Optimal. Leaf size=11 \[ -\frac {\cot (a+b x)}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3852, 8}
\begin {gather*} -\frac {\cot (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \, dx &=-\frac {\text {Subst}(\int 1 \, dx,x,\cot (a+b x))}{b}\\ &=-\frac {\cot (a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\cot (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 12, normalized size = 1.09
method | result | size |
derivativedivides | \(-\frac {\cot \left (x b +a \right )}{b}\) | \(12\) |
default | \(-\frac {\cot \left (x b +a \right )}{b}\) | \(12\) |
risch | \(-\frac {2 i}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}\) | \(20\) |
norman | \(\frac {-\frac {1}{2 b}+\frac {\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )}{2 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 13, normalized size = 1.18 \begin {gather*} -\frac {1}{b \tan \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.03, size = 19, normalized size = 1.73 \begin {gather*} -\frac {\cos \left (b x + a\right )}{b \sin \left (b x + a\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*} \int \csc ^{2}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 13, normalized size = 1.18 \begin {gather*} -\frac {1}{b \tan \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\mathrm {cot}\left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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