Optimal. Leaf size=46 \[ a^2 x-\frac {\left (a^2-b^2\right ) \coth (c+d x)}{d}-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 46, 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 = {4226, 1816,
213} \begin {gather*} -\frac {\left (a^2-b^2\right ) \coth (c+d x)}{d}+a^2 x-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 1816
Rule 4226
Rubi steps
\begin {align*} \int \coth ^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \left (1-x^2\right )\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+b)^2}{x^4}+\frac {a^2-b^2}{x^2}-\frac {a^2}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\left (a^2-b^2\right ) \coth (c+d x)}{d}-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d}-\frac {a^2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^2 x-\frac {\left (a^2-b^2\right ) \coth (c+d x)}{d}-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(46)=92\).
time = 0.56, size = 160, normalized size = 3.48 \begin {gather*} \frac {\text {csch}(c) \text {csch}^3(c+d x) \left (9 a^2 d x \cosh (d x)-9 a^2 d x \cosh (2 c+d x)-3 a^2 d x \cosh (2 c+3 d x)+3 a^2 d x \cosh (4 c+3 d x)-12 a^2 \sinh (d x)+12 b^2 \sinh (d x)-12 a^2 \sinh (2 c+d x)-12 a b \sinh (2 c+d x)+8 a^2 \sinh (2 c+3 d x)+4 a b \sinh (2 c+3 d x)-4 b^2 \sinh (2 c+3 d x)\right )}{24 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs.
\(2(44)=88\).
time = 2.16, size = 94, normalized size = 2.04
method | result | size |
risch | \(a^{2} x -\frac {4 \left (3 a^{2} {\mathrm e}^{4 d x +4 c}+3 a b \,{\mathrm e}^{4 d x +4 c}-3 a^{2} {\mathrm e}^{2 d x +2 c}+3 b^{2} {\mathrm e}^{2 d x +2 c}+2 a^{2}+a b -b^{2}\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 268 vs.
\(2 (44) = 88\).
time = 0.27, size = 268, normalized size = 5.83 \begin {gather*} \frac {1}{3} \, a^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\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 )}}\right )} + \frac {4}{3} \, b^{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}{3} \, a b {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs.
\(2 (44) = 88\).
time = 0.39, size = 201, normalized size = 4.37 \begin {gather*} -\frac {2 \, {\left (2 \, a^{2} + a b - b^{2}\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (2 \, a^{2} + a b - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, a^{2} d x + 4 \, a^{2} + 2 \, a b - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (a b + b^{2}\right )} \cosh \left (d x + c\right ) + 3 \, {\left (3 \, a^{2} d x - {\left (3 \, a^{2} d x + 4 \, a^{2} + 2 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 4 \, a^{2} + 2 \, a b - 2 \, b^{2}\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.
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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.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (44) = 88\).
time = 0.45, size = 100, normalized size = 2.17 \begin {gather*} \frac {3 \, {\left (d x + c\right )} a^{2} - \frac {4 \, {\left (3 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} + a b - b^{2}\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.
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Mupad [B]
time = 1.42, size = 183, normalized size = 3.98 \begin {gather*} a^2\,x-\frac {\frac {4\,\left (a^2+b\,a\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^2+b\,a\right )}{3\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^2+a\,b\right )}{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 (b^2+a\,b\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2+b\,a\right )}{3\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {4\,\left (a^2+b\,a\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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