Optimal. Leaf size=32 \[ \frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d} \]
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Rubi [A]
time = 0.05, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2785, 2686, 30,
8} \begin {gather*} \frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2686
Rule 2785
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot (c+d x) \csc (c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^2(c+d x) \, dx}{a}\\ &=\frac {\text {Subst}(\int 1 \, dx,x,\csc (c+d x))}{a d}-\frac {\text {Subst}(\int x \, dx,x,\csc (c+d x))}{a d}\\ &=\frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 0.75 \begin {gather*} -\frac {(-2+\csc (c+d x)) \csc (c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 30, normalized size = 0.94
method | result | size |
derivativedivides | \(-\frac {\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{\sin \left (d x +c \right )}}{a d}\) | \(30\) |
default | \(-\frac {\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{\sin \left (d x +c \right )}}{a d}\) | \(30\) |
risch | \(\frac {2 i \left (-i {\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 26, normalized size = 0.81 \begin {gather*} \frac {2 \, \sin \left (d x + c\right ) - 1}{2 \, a d \sin \left (d x + c\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 30, normalized size = 0.94 \begin {gather*} -\frac {2 \, \sin \left (d x + c\right ) - 1}{2 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\cot ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.35, size = 26, normalized size = 0.81 \begin {gather*} \frac {2 \, \sin \left (d x + c\right ) - 1}{2 \, a d \sin \left (d x + c\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.58, size = 23, normalized size = 0.72 \begin {gather*} \frac {\sin \left (c+d\,x\right )-\frac {1}{2}}{a\,d\,{\sin \left (c+d\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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