Optimal. Leaf size=43 \[ a^2 x-\frac {(2 a-b) b \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4213, 398, 212}
\begin {gather*} a^2 x-\frac {b (2 a-b) \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 398
Rule 4213
Rubi steps
\begin {align*} \int \left (a+b \text {csch}^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a-b+b x^2\right )^2}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-(2 a-b) b-b^2 x^2+\frac {a^2}{1-x^2}\right ) \, dx,x,\coth (c+d x)\right )}{d}\\ &=-\frac {(2 a-b) b \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=a^2 x-\frac {(2 a-b) b \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 84, normalized size = 1.95 \begin {gather*} \frac {4 \left (a+b \text {csch}^2(c+d x)\right )^2 \left (3 a^2 (c+d x)-b \coth (c+d x) \left (6 a-2 b+b \text {csch}^2(c+d x)\right )\right ) \sinh ^4(c+d x)}{3 d (a-2 b-a \cosh (2 (c+d x)))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.78, size = 69, normalized size = 1.60
method | result | size |
risch | \(a^{2} x -\frac {4 b \left (3 a \,{\mathrm e}^{4 d x +4 c}-6 a \,{\mathrm e}^{2 d x +2 c}+3 b \,{\mathrm e}^{2 d x +2 c}+3 a -b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (41) = 82\).
time = 0.26, size = 121, normalized size = 2.81 \begin {gather*} a^{2} x + \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 \, 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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 180 vs.
\(2 (41) = 82\).
time = 0.50, size = 180, normalized size = 4.19 \begin {gather*} -\frac {2 \, {\left (3 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (3 \, a b - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, a^{2} d x + 6 \, 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 + 6 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 6 \, 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 81, normalized size = 1.88 \begin {gather*} \frac {3 \, {\left (d x + c\right )} a^{2} - \frac {4 \, {\left (3 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, 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 = 1.50, size = 166, normalized size = 3.86 \begin {gather*} a^2\,x-\frac {\frac {4\,a\,b}{3\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b-b^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-b^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.
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