Optimal. Leaf size=30 \[ -\frac {\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{4 a d} \]
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Rubi [A] time = 0.06, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2833, 12, 37} \[ -\frac {\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{4 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 2833
Rubi steps
\begin {align*} \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^5 (a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^4 \operatorname {Subst}\left (\int \frac {(a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^4(c+d x) (a+a \sin (c+d x))^4}{4 a d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.67 \[ -\frac {a^3 (\csc (c+d x)+1)^4}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 72, normalized size = 2.40 \[ \frac {6 \, a^{3} \cos \left (d x + c\right )^{2} - 7 \, a^{3} + 4 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 54, normalized size = 1.80 \[ -\frac {4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{4 \, d \sin \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 49, normalized size = 1.63 \[ \frac {a^{3} \left (-\frac {1}{\sin \left (d x +c \right )}-\frac {3}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{\sin \left (d x +c \right )^{3}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 54, normalized size = 1.80 \[ -\frac {4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{4 \, d \sin \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.61, size = 54, normalized size = 1.80 \[ -\frac {4\,a^3\,{\sin \left (c+d\,x\right )}^3+6\,a^3\,{\sin \left (c+d\,x\right )}^2+4\,a^3\,\sin \left (c+d\,x\right )+a^3}{4\,d\,{\sin \left (c+d\,x\right )}^4} \]
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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