Optimal. Leaf size=72 \[ \frac {a (a-2 b) \coth (c+d x)}{d}-\frac {a^2 \coth ^3(c+d x)}{3 d}-\frac {(2 a-b) b \tanh (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3744, 459}
\begin {gather*} -\frac {a^2 \coth ^3(c+d x)}{3 d}-\frac {b (2 a-b) \tanh (c+d x)}{d}+\frac {a (a-2 b) \coth (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rule 3744
Rubi steps
\begin {align*} \int \text {csch}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (a+b x^2\right )^2}{x^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (b (-2 a+b)+\frac {a^2}{x^4}-\frac {a (a-2 b)}{x^2}-b^2 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 {(2 a-b) b \tanh (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 59, normalized size = 0.82 \begin {gather*} \frac {-a \coth (c+d x) \left (-2 a+6 b+a \text {csch}^2(c+d x)\right )+b \left (-6 a+2 b+b \text {sech}^2(c+d x)\right ) \tanh (c+d x)}{3 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. \(156\) vs.
\(2(68)=136\).
time = 2.55, size = 157, normalized size = 2.18
method | result | size |
risch | \(-\frac {4 \left (3 a^{2} {\mathrm e}^{8 d x +8 c}+6 a b \,{\mathrm e}^{8 d x +8 c}+3 b^{2} {\mathrm e}^{8 d x +8 c}+8 a^{2} {\mathrm e}^{6 d x +6 c}-8 b^{2} {\mathrm e}^{6 d x +6 c}+6 a^{2} {\mathrm e}^{4 d x +4 c}-12 a b \,{\mathrm e}^{4 d x +4 c}+6 b^{2} {\mathrm e}^{4 d x +4 c}-a^{2}+6 a b -b^{2}\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3} \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}\) | \(157\) |
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. 210 vs.
\(2 (68) = 136\).
time = 0.28, size = 210, normalized size = 2.92 \begin {gather*} \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^{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 {8 \, a b}{d {\left (e^{\left (-4 \, d x - 4 \, 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. 393 vs.
\(2 (68) = 136\).
time = 0.34, size = 393, normalized size = 5.46 \begin {gather*} -\frac {8 \, {\left ({\left (a^{2} + 6 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 8 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + 6 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 6 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, a^{2} - 6 \, a b + 3 \, b^{2} + 8 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{3 \, {\left (d \cosh \left (d x + c\right )^{8} + 56 \, d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{5} + 28 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{6} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} - 4 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} - 2 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} - 6 \, d \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (d \cosh \left (d x + c\right )^{7} - d \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + 3 \, d\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 \tanh ^{2}{\left (c + d x \right )}\right )^{2} \operatorname {csch}^{4}{\left (c + d x \right )}\, dx \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. 143 vs.
\(2 (68) = 136\).
time = 0.46, size = 143, normalized size = 1.99 \begin {gather*} -\frac {4 \, {\left (3 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 6 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 3 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 8 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} - 8 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 6 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 12 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - a^{2} + 6 \, a b - b^{2}\right )}}{3 \, d {\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.08, size = 143, normalized size = 1.99 \begin {gather*} -\frac {4\,\left (6\,a\,b-a^2-b^2+6\,a^2\,{\mathrm {e}}^{4\,c+4\,d\,x}+8\,a^2\,{\mathrm {e}}^{6\,c+6\,d\,x}+3\,a^2\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}-8\,b^2\,{\mathrm {e}}^{6\,c+6\,d\,x}+3\,b^2\,{\mathrm {e}}^{8\,c+8\,d\,x}-12\,a\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}+6\,a\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}\right )}{3\,d\,{\left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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