Optimal. Leaf size=13 \[ x-\frac {\coth (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 8} \[ x-\frac {\coth (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \coth ^2(a+b x) \, dx &=-\frac {\coth (a+b x)}{b}+\int 1 \, dx\\ &=x-\frac {\coth (a+b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 27, normalized size = 2.08 \[ -\frac {\coth (a+b x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.40, size = 33, normalized size = 2.54 \[ \frac {{\left (b x + 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \sinh \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 24, normalized size = 1.85 \[ \frac {b x + a - \frac {2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.12, size = 18, normalized size = 1.38 \[ \frac {b x +a -\coth \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 25, normalized size = 1.92 \[ x + \frac {a}{b} + \frac {2}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 20, normalized size = 1.54 \[ x-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh ^{2}{\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________