Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\text {sech}(a+b x)}{x},x\right )-\text {Int}\left (\frac {\text {sech}^3(a+b x)}{x},x\right ) \]
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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 = {} \[ \int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x} \, dx &=\int \frac {\text {sech}(a+b x)}{x} \, dx-\int \frac {\text {sech}^3(a+b x)}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 16.71, size = 0, normalized size = 0.00 \[ \int \frac {\text {sech}(a+b x) \tanh ^2(a+b x)}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.88, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\left (b x +a \right )^{3} \left (\sinh ^{2}\left (b x +a \right )\right )}{x}\, dx \]
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 \[ -\frac {{\left (b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} - {\left (b x e^{a} + e^{a}\right )} e^{\left (b x\right )}}{b^{2} x^{2} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} x^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2} x^{2}} + 2 \, \int \frac {{\left (b^{2} x^{2} e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{2 \, {\left (b^{2} x^{3} e^{\left (2 \, b x + 2 \, a\right )} + b^{2} x^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{x\,{\mathrm {cosh}\left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \frac {\sinh ^{2}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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