Optimal. Leaf size=14 \[ -\frac {\log (\cos (a+b x))}{2 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4373, 3556}
\begin {gather*} -\frac {\log (\cos (a+b x))}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 4373
Rubi steps
\begin {align*} \int \csc (2 a+2 b x) \sin ^2(a+b x) \, dx &=\frac {1}{2} \int \tan (a+b x) \, dx\\ &=-\frac {\log (\cos (a+b x))}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\log (\cos (a+b x))}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 13, normalized size = 0.93
method | result | size |
default | \(-\frac {\ln \left (\cos \left (x b +a \right )\right )}{2 b}\) | \(13\) |
risch | \(\frac {i x}{2}+\frac {i a}{b}-\frac {\ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{2 b}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs.
\(2 (12) = 24\).
time = 0.27, size = 55, normalized size = 3.93 \begin {gather*} -\frac {\log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.92, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\log \left (-\cos \left (b x + a\right )\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 18, normalized size = 1.29 \begin {gather*} -\frac {\log \left (-\sin \left (b x + a\right )^{2} + 1\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 12, normalized size = 0.86 \begin {gather*} -\frac {\ln \left (\cos \left (a+b\,x\right )\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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