Optimal. Leaf size=51 \[ -\frac {\cot ^4(c+d x)}{4 a d}-\frac {\csc (c+d x)}{a d}+\frac {\csc ^3(c+d x)}{3 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2785, 2687, 30,
2686} \begin {gather*} -\frac {\cot ^4(c+d x)}{4 a d}+\frac {\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 2686
Rule 2687
Rule 2785
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^3(c+d x) \csc (c+d x) \, dx}{a}+\frac {\int \cot ^3(c+d x) \csc ^2(c+d x) \, dx}{a}\\ &=-\frac {\text {Subst}\left (\int x^3 \, dx,x,-\cot (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac {\cot ^4(c+d x)}{4 a d}-\frac {\csc (c+d x)}{a d}+\frac {\csc ^3(c+d x)}{3 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 30, normalized size = 0.59 \begin {gather*} -\frac {(-1+\csc (c+d x))^3 (5+3 \csc (c+d x))}{12 a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.24, size = 49, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {-\frac {1}{\sin \left (d x +c \right )}-\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{3 \sin \left (d x +c \right )^{3}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}}{d a}\) | \(49\) |
default | \(\frac {-\frac {1}{\sin \left (d x +c \right )}-\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{3 \sin \left (d x +c \right )^{3}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}}{d a}\) | \(49\) |
risch | \(-\frac {2 i \left (-3 i {\mathrm e}^{6 i \left (d x +c \right )}+3 \,{\mathrm e}^{7 i \left (d x +c \right )}-5 \,{\mathrm e}^{5 i \left (d x +c \right )}-3 i {\mathrm e}^{2 i \left (d x +c \right )}+5 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 46, normalized size = 0.90 \begin {gather*} -\frac {12 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) + 3}{12 \, a d \sin \left (d x + c\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 63, normalized size = 1.24 \begin {gather*} -\frac {6 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 3}{12 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, 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 ^{5}{\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 = 5.30, size = 46, normalized size = 0.90 \begin {gather*} -\frac {12 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) + 3}{12 \, a d \sin \left (d x + c\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 6.56, size = 45, normalized size = 0.88 \begin {gather*} \frac {-{\sin \left (c+d\,x\right )}^3+\frac {{\sin \left (c+d\,x\right )}^2}{2}+\frac {\sin \left (c+d\,x\right )}{3}-\frac {1}{4}}{a\,d\,{\sin \left (c+d\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________