Optimal. Leaf size=67 \[ \frac {1}{2} \text {Int}\left (\frac {\text {csch}^2(a+b x) \sec (\coth (a+b x))}{-1+\coth (a+b x)},x\right )-\frac {1}{2} \text {Int}\left (\frac {\text {csch}^2(a+b x) \sec (\coth (a+b x))}{1+\coth (a+b x)},x\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \sec (\coth (a+b x)) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \sec (\coth (a+b x)) \, dx &=\frac {\text {Subst}\left (\int \frac {\sec (x)}{1-x^2} \, dx,x,\coth (a+b x)\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {\sec (x)}{2 (-1+x)}+\frac {\sec (x)}{2 (1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{b}\\ &=-\frac {\text {Subst}\left (\int \frac {\sec (x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {\sec (x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b}\\ \end {align*}
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Mathematica [A]
time = 4.15, size = 0, normalized size = 0.00 \begin {gather*} \int \sec (\coth (a+b x)) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \sec \left (\coth \left (b x +a \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sec {\left (\coth {\left (a + b x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\cos \left (\mathrm {coth}\left (a+b\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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