Optimal. Leaf size=23 \[ -\frac {\coth (a+b x)}{b}-\frac {\tanh (a+b x)}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2700, 14}
\begin {gather*} -\frac {\tanh (a+b x)}{b}-\frac {\coth (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2700
Rubi steps
\begin {align*} \int \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx &=\frac {i \text {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {i \text {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac {\coth (a+b x)}{b}-\frac {\tanh (a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 0.57 \begin {gather*} -\frac {2 \coth (2 (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.88, size = 32, normalized size = 1.39
method | result | size |
risch | \(-\frac {4}{b \left ({\mathrm e}^{2 b x +2 a}+1\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right )}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 18, normalized size = 0.78 \begin {gather*} \frac {4}{b {\left (e^{\left (-4 \, b x - 4 \, a\right )} - 1\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. 81 vs.
\(2 (23) = 46\).
time = 0.38, size = 81, normalized size = 3.52 \begin {gather*} -\frac {4}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - b} \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 \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{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 = 0.41, size = 18, normalized size = 0.78 \begin {gather*} -\frac {4}{b {\left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 18, normalized size = 0.78 \begin {gather*} -\frac {4}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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