Optimal. Leaf size=55 \[ -\frac {\csc ^2(c+d x)}{2 a^2 d}+\frac {2 \csc ^3(c+d x)}{3 a^2 d}-\frac {\csc ^4(c+d x)}{4 a^2 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 55, 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, 45}
\begin {gather*} -\frac {\csc ^4(c+d x)}{4 a^2 d}+\frac {2 \csc ^3(c+d x)}{3 a^2 d}-\frac {\csc ^2(c+d x)}{2 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2786
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^2}{x^5}-\frac {2 a}{x^4}+\frac {1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^2(c+d x)}{2 a^2 d}+\frac {2 \csc ^3(c+d x)}{3 a^2 d}-\frac {\csc ^4(c+d x)}{4 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 38, normalized size = 0.69 \begin {gather*} \frac {\csc ^4(c+d x) (-6+3 \cos (2 (c+d x))+8 \sin (c+d x))}{12 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 39, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {\frac {2}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}}{d \,a^{2}}\) | \(39\) |
default | \(\frac {\frac {2}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}}{d \,a^{2}}\) | \(39\) |
risch | \(\frac {2 \,{\mathrm e}^{6 i \left (d x +c \right )}-8 \,{\mathrm e}^{4 i \left (d x +c \right )}-\frac {16 i {\mathrm e}^{5 i \left (d x +c \right )}}{3}+2 \,{\mathrm e}^{2 i \left (d x +c \right )}+\frac {16 i {\mathrm e}^{3 i \left (d x +c \right )}}{3}}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 36, normalized size = 0.65 \begin {gather*} -\frac {6 \, \sin \left (d x + c\right )^{2} - 8 \, \sin \left (d x + c\right ) + 3}{12 \, a^{2} d \sin \left (d x + c\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 57, normalized size = 1.04 \begin {gather*} \frac {6 \, \cos \left (d x + c\right )^{2} + 8 \, \sin \left (d x + c\right ) - 9}{12 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\cot ^{5}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.70, size = 36, normalized size = 0.65 \begin {gather*} -\frac {6 \, \sin \left (d x + c\right )^{2} - 8 \, \sin \left (d x + c\right ) + 3}{12 \, a^{2} d \sin \left (d x + c\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.33, size = 36, normalized size = 0.65 \begin {gather*} -\frac {\frac {{\sin \left (c+d\,x\right )}^2}{2}-\frac {2\,\sin \left (c+d\,x\right )}{3}+\frac {1}{4}}{a^2\,d\,{\sin \left (c+d\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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