Optimal. Leaf size=34 \[ a x-\frac {a \coth (c+d x)}{d}-\frac {(a+b) \coth ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4226, 1816,
213} \begin {gather*} -\frac {(a+b) \coth ^3(c+d x)}{3 d}-\frac {a \coth (c+d x)}{d}+a x \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 ) \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \left (1-x^2\right )}{x^4 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a+b}{x^4}+\frac {a}{x^2}-\frac {a}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a \coth (c+d x)}{d}-\frac {(a+b) \coth ^3(c+d x)}{3 d}-\frac {a \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a x-\frac {a \coth (c+d x)}{d}-\frac {(a+b) \coth ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 49, normalized size = 1.44 \begin {gather*} -\frac {b \coth ^3(c+d x)}{3 d}-\frac {a \coth ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\tanh ^2(c+d x)\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.12, size = 64, normalized size = 1.88
method | result | size |
risch | \(a x -\frac {2 \left (6 a \,{\mathrm e}^{4 d x +4 c}+3 b \,{\mathrm e}^{4 d x +4 c}-6 a \,{\mathrm e}^{2 d x +2 c}+4 a +b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\) | \(64\) |
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. 170 vs.
\(2 (32) = 64\).
time = 0.27, size = 170, normalized size = 5.00 \begin {gather*} \frac {1}{3} \, a {\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 {2}{3} \, 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. 140 vs.
\(2 (32) = 64\).
time = 0.38, size = 140, normalized size = 4.12 \begin {gather*} -\frac {{\left (4 \, a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (4 \, a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, a d x + 4 \, a + b\right )} \sinh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) + 3 \, {\left (3 \, a d x - {\left (3 \, a d x + 4 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 4 \, 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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \coth ^{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. 70 vs.
\(2 (32) = 64\).
time = 0.43, size = 70, normalized size = 2.06 \begin {gather*} \frac {3 \, {\left (d x + c\right )} a - \frac {2 \, {\left (6 \, a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, 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.
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Mupad [B]
time = 1.49, size = 161, normalized size = 4.74 \begin {gather*} a\,x-\frac {\frac {2\,b}{3\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a+b\right )}{3\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {2\,\left (2\,a+b\right )}{3\,d}+\frac {4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (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 {2\,\left (2\,a+b\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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