Optimal. Leaf size=14 \[ \frac {\tan (c+d x)}{d}-x \]
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Rubi [A] time = 0.15, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {321, 203} \[ \frac {\tan (c+d x)}{d}-x \]
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\tan (c+d x)}{d}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-x+\frac {\tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.64 \[ \frac {\tan (c+d x)}{d}-\frac {\tan ^{-1}(\tan (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 31, normalized size = 2.21 \[ -\frac {d x \cos \left (d x + c\right ) - \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 18, normalized size = 1.29 \[ -\frac {d x + c - \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 24, normalized size = 1.71 \[ \frac {\tan \left (d x +c \right )}{d}-\frac {\arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 64, normalized size = 4.57 \[ -\frac {2 \, {\left (\frac {\sin \left (d x + c\right )}{{\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 33, normalized size = 2.36 \[ -x-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \frac {\sin {\left (c + d x \right )}}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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