Optimal. Leaf size=84 \[ -\frac {\cot ^8(c+d x)}{8 a d}-\frac {\csc (c+d x)}{a d}+\frac {\csc ^3(c+d x)}{a d}-\frac {3 \csc ^5(c+d x)}{5 a d}+\frac {\csc ^7(c+d x)}{7 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2785, 2687, 30,
2686, 200} \begin {gather*} -\frac {\cot ^8(c+d x)}{8 a d}+\frac {\csc ^7(c+d x)}{7 a d}-\frac {3 \csc ^5(c+d x)}{5 a d}+\frac {\csc ^3(c+d x)}{a d}-\frac {\csc (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 200
Rule 2686
Rule 2687
Rule 2785
Rubi steps
\begin {align*} \int \frac {\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^7(c+d x) \csc (c+d x) \, dx}{a}+\frac {\int \cot ^7(c+d x) \csc ^2(c+d x) \, dx}{a}\\ &=-\frac {\text {Subst}\left (\int x^7 \, dx,x,-\cot (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac {\cot ^8(c+d x)}{8 a d}+\frac {\text {Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac {\cot ^8(c+d x)}{8 a d}-\frac {\csc (c+d x)}{a d}+\frac {\csc ^3(c+d x)}{a d}-\frac {3 \csc ^5(c+d x)}{5 a d}+\frac {\csc ^7(c+d x)}{7 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.13, size = 77, normalized size = 0.92 \begin {gather*} \frac {\csc ^8(c+d x) (-245 \cos (2 (c+d x))-35 \cos (6 (c+d x))-513 \sin (c+d x)+371 \sin (3 (c+d x))-105 \sin (5 (c+d x))+35 \sin (7 (c+d x)))}{2240 a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.32, size = 87, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{\sin \left (d x +c \right )}+\frac {1}{2 \sin \left (d x +c \right )^{6}}-\frac {3}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{\sin \left (d x +c \right )^{3}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}}{d a}\) | \(87\) |
default | \(\frac {\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{\sin \left (d x +c \right )}+\frac {1}{2 \sin \left (d x +c \right )^{6}}-\frac {3}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{\sin \left (d x +c \right )^{3}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}}{d a}\) | \(87\) |
risch | \(-\frac {2 i \left (-35 i {\mathrm e}^{14 i \left (d x +c \right )}+35 \,{\mathrm e}^{15 i \left (d x +c \right )}-105 \,{\mathrm e}^{13 i \left (d x +c \right )}-245 i {\mathrm e}^{10 i \left (d x +c \right )}+371 \,{\mathrm e}^{11 i \left (d x +c \right )}-513 \,{\mathrm e}^{9 i \left (d x +c \right )}-245 i {\mathrm e}^{6 i \left (d x +c \right )}+513 \,{\mathrm e}^{7 i \left (d x +c \right )}-371 \,{\mathrm e}^{5 i \left (d x +c \right )}-35 i {\mathrm e}^{2 i \left (d x +c \right )}+105 \,{\mathrm e}^{3 i \left (d x +c \right )}-35 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{35 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 86, normalized size = 1.02 \begin {gather*} -\frac {280 \, \sin \left (d x + c\right )^{7} - 140 \, \sin \left (d x + c\right )^{6} - 280 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} + 168 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 40 \, \sin \left (d x + c\right ) + 35}{280 \, a 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 = 127, normalized size = 1.51 \begin {gather*} -\frac {140 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{4} + 140 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (35 \, \cos \left (d x + c\right )^{6} - 70 \, \cos \left (d x + c\right )^{4} + 56 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) - 35}{280 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a 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 {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.22, size = 86, normalized size = 1.02 \begin {gather*} -\frac {280 \, \sin \left (d x + c\right )^{7} - 140 \, \sin \left (d x + c\right )^{6} - 280 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} + 168 \, \sin \left (d x + c\right )^{3} - 140 \, \sin \left (d x + c\right )^{2} - 40 \, \sin \left (d x + c\right ) + 35}{280 \, a 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 = 6.77, size = 83, normalized size = 0.99 \begin {gather*} \frac {-{\sin \left (c+d\,x\right )}^7+\frac {{\sin \left (c+d\,x\right )}^6}{2}+{\sin \left (c+d\,x\right )}^5-\frac {3\,{\sin \left (c+d\,x\right )}^4}{4}-\frac {3\,{\sin \left (c+d\,x\right )}^3}{5}+\frac {{\sin \left (c+d\,x\right )}^2}{2}+\frac {\sin \left (c+d\,x\right )}{7}-\frac {1}{8}}{a\,d\,{\sin \left (c+d\,x\right )}^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________