Optimal. Leaf size=10 \[ \frac {\tan (c+d x)}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {8} \[ \frac {\tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rubi steps
\begin {align*} \int \frac {\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx &=\frac {\operatorname {Subst}(\int 1 \, dx,x,\tan (c+d x))}{d}\\ &=\frac {\tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 10, normalized size = 1.00 \[ \frac {\tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 18, normalized size = 1.80 \[ \frac {\sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 10, normalized size = 1.00 \[ \frac {\tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 11, normalized size = 1.10 \[ \frac {\tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 44, normalized size = 4.40 \[ -\frac {2 \, \sin \left (d x + c\right )}{d {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 29, normalized size = 2.90 \[ -\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \frac {\csc {\left (c + d x \right )}}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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