Optimal. Leaf size=73 \[ -\frac {\csc ^7(c+d x)}{7 a d}+\frac {\csc ^6(c+d x)}{6 a d}+\frac {\csc ^5(c+d x)}{5 a d}-\frac {\csc ^4(c+d x)}{4 a d} \]
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Rubi [A] time = 0.11, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 75} \[ -\frac {\csc ^7(c+d x)}{7 a d}+\frac {\csc ^6(c+d x)}{6 a d}+\frac {\csc ^5(c+d x)}{5 a d}-\frac {\csc ^4(c+d x)}{4 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^8 (a-x)^2 (a+x)}{x^8} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)}{x^8} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {a^3}{x^8}-\frac {a^2}{x^7}-\frac {a}{x^6}+\frac {1}{x^5}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^4(c+d x)}{4 a d}+\frac {\csc ^5(c+d x)}{5 a d}+\frac {\csc ^6(c+d x)}{6 a d}-\frac {\csc ^7(c+d x)}{7 a d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 48, normalized size = 0.66 \[ \frac {\csc ^4(c+d x) \left (-60 \csc ^3(c+d x)+70 \csc ^2(c+d x)+84 \csc (c+d x)-105\right )}{420 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 84, normalized size = 1.15 \[ \frac {84 \, \cos \left (d x + c\right )^{2} - 35 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 24}{420 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 46, normalized size = 0.63 \[ -\frac {105 \, \sin \left (d x + c\right )^{3} - 84 \, \sin \left (d x + c\right )^{2} - 70 \, \sin \left (d x + c\right ) + 60}{420 \, a d \sin \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 49, normalized size = 0.67 \[ \frac {\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 46, normalized size = 0.63 \[ -\frac {105 \, \sin \left (d x + c\right )^{3} - 84 \, \sin \left (d x + c\right )^{2} - 70 \, \sin \left (d x + c\right ) + 60}{420 \, a d \sin \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.96, size = 46, normalized size = 0.63 \[ \frac {-105\,{\sin \left (c+d\,x\right )}^3+84\,{\sin \left (c+d\,x\right )}^2+70\,\sin \left (c+d\,x\right )-60}{420\,a\,d\,{\sin \left (c+d\,x\right )}^7} \]
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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