Optimal. Leaf size=73 \[ -\frac {\csc ^6(c+d x)}{6 a^3 d}+\frac {3 \csc ^5(c+d x)}{5 a^3 d}-\frac {3 \csc ^4(c+d x)}{4 a^3 d}+\frac {\csc ^3(c+d x)}{3 a^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 73, 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 = {2707, 43} \[ -\frac {\csc ^6(c+d x)}{6 a^3 d}+\frac {3 \csc ^5(c+d x)}{5 a^3 d}-\frac {3 \csc ^4(c+d x)}{4 a^3 d}+\frac {\csc ^3(c+d x)}{3 a^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2707
Rubi steps
\begin {align*} \int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^3}{x^7}-\frac {3 a^2}{x^6}+\frac {3 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^3 d}-\frac {3 \csc ^4(c+d x)}{4 a^3 d}+\frac {3 \csc ^5(c+d x)}{5 a^3 d}-\frac {\csc ^6(c+d x)}{6 a^3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 48, normalized size = 0.66 \[ \frac {\csc ^3(c+d x) \left (-10 \csc ^3(c+d x)+36 \csc ^2(c+d x)-45 \csc (c+d x)+20\right )}{60 a^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 84, normalized size = 1.15 \[ -\frac {45 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 14\right )} \sin \left (d x + c\right ) - 55}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.81, size = 46, normalized size = 0.63 \[ \frac {20 \, \sin \left (d x + c\right )^{3} - 45 \, \sin \left (d x + c\right )^{2} + 36 \, \sin \left (d x + c\right ) - 10}{60 \, a^{3} d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.33, size = 49, normalized size = 0.67 \[ \frac {-\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {3}{5 \sin \left (d x +c \right )^{5}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 46, normalized size = 0.63 \[ \frac {20 \, \sin \left (d x + c\right )^{3} - 45 \, \sin \left (d x + c\right )^{2} + 36 \, \sin \left (d x + c\right ) - 10}{60 \, a^{3} d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.69, size = 46, normalized size = 0.63 \[ \frac {20\,{\sin \left (c+d\,x\right )}^3-45\,{\sin \left (c+d\,x\right )}^2+36\,\sin \left (c+d\,x\right )-10}{60\,a^3\,d\,{\sin \left (c+d\,x\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{7}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________