Optimal. Leaf size=50 \[ \frac {1}{2} (4 a-b) b x-\frac {a^2 \coth (c+d x)}{d}+\frac {b^2 \cosh (c+d x) \sinh (c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 64, normalized size of antiderivative = 1.28, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3266, 473, 393,
212} \begin {gather*} \frac {\left (2 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {a^2 \cosh ^2(c+d x) \coth (c+d x)}{d}+\frac {1}{2} b x (4 a-b) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 393
Rule 473
Rule 3266
Rubi steps
\begin {align*} \int \text {csch}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a-(a-b) x^2\right )^2}{x^2 \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a^2 \cosh ^2(c+d x) \coth (c+d x)}{d}+\frac {\text {Subst}\left (\int \frac {a (a+2 b)+(a-b)^2 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a^2 \cosh ^2(c+d x) \coth (c+d x)}{d}+\frac {\left (2 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {((4 a-b) b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {1}{2} (4 a-b) b x-\frac {a^2 \cosh ^2(c+d x) \coth (c+d x)}{d}+\frac {\left (2 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 56, normalized size = 1.12 \begin {gather*} 2 a b x+\frac {b^2 (-c-d x)}{2 d}-\frac {a^2 \coth (c+d x)}{d}+\frac {b^2 \sinh (2 (c+d x))}{4 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.29, size = 68, normalized size = 1.36
method | result | size |
risch | \(2 a b x -\frac {b^{2} x}{2}+\frac {{\mathrm e}^{2 d x +2 c} b^{2}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} b^{2}}{8 d}-\frac {2 a^{2}}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 63, normalized size = 1.26 \begin {gather*} -\frac {1}{8} \, b^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + 2 \, a b x + \frac {2 \, a^{2}}{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 [A]
time = 0.49, size = 89, normalized size = 1.78 \begin {gather*} \frac {b^{2} \cosh \left (d x + c\right )^{3} + 3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (8 \, a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) + 4 \, {\left ({\left (4 \, a b - b^{2}\right )} d x + 2 \, a^{2}\right )} \sinh \left (d x + c\right )}{8 \, d \sinh \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs.
\(2 (46) = 92\).
time = 0.43, size = 135, normalized size = 2.70 \begin {gather*} \frac {b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, {\left (4 \, a b - b^{2}\right )} {\left (d x + c\right )} - \frac {4 \, a b e^{\left (4 \, d x + 4 \, c\right )} - b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{2}}{e^{\left (4 \, d x + 4 \, c\right )} - e^{\left (2 \, d x + 2 \, c\right )}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.66, size = 67, normalized size = 1.34 \begin {gather*} \frac {b\,x\,\left (4\,a-b\right )}{2}-\frac {2\,a^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {b^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}+\frac {b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________