Optimal. Leaf size=76 \[ \frac {2 b \text {csch}(x)}{a^3}-\frac {\text {csch}^2(x)}{2 a^2}+\frac {\left (a^2+3 b^2\right ) \log (\sinh (x))}{a^4}-\frac {\left (a^2+3 b^2\right ) \log (a+b \sinh (x))}{a^4}+\frac {a^2+b^2}{a^3 (a+b \sinh (x))} \]
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Rubi [A]
time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2800, 908}
\begin {gather*} \frac {2 b \text {csch}(x)}{a^3}-\frac {\text {csch}^2(x)}{2 a^2}+\frac {\left (a^2+3 b^2\right ) \log (\sinh (x))}{a^4}-\frac {\left (a^2+3 b^2\right ) \log (a+b \sinh (x))}{a^4}+\frac {a^2+b^2}{a^3 (a+b \sinh (x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 908
Rule 2800
Rubi steps
\begin {align*} \int \frac {\coth ^3(x)}{(a+b \sinh (x))^2} \, dx &=-\text {Subst}\left (\int \frac {-b^2-x^2}{x^3 (a+x)^2} \, dx,x,b \sinh (x)\right )\\ &=-\text {Subst}\left (\int \left (-\frac {b^2}{a^2 x^3}+\frac {2 b^2}{a^3 x^2}+\frac {-a^2-3 b^2}{a^4 x}+\frac {a^2+b^2}{a^3 (a+x)^2}+\frac {a^2+3 b^2}{a^4 (a+x)}\right ) \, dx,x,b \sinh (x)\right )\\ &=\frac {2 b \text {csch}(x)}{a^3}-\frac {\text {csch}^2(x)}{2 a^2}+\frac {\left (a^2+3 b^2\right ) \log (\sinh (x))}{a^4}-\frac {\left (a^2+3 b^2\right ) \log (a+b \sinh (x))}{a^4}+\frac {a^2+b^2}{a^3 (a+b \sinh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 73, normalized size = 0.96 \begin {gather*} \frac {4 a b \text {csch}(x)-a^2 \text {csch}^2(x)+2 \left (a^2+3 b^2\right ) \log (\sinh (x))-2 \left (a^2+3 b^2\right ) \log (a+b \sinh (x))+\frac {2 a \left (a^2+b^2\right )}{a+b \sinh (x)}}{2 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 142, normalized size = 1.87
method | result | size |
default | \(-\frac {\frac {a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2}+4 b \tanh \left (\frac {x}{2}\right )}{4 a^{3}}-\frac {2 \left (\frac {\left (-a^{2} b -b^{3}\right ) \tanh \left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 b \tanh \left (\frac {x}{2}\right )-a}+\frac {\left (a^{2}+3 b^{2}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 b \tanh \left (\frac {x}{2}\right )-a \right )}{2}\right )}{a^{4}}-\frac {1}{8 a^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (4 a^{2}+12 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tanh \left (\frac {x}{2}\right )}\) | \(142\) |
risch | \(\frac {2 \,{\mathrm e}^{x} \left (a^{2} {\mathrm e}^{4 x}+3 b^{2} {\mathrm e}^{4 x}+3 a b \,{\mathrm e}^{3 x}-4 a^{2} {\mathrm e}^{2 x}-6 b^{2} {\mathrm e}^{2 x}-3 b \,{\mathrm e}^{x} a +a^{2}+3 b^{2}\right )}{a^{3} \left ({\mathrm e}^{2 x}-1\right )^{2} \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{a^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 x}-1\right ) b^{2}}{a^{4}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{a^{2}}-\frac {3 \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) b^{2}}{a^{4}}\) | \(161\) |
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. 202 vs.
\(2 (74) = 148\).
time = 0.29, size = 202, normalized size = 2.66 \begin {gather*} \frac {2 \, {\left (3 \, a b e^{\left (-2 \, x\right )} - 3 \, a b e^{\left (-4 \, x\right )} + {\left (a^{2} + 3 \, b^{2}\right )} e^{\left (-x\right )} - 2 \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} e^{\left (-3 \, x\right )} + {\left (a^{2} + 3 \, b^{2}\right )} e^{\left (-5 \, x\right )}\right )}}{2 \, a^{4} e^{\left (-x\right )} - 3 \, a^{3} b e^{\left (-2 \, x\right )} - 4 \, a^{4} e^{\left (-3 \, x\right )} + 3 \, a^{3} b e^{\left (-4 \, x\right )} + 2 \, a^{4} e^{\left (-5 \, x\right )} - a^{3} b e^{\left (-6 \, x\right )} + a^{3} b} - \frac {{\left (a^{2} + 3 \, b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{4}} + \frac {{\left (a^{2} + 3 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{4}} + \frac {{\left (a^{2} + 3 \, b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{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. 1463 vs.
\(2 (74) = 148\).
time = 0.45, size = 1463, normalized size = 19.25 \begin {gather*} \text {Too large to display} \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 \frac {\coth ^{3}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, 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. 190 vs.
\(2 (74) = 148\).
time = 0.45, size = 190, normalized size = 2.50 \begin {gather*} \frac {{\left (a^{2} + 3 \, b^{2}\right )} \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a^{4}} - \frac {{\left (a^{2} b + 3 \, b^{3}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{4} b} + \frac {a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )} + 3 \, b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} - 4 \, a^{3} - 8 \, a b^{2}}{{\left (b {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )} a^{4}} - \frac {3 \, a^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 9 \, b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 8 \, a b {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, a^{2}}{2 \, a^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.63, size = 1375, normalized size = 18.09 \begin {gather*} \frac {2\,{\mathrm {e}}^x\,\left (a^5\,b^3+2\,a^3\,b^5+a\,b^7\right )}{a^4\,\left (a^2\,b^3+b^5\right )\,\left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )}-\frac {2}{a^2\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {\left (2\,\mathrm {atan}\left (\left (4\,a^9\,b\,\sqrt {{\left (a^2+3\,b^2\right )}^2}\,\sqrt {-a^8}+12\,a^5\,b^5\,\sqrt {{\left (a^2+3\,b^2\right )}^2}\,\sqrt {-a^8}+16\,a^7\,b^3\,\sqrt {{\left (a^2+3\,b^2\right )}^2}\,\sqrt {-a^8}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {{\left (a^2+2\,b^2\right )}^2}{16\,a^{10}\,b^2\,{\left (a^4+4\,a^2\,b^2+3\,b^4\right )}^2}-\frac {1}{16\,a^6\,b^2\,{\left (a^2+3\,b^2\right )}^2\,{\left (a^2+b^2\right )}^2}\right )+\frac {a^2+2\,b^2}{8\,a^9\,b\,{\left (a^4+4\,a^2\,b^2+3\,b^4\right )}^2}+\frac {1}{8\,a^7\,b\,{\left (a^2+3\,b^2\right )}^2\,{\left (a^2+b^2\right )}^2}\right )\right )-2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {-a^8}\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}+2\,b^2\,\sqrt {-a^8}\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}{2\,a^4\,\left (a^4+4\,a^2\,b^2+3\,b^4\right )}+\frac {\left (a^8+3\,a^6\,b^2\right )\,\sqrt {-a^8}}{2\,a^8\,\sqrt {{\left (a^2+3\,b^2\right )}^2}\,\left (a^2+b^2\right )}-\frac {a^8\,b^2\,{\mathrm {e}}^{2\,x}\,\sqrt {-a^8}\,\left (\frac {4\,\left (a^2+2\,b^2\right )\,\left (a^4+6\,a^2\,b^2+9\,b^4\right )}{a^{12}\,b^2\,\left (a^4+4\,a^2\,b^2+3\,b^4\right )}+\frac {4\,\left (a^2\,\sqrt {-a^8}\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}+2\,b^2\,\sqrt {-a^8}\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}\right )\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}{a^{12}\,b^2\,\sqrt {-a^8}\,\left (a^4+4\,a^2\,b^2+3\,b^4\right )}+\frac {2\,\left (2\,a^7\,b+6\,a^5\,b^3\right )\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}{a^{15}\,b^3\,\sqrt {{\left (a^2+3\,b^2\right )}^2}\,\left (a^2+b^2\right )}+\frac {4\,\left (a^8+3\,a^6\,b^2\right )\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}{a^{16}\,b^2\,\sqrt {{\left (a^2+3\,b^2\right )}^2}\,\left (a^2+b^2\right )}\right )}{8\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}+\frac {a^8\,b^2\,{\mathrm {e}}^{3\,x}\,\left (\frac {2\,\left (a^8+3\,a^6\,b^2\right )\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}{a^{15}\,b^3\,\sqrt {{\left (a^2+3\,b^2\right )}^2}\,\left (a^2+b^2\right )}-\frac {2\,\left (a^2+2\,b^2\right )\,\left (a^2\,\sqrt {-a^8}\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}+2\,b^2\,\sqrt {-a^8}\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}\right )\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}{a^{13}\,b^3\,\sqrt {-a^8}\,\left (a^4+4\,a^2\,b^2+3\,b^4\right )}\right )\,\sqrt {-a^8}}{8\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}-\frac {a^8\,b^2\,{\mathrm {e}}^x\,\sqrt {-a^8}\,\left (\frac {8\,\left (a^4+6\,a^2\,b^2+9\,b^4\right )}{a^{11}\,b\,\left (a^4+4\,a^2\,b^2+3\,b^4\right )}-\frac {4\,\left (2\,a^7\,b+6\,a^5\,b^3\right )\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}{a^{16}\,b^2\,\sqrt {{\left (a^2+3\,b^2\right )}^2}\,\left (a^2+b^2\right )}+\frac {2\,\left (a^8+3\,a^6\,b^2\right )\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}{a^{15}\,b^3\,\sqrt {{\left (a^2+3\,b^2\right )}^2}\,\left (a^2+b^2\right )}-\frac {2\,\left (a^2+2\,b^2\right )\,\left (a^2\,\sqrt {-a^8}\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}+2\,b^2\,\sqrt {-a^8}\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}\right )\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}{a^{13}\,b^3\,\sqrt {-a^8}\,\left (a^4+4\,a^2\,b^2+3\,b^4\right )}\right )}{8\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}\right )\right )\,\sqrt {a^4+6\,a^2\,b^2+9\,b^4}}{\sqrt {-a^8}}-\frac {\frac {2}{a^2}-\frac {4\,b\,{\mathrm {e}}^x}{a^3}}{{\mathrm {e}}^{2\,x}-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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