Optimal. Leaf size=52 \[ -\frac {a (a+b) \text {csch}^2(c+d x)}{d}-\frac {(a+b)^2 \text {csch}^4(c+d x)}{4 d}+\frac {a^2 \log (\sinh (c+d x))}{d} \]
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Rubi [A]
time = 0.07, antiderivative size = 52, 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 = {4223, 455, 45}
\begin {gather*} \frac {a^2 \log (\sinh (c+d x))}{d}-\frac {(a+b)^2 \text {csch}^4(c+d x)}{4 d}-\frac {a (a+b) \text {csch}^2(c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rule 4223
Rubi steps
\begin {align*} \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx &=-\frac {\text {Subst}\left (\int \frac {x \left (b+a x^2\right )^2}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {(b+a x)^2}{(1-x)^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {(a+b)^2}{(-1+x)^3}-\frac {2 a (a+b)}{(-1+x)^2}-\frac {a^2}{-1+x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {a (a+b) \text {csch}^2(c+d x)}{d}-\frac {(a+b)^2 \text {csch}^4(c+d x)}{4 d}+\frac {a^2 \log (\sinh (c+d x))}{d}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 77, normalized size = 1.48 \begin {gather*} -\frac {\left (b+a \cosh ^2(c+d x)\right )^2 \left (4 a (a+b) \text {csch}^2(c+d x)+(a+b)^2 \text {csch}^4(c+d x)-4 a^2 \log (\sinh (c+d x))\right )}{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 = 84, normalized size = 1.62
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}\right )+2 a b \left (-\frac {\cosh ^{2}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{4}}+\frac {1}{4 \sinh \left (d x +c \right )^{4}}\right )-\frac {b^{2}}{4 \sinh \left (d x +c \right )^{4}}}{d}\) | \(84\) |
default | \(\frac {a^{2} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}\right )+2 a b \left (-\frac {\cosh ^{2}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{4}}+\frac {1}{4 \sinh \left (d x +c \right )^{4}}\right )-\frac {b^{2}}{4 \sinh \left (d x +c \right )^{4}}}{d}\) | \(84\) |
risch | \(-a^{2} x -\frac {2 a^{2} c}{d}-\frac {4 \,{\mathrm e}^{2 d x +2 c} \left (a^{2} {\mathrm e}^{4 d x +4 c}+a b \,{\mathrm e}^{4 d x +4 c}-a^{2} {\mathrm e}^{2 d x +2 c}+b^{2} {\mathrm e}^{2 d x +2 c}+a^{2}+a b \right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{2}}{d}\) | \(122\) |
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. 282 vs.
\(2 (50) = 100\).
time = 0.27, size = 282, normalized size = 5.42 \begin {gather*} a^{2} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 4 \, a b {\left (\frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} + \frac {e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - \frac {4 \, b^{2}}{d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{4}} \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. 1252 vs.
\(2 (50) = 100\).
time = 0.70, size = 1252, normalized size = 24.08 \begin {gather*} -\frac {a^{2} d x \cosh \left (d x + c\right )^{8} + 8 \, a^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + a^{2} d x \sinh \left (d x + c\right )^{8} - 4 \, {\left (a^{2} d x - a^{2} - a b\right )} \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{2} - a^{2} d x + a^{2} + a b\right )} \sinh \left (d x + c\right )^{6} + 8 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{3} - 3 \, {\left (a^{2} d x - a^{2} - a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 2 \, {\left (3 \, a^{2} d x - 2 \, a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, a^{2} d x \cosh \left (d x + c\right )^{4} + 3 \, a^{2} d x - 30 \, {\left (a^{2} d x - a^{2} - a b\right )} \cosh \left (d x + c\right )^{2} - 2 \, a^{2} + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + a^{2} d x + 8 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{5} - 10 \, {\left (a^{2} d x - a^{2} - a b\right )} \cosh \left (d x + c\right )^{3} + {\left (3 \, a^{2} d x - 2 \, a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 4 \, {\left (a^{2} d x - a^{2} - a b\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{6} - 15 \, {\left (a^{2} d x - a^{2} - a b\right )} \cosh \left (d x + c\right )^{4} - a^{2} d x + 3 \, {\left (3 \, a^{2} d x - 2 \, a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} + a b\right )} \sinh \left (d x + c\right )^{2} - {\left (a^{2} \cosh \left (d x + c\right )^{8} + 8 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + a^{2} \sinh \left (d x + c\right )^{8} - 4 \, a^{2} \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{2} - a^{2}\right )} \sinh \left (d x + c\right )^{6} + 6 \, a^{2} \cosh \left (d x + c\right )^{4} + 8 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{3} - 3 \, a^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 2 \, {\left (35 \, a^{2} \cosh \left (d x + c\right )^{4} - 30 \, a^{2} \cosh \left (d x + c\right )^{2} + 3 \, a^{2}\right )} \sinh \left (d x + c\right )^{4} - 4 \, a^{2} \cosh \left (d x + c\right )^{2} + 8 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{5} - 10 \, a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{6} - 15 \, a^{2} \cosh \left (d x + c\right )^{4} + 9 \, a^{2} \cosh \left (d x + c\right )^{2} - a^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 8 \, {\left (a^{2} \cosh \left (d x + c\right )^{7} - 3 \, a^{2} \cosh \left (d x + c\right )^{5} + 3 \, a^{2} \cosh \left (d x + c\right )^{3} - a^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 8 \, {\left (a^{2} d x \cosh \left (d x + c\right )^{7} - 3 \, {\left (a^{2} d x - a^{2} - a b\right )} \cosh \left (d x + c\right )^{5} + {\left (3 \, a^{2} d x - 2 \, a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} d x - a^{2} - a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{8} + 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 )^{6} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{6} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{3} - 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 6 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} - 30 \, d \cosh \left (d x + c\right )^{2} + 3 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} - 10 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 4 \, d \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} - 15 \, d \cosh \left (d x + c\right )^{4} + 9 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (d \cosh \left (d x + c\right )^{7} - 3 \, d \cosh \left (d x + c\right )^{5} + 3 \, d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \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. 150 vs.
\(2 (50) = 100\).
time = 0.46, size = 150, normalized size = 2.88 \begin {gather*} -\frac {12 \, {\left (d x + c\right )} a^{2} - 12 \, a^{2} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {25 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} - 52 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 48 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 102 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 52 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 48 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.47, size = 207, normalized size = 3.98 \begin {gather*} \frac {a^2\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )}{d}-\frac {4\,\left (a^2+2\,a\,b+b^2\right )}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {4\,\left (2\,a^2+3\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-a^2\,x-\frac {4\,\left (a^2+b\,a\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {8\,\left (a^2+2\,a\,b+b^2\right )}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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