^6}{6 a^7 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 66, normalized size = 0.97 \[ -\frac {\sin (c+d x) \left (5 \sin ^5(c+d x)-6 \sin ^4(c+d x)-15 \sin ^3(c+d x)+20 \sin ^2(c+d x)+15 \sin (c+d x)-30\right )}{30 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 49, normalized size = 0.72 \[ \frac {5 \, \cos \left (d x + c\right )^{6} + 2 \, {\left (3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right )}{30 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 67, normalized size = 0.99 \[ -\frac {5 \, \sin \left (d x + c\right )^{6} - 6 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} + 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right )}{30 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.31, size = 65, normalized size = 0.96 \[ \frac {-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}-\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 67, normalized size = 0.99 \[ -\frac {5 \, \sin \left (d x + c\right )^{6} - 6 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} + 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right )}{30 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.14, size = 80, normalized size = 1.18 \[ \frac {\frac {\sin \left (c+d\,x\right )}{a}-\frac {{\sin \left (c+d\,x\right )}^2}{2\,a}-\frac {2\,{\sin \left (c+d\,x\right )}^3}{3\,a}+\frac {{\sin \left (c+d\,x\right )}^4}{2\,a}+\frac {{\sin \left (c+d\,x\right )}^5}{5\,a}-\frac {{\sin \left (c+d\,x\right )}^6}{6\,a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 49.95, size = 1096, normalized size = 16.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________