Optimal. Leaf size=18 \[ a x-\frac {(a+b) \coth (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4226, 1816,
213} \begin {gather*} a x-\frac {(a+b) \coth (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 213
Rule 1816
Rule 4226
Rubi steps
\begin {align*} \int \coth ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \left (1-x^2\right )}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a+b}{x^2}-\frac {a}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a+b) \coth (c+d x)}{d}-\frac {a \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a x-\frac {(a+b) \coth (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, size = 41, normalized size = 2.28 \begin {gather*} -\frac {b \coth (c+d x)}{d}-\frac {a \coth (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(42\) vs.
\(2(18)=36\).
time = 1.76, size = 43, normalized size = 2.39
method | result | size |
risch | \(a x -\frac {2 a}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {2 b}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs.
\(2 (18) = 36\).
time = 0.29, size = 47, normalized size = 2.61 \begin {gather*} a {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + \frac {2 \, b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (18) = 36\).
time = 0.37, size = 39, normalized size = 2.17 \begin {gather*} -\frac {{\left (a + b\right )} \cosh \left (d x + c\right ) - {\left (a d x + a + b\right )} \sinh \left (d x + c\right )}{d \sinh \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \coth ^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 30, normalized size = 1.67 \begin {gather*} \frac {{\left (d x + c\right )} a - \frac {2 \, {\left (a + b\right )}}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.11, size = 25, normalized size = 1.39 \begin {gather*} a\,x-\frac {2\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________