Optimal. Leaf size=68 \[ -\frac {\cot ^6(c+d x)}{6 a d}+\frac {\csc (c+d x)}{a d}-\frac {2 \csc ^3(c+d x)}{3 a d}+\frac {\csc ^5(c+d x)}{5 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 68, 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 ^6(c+d x)}{6 a d}+\frac {\csc ^5(c+d x)}{5 a d}-\frac {2 \csc ^3(c+d x)}{3 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 ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^5(c+d x) \csc (c+d x) \, dx}{a}+\frac {\int \cot ^5(c+d x) \csc ^2(c+d x) \, dx}{a}\\ &=-\frac {\text {Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac {\cot ^6(c+d x)}{6 a d}+\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac {\cot ^6(c+d x)}{6 a d}+\frac {\csc (c+d x)}{a d}-\frac {2 \csc ^3(c+d x)}{3 a d}+\frac {\csc ^5(c+d x)}{5 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 61, normalized size = 0.90 \begin {gather*} \frac {\csc ^6(c+d x) (-15 \cos (4 (c+d x))+78 \sin (c+d x)-5 (5+7 \sin (3 (c+d x))-3 \sin (5 (c+d x))))}{240 a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.24, size = 67, normalized size = 0.99
method | result | size |
derivativedivides | \(\frac {\frac {1}{2 \sin \left (d x +c \right )^{4}}+\frac {1}{\sin \left (d x +c \right )}-\frac {2}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}}{d a}\) | \(67\) |
default | \(\frac {\frac {1}{2 \sin \left (d x +c \right )^{4}}+\frac {1}{\sin \left (d x +c \right )}-\frac {2}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}}{d a}\) | \(67\) |
risch | \(\frac {2 i \left (-15 i {\mathrm e}^{10 i \left (d x +c \right )}+15 \,{\mathrm e}^{11 i \left (d x +c \right )}-35 \,{\mathrm e}^{9 i \left (d x +c \right )}-50 i {\mathrm e}^{6 i \left (d x +c \right )}+78 \,{\mathrm e}^{7 i \left (d x +c \right )}-78 \,{\mathrm e}^{5 i \left (d x +c \right )}-15 i {\mathrm e}^{2 i \left (d x +c \right )}+35 \,{\mathrm e}^{3 i \left (d x +c \right )}-15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 66, normalized size = 0.97 \begin {gather*} \frac {30 \, \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} + 6 \, \sin \left (d x + c\right ) - 5}{30 \, a d \sin \left (d x + c\right )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 96, normalized size = 1.41 \begin {gather*} \frac {15 \, \cos \left (d x + c\right )^{4} - 15 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 5}{30 \, {\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 )}} \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 ^{7}{\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 = 4.26, size = 66, normalized size = 0.97 \begin {gather*} \frac {30 \, \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} + 6 \, \sin \left (d x + c\right ) - 5}{30 \, a d \sin \left (d x + c\right )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 6.79, size = 63, normalized size = 0.93 \begin {gather*} \frac {{\sin \left (c+d\,x\right )}^5-\frac {{\sin \left (c+d\,x\right )}^4}{2}-\frac {2\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {{\sin \left (c+d\,x\right )}^2}{2}+\frac {\sin \left (c+d\,x\right )}{5}-\frac {1}{6}}{a\,d\,{\sin \left (c+d\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________