Optimal. Leaf size=60 \[ a^3 x-\frac {(a-2 b) (a+b)^2 \coth (c+d x)}{d}-\frac {(a+b)^3 \coth ^3(c+d x)}{3 d}+\frac {b^3 \tanh (c+d x)}{d} \]
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Rubi [A]
time = 0.07, antiderivative size = 60, 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*} a^3 x-\frac {(a+b)^3 \coth ^3(c+d x)}{3 d}-\frac {(a-2 b) (a+b)^2 \coth (c+d x)}{d}+\frac {b^3 \tanh (c+d x)}{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 )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \left (1-x^2\right )\right )^3}{x^4 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (b^3+\frac {(a+b)^3}{x^4}+\frac {(a-2 b) (a+b)^2}{x^2}-\frac {a^3}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a-2 b) (a+b)^2 \coth (c+d x)}{d}-\frac {(a+b)^3 \coth ^3(c+d x)}{3 d}+\frac {b^3 \tanh (c+d x)}{d}-\frac {a^3 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^3 x-\frac {(a-2 b) (a+b)^2 \coth (c+d x)}{d}-\frac {(a+b)^3 \coth ^3(c+d x)}{3 d}+\frac {b^3 \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(343\) vs. \(2(60)=120\).
time = 1.15, size = 343, normalized size = 5.72 \begin {gather*} \frac {\text {csch}(c) \text {csch}^3(c+d x) \text {sech}(c) \text {sech}(c+d x) \left (6 a^3 d x \cosh (2 d x)-3 a^3 d x \cosh (2 (c+2 d x))-6 a^3 d x \cosh (4 c+2 d x)+3 a^3 d x \cosh (6 c+4 d x)-18 a^2 b \sinh (2 c)-36 a b^2 \sinh (2 c)-4 a^3 \sinh (2 d x)+6 a^2 b \sinh (2 d x)+24 a b^2 \sinh (2 d x)+32 b^3 \sinh (2 d x)-16 a^3 \sinh (2 (c+d x))-12 a^2 b \sinh (2 (c+d x))+24 a b^2 \sinh (2 (c+d x))+8 b^3 \sinh (2 (c+d x))+8 a^3 \sinh (4 (c+d x))+6 a^2 b \sinh (4 (c+d x))-12 a b^2 \sinh (4 (c+d x))-4 b^3 \sinh (4 (c+d x))+8 a^3 \sinh (2 (c+2 d x))+6 a^2 b \sinh (2 (c+2 d x))-12 a b^2 \sinh (2 (c+2 d x))-16 b^3 \sinh (2 (c+2 d x))-12 a^3 \sinh (4 c+2 d x)-18 a^2 b \sinh (4 c+2 d x)\right )}{96 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. \(177\) vs.
\(2(58)=116\).
time = 2.65, size = 178, normalized size = 2.97
method | result | size |
risch | \(a^{3} x -\frac {2 \left (6 a^{3} {\mathrm e}^{6 d x +6 c}+9 a^{2} b \,{\mathrm e}^{6 d x +6 c}+9 a^{2} b \,{\mathrm e}^{4 d x +4 c}+18 a \,b^{2} {\mathrm e}^{4 d x +4 c}-2 a^{3} {\mathrm e}^{2 d x +2 c}+3 a^{2} b \,{\mathrm e}^{2 d x +2 c}+12 a \,b^{2} {\mathrm e}^{2 d x +2 c}+16 b^{3} {\mathrm e}^{2 d x +2 c}+4 a^{3}+3 a^{2} b -6 a \,b^{2}-8 b^{3}\right )}{3 d \left (1+{\mathrm e}^{2 d x +2 c}\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\) | \(178\) |
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. 366 vs.
\(2 (58) = 116\).
time = 0.28, size = 366, normalized size = 6.10 \begin {gather*} \frac {1}{3} \, a^{3} {\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 )} + 4 \, a 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 {16}{3} \, b^{3} {\left (\frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} - \frac {1}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 2 \, a^{2} 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. 354 vs.
\(2 (58) = 116\).
time = 0.36, size = 354, normalized size = 5.90 \begin {gather*} -\frac {{\left (4 \, a^{3} + 3 \, a^{2} b - 6 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} - 4 \, {\left (3 \, a^{3} d x + 4 \, a^{3} + 3 \, a^{2} b - 6 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (4 \, a^{3} + 3 \, a^{2} b - 6 \, a b^{2} - 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{4} + 9 \, a^{2} b + 18 \, a b^{2} + 4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{3} + 6 \, a^{2} b + 6 \, a b^{2} + 8 \, b^{3} + 3 \, {\left (4 \, a^{3} + 3 \, a^{2} b - 6 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 4 \, {\left ({\left (3 \, a^{3} d x + 4 \, a^{3} + 3 \, a^{2} b - 6 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (3 \, a^{3} d x + 4 \, a^{3} + 3 \, a^{2} b - 6 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{12 \, {\left (d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\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. 158 vs.
\(2 (58) = 116\).
time = 0.46, size = 158, normalized size = 2.63 \begin {gather*} \frac {3 \, {\left (d x + c\right )} a^{3} - \frac {6 \, b^{3}}{e^{\left (2 \, d x + 2 \, c\right )} + 1} - \frac {2 \, {\left (6 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a^{3} + 3 \, a^{2} b - 6 \, a b^{2} - 5 \, b^{3}\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 = 1.48, size = 260, normalized size = 4.33 \begin {gather*} a^3\,x-\frac {\frac {2\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^3+3\,a^2\,b-b^3\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^3+3\,a^2\,b-b^3\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2\,a^3+3\,a^2\,b-b^3\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{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\,b^3}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,\left (2\,a^3+3\,a^2\,b-b^3\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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