Optimal. Leaf size=30 \[ -\frac {\cot ^2(a+b x)}{4 b}+\frac {\log (\tan (a+b x))}{2 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4373, 2700, 14}
\begin {gather*} \frac {\log (\tan (a+b x))}{2 b}-\frac {\cot ^2(a+b x)}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2700
Rule 4373
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \csc (2 a+2 b x) \, dx &=\frac {1}{2} \int \csc ^3(a+b x) \sec (a+b x) \, dx\\ &=\frac {\text {Subst}\left (\int \frac {1+x^2}{x^3} \, dx,x,\tan (a+b x)\right )}{2 b}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{x^3}+\frac {1}{x}\right ) \, dx,x,\tan (a+b x)\right )}{2 b}\\ &=-\frac {\cot ^2(a+b x)}{4 b}+\frac {\log (\tan (a+b x))}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 34, normalized size = 1.13 \begin {gather*} -\frac {\csc ^2(a+b x)+2 \log (\cos (a+b x))-2 \log (\sin (a+b x))}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 24, normalized size = 0.80
method | result | size |
default | \(\frac {-\frac {1}{2 \sin \left (x b +a \right )^{2}}+\ln \left (\tan \left (x b +a \right )\right )}{2 b}\) | \(24\) |
risch | \(\frac {{\mathrm e}^{2 i \left (x b +a \right )}}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{2 b}\) | \(62\) |
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. 656 vs.
\(2 (26) = 52\).
time = 0.29, size = 656, normalized size = 21.87 \begin {gather*} \frac {4 \, \cos \left (4 \, b x + 4 \, a\right ) \cos \left (2 \, b x + 2 \, a\right ) - 8 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )^{2} - 4 \, \cos \left (2 \, b x + 2 \, a\right )^{2} - \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 4 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \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 ) - {\left (2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )^{2} - 4 \, \cos \left (2 \, b x + 2 \, a\right )^{2} - \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 4 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )^{2} - 4 \, \cos \left (2 \, b x + 2 \, a\right )^{2} - \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 4 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + 4 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 8 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, \cos \left (2 \, b x + 2 \, a\right )}{4 \, {\left (b \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (4 \, b x + 4 \, a\right )^{2} - 4 \, b \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, {\left (2 \, b \cos \left (2 \, b x + 2 \, a\right ) - b\right )} \cos \left (4 \, b x + 4 \, a\right ) - 4 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (26) = 52\).
time = 4.32, size = 65, normalized size = 2.17 \begin {gather*} -\frac {{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) - {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - 1}{4 \, {\left (b \cos \left (b x + a\right )^{2} - b\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 )} \csc {\left (2 a + 2 b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 37, normalized size = 1.23 \begin {gather*} -\frac {\frac {1}{\sin \left (b x + a\right )^{2}} + \log \left (-\sin \left (b x + a\right )^{2} + 1\right ) - 2 \, \log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 36, normalized size = 1.20 \begin {gather*} -\frac {\frac {\ln \left (\cos \left (a+b\,x\right )\right )}{2}-\frac {\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{4}+\frac {1}{4\,{\sin \left (a+b\,x\right )}^2}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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