Optimal. Leaf size=73 \[ -\frac {a^3 \csc ^6(c+d x)}{6 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {3 a^3 \csc ^4(c+d x)}{4 d}-\frac {a^3 \csc ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ -\frac {a^3 \csc ^6(c+d x)}{6 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {3 a^3 \csc ^4(c+d x)}{4 d}-\frac {a^3 \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^7 (a+x)^3}{x^7} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^6 \operatorname {Subst}\left (\int \frac {(a+x)^3}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^6 \operatorname {Subst}\left (\int \left (\frac {a^3}{x^7}+\frac {3 a^2}{x^6}+\frac {3 a}{x^5}+\frac {1}{x^4}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a^3 \csc ^3(c+d x)}{3 d}-\frac {3 a^3 \csc ^4(c+d x)}{4 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^6(c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 1.00 \[ -\frac {a^3 \csc ^6(c+d x)}{6 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {3 a^3 \csc ^4(c+d x)}{4 d}-\frac {a^3 \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 86, normalized size = 1.18 \[ -\frac {45 \, a^{3} \cos \left (d x + c\right )^{2} - 55 \, a^{3} + 4 \, {\left (5 \, a^{3} \cos \left (d x + c\right )^{2} - 14 \, a^{3}\right )} \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 56, normalized size = 0.77 \[ -\frac {20 \, a^{3} \sin \left (d x + c\right )^{3} + 45 \, a^{3} \sin \left (d x + c\right )^{2} + 36 \, a^{3} \sin \left (d x + c\right ) + 10 \, a^{3}}{60 \, d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 49, normalized size = 0.67 \[ \frac {a^{3} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {3}{5 \sin \left (d x +c \right )^{5}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 56, normalized size = 0.77 \[ -\frac {20 \, a^{3} \sin \left (d x + c\right )^{3} + 45 \, a^{3} \sin \left (d x + c\right )^{2} + 36 \, a^{3} \sin \left (d x + c\right ) + 10 \, a^{3}}{60 \, d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.60, size = 56, normalized size = 0.77 \[ -\frac {a^3\,\left (-20\,{\sin \left (c+d\,x\right )}^6+20\,{\sin \left (c+d\,x\right )}^3+45\,{\sin \left (c+d\,x\right )}^2+36\,\sin \left (c+d\,x\right )+10\right )}{60\,d\,{\sin \left (c+d\,x\right )}^6} \]
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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