Optimal. Leaf size=51 \[ -\frac {\cot ^4(c+d x)}{4 a d}+\frac {\csc ^3(c+d x)}{3 a d}-\frac {\csc (c+d x)}{a d} \]
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Rubi [A] time = 0.09, 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 = {2706, 2607, 30, 2606} \[ -\frac {\cot ^4(c+d x)}{4 a d}+\frac {\csc ^3(c+d x)}{3 a d}-\frac {\csc (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2607
Rule 2706
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 {\operatorname {Subst}\left (\int x^3 \, dx,x,-\cot (c+d x)\right )}{a d}+\frac {\operatorname {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*}
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Mathematica [A] time = 0.05, size = 30, normalized size = 0.59 \[ -\frac {(\csc (c+d x)-1)^3 (3 \csc (c+d x)+5)}{12 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 63, normalized size = 1.24 \[ -\frac {6 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 3}{12 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 46, normalized size = 0.90 \[ -\frac {12 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) + 3}{12 \, a d \sin \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 49, normalized size = 0.96 \[ \frac {-\frac {1}{\sin \left (d x +c \right )}+\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{3 \sin \left (d x +c \right )^{3}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 46, normalized size = 0.90 \[ -\frac {12 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) + 3}{12 \, a d \sin \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.56, size = 45, normalized size = 0.88 \[ \frac {-{\sin \left (c+d\,x\right )}^3+\frac {{\sin \left (c+d\,x\right )}^2}{2}+\frac {\sin \left (c+d\,x\right )}{3}-\frac {1}{4}}{a\,d\,{\sin \left (c+d\,x\right )}^4} \]
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 \[ \frac {\int \frac {\cot ^{5}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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