Optimal. Leaf size=91 \[ -\frac {\cot ^8(c+d x)}{8 a d}-\frac {\cot ^6(c+d x)}{6 a d}+\frac {\csc ^7(c+d x)}{7 a d}-\frac {2 \csc ^5(c+d x)}{5 a d}+\frac {\csc ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.16, 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, 2607, 14, 2606, 270} \[ -\frac {\cot ^8(c+d x)}{8 a d}-\frac {\cot ^6(c+d x)}{6 a d}+\frac {\csc ^7(c+d x)}{7 a d}-\frac {2 \csc ^5(c+d x)}{5 a d}+\frac {\csc ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2606
Rule 2607
Rule 2835
Rubi steps
\begin {align*} \int \frac {\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^5(c+d x) \csc ^3(c+d x) \, dx}{a}+\frac {\int \cot ^5(c+d x) \csc ^4(c+d x) \, dx}{a}\\ &=\frac {\operatorname {Subst}\left (\int x^2 \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^2-2 x^4+x^6\right ) \, dx,x,\csc (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 {\cot ^6(c+d x)}{6 a d}-\frac {\cot ^8(c+d x)}{8 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 = 0.14, size = 68, normalized size = 0.75 \[ \frac {\csc ^3(c+d x) \left (-105 \csc ^5(c+d x)+120 \csc ^4(c+d x)+280 \csc ^3(c+d x)-336 \csc ^2(c+d x)-210 \csc (c+d x)+280\right )}{840 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 107, normalized size = 1.18 \[ -\frac {210 \, \cos \left (d x + c\right )^{4} - 140 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 28 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 35}{840 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 66, normalized size = 0.73 \[ \frac {280 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} + 120 \, \sin \left (d x + c\right ) - 105}{840 \, a d \sin \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 69, normalized size = 0.76 \[ \frac {\frac {1}{3 \sin \left (d x +c \right )^{6}}-\frac {2}{5 \sin \left (d x +c \right )^{5}}+\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{3 \sin \left (d x +c \right )^{3}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 66, normalized size = 0.73 \[ \frac {280 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} + 120 \, \sin \left (d x + c\right ) - 105}{840 \, a d \sin \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.04, size = 66, normalized size = 0.73 \[ \frac {280\,{\sin \left (c+d\,x\right )}^5-210\,{\sin \left (c+d\,x\right )}^4-336\,{\sin \left (c+d\,x\right )}^3+280\,{\sin \left (c+d\,x\right )}^2+120\,\sin \left (c+d\,x\right )-105}{840\,a\,d\,{\sin \left (c+d\,x\right )}^8} \]
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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