Optimal. Leaf size=91 \[ \frac {\cot ^8(c+d x)}{8 a d}+\frac {\cot ^6(c+d x)}{6 a d}-\frac {\csc ^9(c+d x)}{9 a d}+\frac {2 \csc ^7(c+d x)}{7 a d}-\frac {\csc ^5(c+d x)}{5 a d} \]
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Rubi [A] time = 0.16, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2835, 2606, 270, 2607, 14} \[ \frac {\cot ^8(c+d x)}{8 a d}+\frac {\cot ^6(c+d x)}{6 a d}-\frac {\csc ^9(c+d x)}{9 a d}+\frac {2 \csc ^7(c+d x)}{7 a d}-\frac {\csc ^5(c+d x)}{5 a d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2606
Rule 2607
Rule 2835
Rubi steps
\begin {align*} \int \frac {\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^5(c+d x) \csc ^4(c+d x) \, dx}{a}+\frac {\int \cot ^5(c+d x) \csc ^5(c+d x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a d}+\frac {\operatorname {Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,-\cot (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{a d}\\ &=\frac {\cot ^6(c+d x)}{6 a d}+\frac {\cot ^8(c+d x)}{8 a d}-\frac {\csc ^5(c+d x)}{5 a d}+\frac {2 \csc ^7(c+d x)}{7 a d}-\frac {\csc ^9(c+d x)}{9 a d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 68, normalized size = 0.75 \[ \frac {\csc ^4(c+d x) \left (-280 \csc ^5(c+d x)+315 \csc ^4(c+d x)+720 \csc ^3(c+d x)-840 \csc ^2(c+d x)-504 \csc (c+d x)+630\right )}{2520 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 115, normalized size = 1.26 \[ -\frac {504 \, \cos \left (d x + c\right )^{4} - 288 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 64}{2520 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 66, normalized size = 0.73 \[ \frac {630 \, \sin \left (d x + c\right )^{5} - 504 \, \sin \left (d x + c\right )^{4} - 840 \, \sin \left (d x + c\right )^{3} + 720 \, \sin \left (d x + c\right )^{2} + 315 \, \sin \left (d x + c\right ) - 280}{2520 \, a d \sin \left (d x + c\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 69, normalized size = 0.76 \[ \frac {-\frac {1}{3 \sin \left (d x +c \right )^{6}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}+\frac {2}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{9 \sin \left (d x +c \right )^{9}}+\frac {1}{8 \sin \left (d x +c \right )^{8}}+\frac {1}{4 \sin \left (d x +c \right )^{4}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 66, normalized size = 0.73 \[ \frac {630 \, \sin \left (d x + c\right )^{5} - 504 \, \sin \left (d x + c\right )^{4} - 840 \, \sin \left (d x + c\right )^{3} + 720 \, \sin \left (d x + c\right )^{2} + 315 \, \sin \left (d x + c\right ) - 280}{2520 \, a d \sin \left (d x + c\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.04, size = 65, normalized size = 0.71 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^5}{4}-\frac {{\sin \left (c+d\,x\right )}^4}{5}-\frac {{\sin \left (c+d\,x\right )}^3}{3}+\frac {2\,{\sin \left (c+d\,x\right )}^2}{7}+\frac {\sin \left (c+d\,x\right )}{8}-\frac {1}{9}}{a\,d\,{\sin \left (c+d\,x\right )}^9} \]
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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