Optimal. Leaf size=40 \[ b^2 x+\frac {a (a-2 b) \coth (c+d x)}{d}-\frac {a^2 \coth ^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3266, 472, 213}
\begin {gather*} -\frac {a^2 \coth ^3(c+d x)}{3 d}+\frac {a (a-2 b) \coth (c+d x)}{d}+b^2 x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 213
Rule 472
Rule 3266
Rubi steps
\begin {align*} \int \text {csch}^4(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^4 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^2}{x^4}-\frac {a (a-2 b)}{x^2}-\frac {b^2}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a (a-2 b) \coth (c+d x)}{d}-\frac {a^2 \coth ^3(c+d x)}{3 d}-\frac {b^2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=b^2 x+\frac {a (a-2 b) \coth (c+d x)}{d}-\frac {a^2 \coth ^3(c+d x)}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(40)=80\).
time = 0.49, size = 85, normalized size = 2.12 \begin {gather*} \frac {4 \left (b+a \text {csch}^2(c+d x)\right )^2 \left (3 b^2 (c+d x)-a \coth (c+d x) \left (-2 a+6 b+a \text {csch}^2(c+d x)\right )\right ) \sinh ^4(c+d x)}{3 d (2 a-b+b \cosh (2 (c+d x)))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.36, size = 69, normalized size = 1.72
method | result | size |
risch | \(b^{2} x -\frac {4 a \left (3 b \,{\mathrm e}^{4 d x +4 c}+3 a \,{\mathrm e}^{2 d x +2 c}-6 b \,{\mathrm e}^{2 d x +2 c}-a +3 b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\) | \(69\) |
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. 121 vs.
\(2 (38) = 76\).
time = 0.28, size = 121, normalized size = 3.02 \begin {gather*} b^{2} x + \frac {4}{3} \, a^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {4 \, a 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. 174 vs.
\(2 (38) = 76\).
time = 0.42, size = 174, normalized size = 4.35 \begin {gather*} \frac {2 \, {\left (a^{2} - 3 \, a b\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (a^{2} - 3 \, a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (3 \, b^{2} d x - 2 \, a^{2} + 6 \, a b\right )} \sinh \left (d x + c\right )^{3} - 6 \, {\left (a^{2} - a b\right )} \cosh \left (d x + c\right ) - 3 \, {\left (3 \, b^{2} d x - {\left (3 \, b^{2} d x - 2 \, a^{2} + 6 \, a b\right )} \cosh \left (d x + c\right )^{2} - 2 \, a^{2} + 6 \, a b\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \sinh \left (d x + c\right )^{3} + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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. 81 vs.
\(2 (38) = 76\).
time = 0.43, size = 81, normalized size = 2.02 \begin {gather*} \frac {3 \, {\left (d x + c\right )} b^{2} - \frac {4 \, {\left (3 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} - a^{2} + 3 \, a b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.62, size = 166, normalized size = 4.15 \begin {gather*} b^2\,x-\frac {\frac {4\,a\,b}{3\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b-a^2\right )}{3\,d}+\frac {4\,a\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}+\frac {\frac {4\,\left (a\,b-a^2\right )}{3\,d}-\frac {4\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {4\,a\,b}{3\,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]
________________________________________________________________________________________