Optimal. Leaf size=73 \[ \frac {\cot ^6(c+d x)}{6 a d}-\frac {\csc ^7(c+d x)}{7 a d}+\frac {2 \csc ^5(c+d x)}{5 a d}-\frac {\csc ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.14, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2835, 2606, 270, 2607, 30} \[ \frac {\cot ^6(c+d x)}{6 a d}-\frac {\csc ^7(c+d x)}{7 a d}+\frac {2 \csc ^5(c+d x)}{5 a d}-\frac {\csc ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 270
Rule 2606
Rule 2607
Rule 2835
Rubi steps
\begin {align*} \int \frac {\cot ^7(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^5(c+d x) \csc ^2(c+d x) \, dx}{a}+\frac {\int \cot ^5(c+d x) \csc ^3(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 x^2 \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 (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a d}\\ &=\frac {\cot ^6(c+d x)}{6 a d}-\frac {\csc ^3(c+d x)}{3 a d}+\frac {2 \csc ^5(c+d x)}{5 a d}-\frac {\csc ^7(c+d x)}{7 a d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 68, normalized size = 0.93 \[ \frac {\csc ^2(c+d x) \left (-30 \csc ^5(c+d x)+35 \csc ^4(c+d x)+84 \csc ^3(c+d x)-105 \csc ^2(c+d x)-70 \csc (c+d x)+105\right )}{210 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 104, normalized size = 1.42 \[ \frac {70 \, \cos \left (d x + c\right )^{4} - 56 \, \cos \left (d x + c\right )^{2} - 35 \, {\left (3 \, \cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 16}{210 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 66, normalized size = 0.90 \[ \frac {105 \, \sin \left (d x + c\right )^{5} - 70 \, \sin \left (d x + c\right )^{4} - 105 \, \sin \left (d x + c\right )^{3} + 84 \, \sin \left (d x + c\right )^{2} + 35 \, \sin \left (d x + c\right ) - 30}{210 \, a d \sin \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 69, normalized size = 0.95 \[ \frac {\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {2}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{2 \sin \left (d x +c \right )^{4}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 66, normalized size = 0.90 \[ \frac {105 \, \sin \left (d x + c\right )^{5} - 70 \, \sin \left (d x + c\right )^{4} - 105 \, \sin \left (d x + c\right )^{3} + 84 \, \sin \left (d x + c\right )^{2} + 35 \, \sin \left (d x + c\right ) - 30}{210 \, a d \sin \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.97, size = 66, normalized size = 0.90 \[ \frac {105\,{\sin \left (c+d\,x\right )}^5-70\,{\sin \left (c+d\,x\right )}^4-105\,{\sin \left (c+d\,x\right )}^3+84\,{\sin \left (c+d\,x\right )}^2+35\,\sin \left (c+d\,x\right )-30}{210\,a\,d\,{\sin \left (c+d\,x\right )}^7} \]
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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