Optimal. Leaf size=109 \[ -\frac {\csc ^3(c+d x)}{3 a^3 d}+\frac {3 \csc ^4(c+d x)}{4 a^3 d}-\frac {2 \csc ^5(c+d x)}{5 a^3 d}-\frac {\csc ^6(c+d x)}{3 a^3 d}+\frac {3 \csc ^7(c+d x)}{7 a^3 d}-\frac {\csc ^8(c+d x)}{8 a^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 109, 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 = {2786, 76}
\begin {gather*} -\frac {\csc ^8(c+d x)}{8 a^3 d}+\frac {3 \csc ^7(c+d x)}{7 a^3 d}-\frac {\csc ^6(c+d x)}{3 a^3 d}-\frac {2 \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 76
Rule 2786
Rubi steps
\begin {align*} \int \frac {\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^4 (a+x)}{x^9} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^5}{x^9}-\frac {3 a^4}{x^8}+\frac {2 a^3}{x^7}+\frac {2 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 {2 \csc ^5(c+d x)}{5 a^3 d}-\frac {\csc ^6(c+d x)}{3 a^3 d}+\frac {3 \csc ^7(c+d x)}{7 a^3 d}-\frac {\csc ^8(c+d x)}{8 a^3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 68, normalized size = 0.62 \begin {gather*} -\frac {\csc ^3(c+d x) \left (280-630 \csc (c+d x)+336 \csc ^2(c+d x)+280 \csc ^3(c+d x)-360 \csc ^4(c+d x)+105 \csc ^5(c+d x)\right )}{840 a^3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.37, size = 69, normalized size = 0.63
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{3 \sin \left (d x +c \right )^{6}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {2}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}+\frac {3}{7 \sin \left (d x +c \right )^{7}}}{d \,a^{3}}\) | \(69\) |
default | \(\frac {-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{3 \sin \left (d x +c \right )^{6}}+\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {2}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}+\frac {3}{7 \sin \left (d x +c \right )^{7}}}{d \,a^{3}}\) | \(69\) |
risch | \(\frac {4 i \left (-315 i {\mathrm e}^{12 i \left (d x +c \right )}+70 \,{\mathrm e}^{13 i \left (d x +c \right )}+700 i {\mathrm e}^{10 i \left (d x +c \right )}-686 \,{\mathrm e}^{11 i \left (d x +c \right )}+70 i {\mathrm e}^{8 i \left (d x +c \right )}+268 \,{\mathrm e}^{9 i \left (d x +c \right )}+700 i {\mathrm e}^{6 i \left (d x +c \right )}-268 \,{\mathrm e}^{7 i \left (d x +c \right )}-315 i {\mathrm e}^{4 i \left (d x +c \right )}+686 \,{\mathrm e}^{5 i \left (d x +c \right )}-70 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{105 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}\) | \(150\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 66, normalized size = 0.61 \begin {gather*} -\frac {280 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4} + 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} - 360 \, \sin \left (d x + c\right ) + 105}{840 \, a^{3} d \sin \left (d x + c\right )^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 117, normalized size = 1.07 \begin {gather*} \frac {630 \, \cos \left (d x + c\right )^{4} - 980 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 112 \, \cos \left (d x + c\right )^{2} + 32\right )} \sin \left (d x + c\right ) + 245}{840 \, {\left (a^{3} d \cos \left (d x + c\right )^{8} - 4 \, a^{3} d \cos \left (d x + c\right )^{6} + 6 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\cot ^{9}{\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 6.48, size = 66, normalized size = 0.61 \begin {gather*} -\frac {280 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4} + 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} - 360 \, \sin \left (d x + c\right ) + 105}{840 \, a^{3} d \sin \left (d x + c\right )^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 6.63, size = 66, normalized size = 0.61 \begin {gather*} -\frac {280\,{\sin \left (c+d\,x\right )}^5-630\,{\sin \left (c+d\,x\right )}^4+336\,{\sin \left (c+d\,x\right )}^3+280\,{\sin \left (c+d\,x\right )}^2-360\,\sin \left (c+d\,x\right )+105}{840\,a^3\,d\,{\sin \left (c+d\,x\right )}^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________