Optimal. Leaf size=68 \[ -\frac {\cot ^6(c+d x)}{6 a d}+\frac {\csc ^5(c+d x)}{5 a d}-\frac {2 \csc ^3(c+d x)}{3 a d}+\frac {\csc (c+d x)}{a d} \]
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Rubi [A] time = 0.09, 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 = {2706, 2607, 30, 2606, 194} \[ -\frac {\cot ^6(c+d x)}{6 a d}+\frac {\csc ^5(c+d x)}{5 a d}-\frac {2 \csc ^3(c+d x)}{3 a d}+\frac {\csc (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 194
Rule 2606
Rule 2607
Rule 2706
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 {\operatorname {Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{a d}+\frac {\operatorname {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 {\operatorname {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*}
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Mathematica [A] time = 0.13, size = 61, normalized size = 0.90 \[ \frac {\csc ^6(c+d x) (78 \sin (c+d x)-5 (7 \sin (3 (c+d x))-3 \sin (5 (c+d x))+5)-15 \cos (4 (c+d x)))}{240 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 96, normalized size = 1.41 \[ \frac {15 \, \cos \left (d x + c\right )^{4} - 15 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 5}{30 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 66, normalized size = 0.97 \[ \frac {30 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 5}{30 \, a d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 67, normalized size = 0.99 \[ \frac {-\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {1}{\sin \left (d x +c \right )}+\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{2 \sin \left (d x +c \right )^{4}}-\frac {2}{3 \sin \left (d x +c \right )^{3}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 66, normalized size = 0.97 \[ \frac {30 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 5}{30 \, a d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.04, size = 63, normalized size = 0.93 \[ \frac {{\sin \left (c+d\,x\right )}^5-\frac {{\sin \left (c+d\,x\right )}^4}{2}-\frac {2\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {{\sin \left (c+d\,x\right )}^2}{2}+\frac {\sin \left (c+d\,x\right )}{5}-\frac {1}{6}}{a\,d\,{\sin \left (c+d\,x\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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