Optimal. Leaf size=73 \[ -\frac {\csc ^6(c+d x)}{6 a d}+\frac {\csc ^5(c+d x)}{5 a d}+\frac {\csc ^4(c+d x)}{4 a d}-\frac {\csc ^3(c+d x)}{3 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 75} \[ -\frac {\csc ^6(c+d x)}{6 a d}+\frac {\csc ^5(c+d x)}{5 a d}+\frac {\csc ^4(c+d x)}{4 a d}-\frac {\csc ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 75
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^7 (a-x)^2 (a+x)}{x^7} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \left (\frac {a^3}{x^7}-\frac {a^2}{x^6}-\frac {a}{x^5}+\frac {1}{x^4}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^4(c+d x)}{4 a d}+\frac {\csc ^5(c+d x)}{5 a d}-\frac {\csc ^6(c+d x)}{6 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 48, normalized size = 0.66 \[ \frac {\csc ^3(c+d x) \left (-10 \csc ^3(c+d x)+12 \csc ^2(c+d x)+15 \csc (c+d x)-20\right )}{60 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 76, normalized size = 1.04 \[ \frac {15 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 5}{60 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 46, normalized size = 0.63 \[ -\frac {20 \, \sin \left (d x + c\right )^{3} - 15 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) + 10}{60 \, a d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.49, size = 49, normalized size = 0.67 \[ \frac {-\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {1}{5 \sin \left (d x +c \right )^{5}}+\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.82, size = 46, normalized size = 0.63 \[ -\frac {20 \, \sin \left (d x + c\right )^{3} - 15 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) + 10}{60 \, a d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 8.94, size = 46, normalized size = 0.63 \[ \frac {-20\,{\sin \left (c+d\,x\right )}^3+15\,{\sin \left (c+d\,x\right )}^2+12\,\sin \left (c+d\,x\right )-10}{60\,a\,d\,{\sin \left (c+d\,x\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________