Optimal. Leaf size=28 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac {\csc (a+b x)}{2 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4373, 2701,
327, 213} \begin {gather*} \frac {\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac {\csc (a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 327
Rule 2701
Rule 4373
Rubi steps
\begin {align*} \int \csc (a+b x) \csc (2 a+2 b x) \, dx &=\frac {1}{2} \int \csc ^2(a+b x) \sec (a+b x) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{2 b}\\ &=-\frac {\csc (a+b x)}{2 b}-\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{2 b}\\ &=\frac {\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac {\csc (a+b x)}{2 b}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 29, normalized size = 1.04 \begin {gather*} -\frac {\csc (a+b x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\sin ^2(a+b x)\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 31, normalized size = 1.11
method | result | size |
default | \(\frac {-\frac {1}{\sin \left (x b +a \right )}+\ln \left (\sec \left (x b +a \right )+\tan \left (x b +a \right )\right )}{2 b}\) | \(31\) |
risch | \(-\frac {i {\mathrm e}^{i \left (x b +a \right )}}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}+\frac {\ln \left (i+{\mathrm e}^{i \left (x b +a \right )}\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-i\right )}{2 b}\) | \(66\) |
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. 233 vs.
\(2 (24) = 48\).
time = 0.51, size = 233, normalized size = 8.32 \begin {gather*} -\frac {{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\frac {\cos \left (b x + 2 \, a\right )^{2} + \cos \left (a\right )^{2} - 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} + 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}}{\cos \left (b x + 2 \, a\right )^{2} + \cos \left (a\right )^{2} + 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} - 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) + 4 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 4 \, \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + 4 \, \sin \left (b x + a\right )}{4 \, {\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, 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. 50 vs.
\(2 (24) = 48\).
time = 2.63, size = 50, normalized size = 1.79 \begin {gather*} \frac {\log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 2}{4 \, 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 {\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 = 0.42, size = 38, normalized size = 1.36 \begin {gather*} -\frac {\frac {2}{\sin \left (b x + a\right )} - \log \left (\sin \left (b x + a\right ) + 1\right ) + \log \left (-\sin \left (b x + a\right ) + 1\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 26, normalized size = 0.93 \begin {gather*} \frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{2\,b}-\frac {1}{2\,b\,\sin \left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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