Optimal. Leaf size=23 \[ -\frac {\cot (c+d x)}{d}-\frac {\csc (c+d x)}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4397, 2669, 3767, 8} \[ -\frac {\cot (c+d x)}{d}-\frac {\csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2669
Rule 3767
Rule 4397
Rubi steps
\begin {align*} \int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx &=\int (1+\cos (c+d x)) \csc ^2(c+d x) \, dx\\ &=-\frac {\csc (c+d x)}{d}+\int \csc ^2(c+d x) \, dx\\ &=-\frac {\csc (c+d x)}{d}-\frac {\operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=-\frac {\cot (c+d x)}{d}-\frac {\csc (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 0.65 \[ -\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 21, normalized size = 0.91 \[ -\frac {\cos \left (d x + c\right ) + 1}{d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 16, normalized size = 0.70 \[ -\frac {1}{d \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 24, normalized size = 1.04 \[ \frac {-\frac {1}{\sin \left (d x +c \right )}-\cot \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 22, normalized size = 0.96 \[ -\frac {\frac {1}{\sin \left (d x + c\right )} + \frac {1}{\tan \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 14, normalized size = 0.61 \[ -\frac {\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.11, size = 27, normalized size = 1.17 \[ \begin {cases} \frac {- \cot {\left (c + d x \right )} - \csc {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (\cot {\relax (c )} + \csc {\relax (c )}\right ) \csc {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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