Optimal. Leaf size=80 \[ \frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {\left (a^2+b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^3 d} \]
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Rubi [A]
time = 0.08, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2800, 908}
\begin {gather*} \frac {b \text {csch}(c+d x)}{a^2 d}+\frac {\left (a^2+b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^3 d}-\frac {\text {csch}^2(c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 908
Rule 2800
Rubi steps
\begin {align*} \int \frac {\coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {-b^2-x^2}{x^3 (a+x)} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {b^2}{a x^3}+\frac {b^2}{a^2 x^2}+\frac {-a^2-b^2}{a^3 x}+\frac {a^2+b^2}{a^3 (a+x)}\right ) \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {\left (a^2+b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 64, normalized size = 0.80 \begin {gather*} \frac {2 a b \text {csch}(c+d x)-a^2 \text {csch}^2(c+d x)+2 \left (a^2+b^2\right ) (\log (\sinh (c+d x))-\log (a+b \sinh (c+d x)))}{2 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.15, size = 143, normalized size = 1.79
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (4 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\left (-4 a^{2}-4 b^{2}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{4 a^{3}}}{d}\) | \(143\) |
default | \(\frac {-\frac {\frac {a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (4 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\left (-4 a^{2}-4 b^{2}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{4 a^{3}}}{d}\) | \(143\) |
risch | \(-\frac {2 \,{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{2 d x +2 c}+a \,{\mathrm e}^{d x +c}+b \right )}{a^{2} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a d}+\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{3} d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a^{3} d}\) | \(159\) |
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. 173 vs.
\(2 (78) = 156\).
time = 0.28, size = 173, normalized size = 2.16 \begin {gather*} -\frac {2 \, {\left (b e^{\left (-d x - c\right )} - a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} - \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{3} d} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \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. 617 vs.
\(2 (78) = 156\).
time = 0.35, size = 617, normalized size = 7.71 \begin {gather*} \frac {2 \, a b \cosh \left (d x + c\right )^{3} + 2 \, a b \sinh \left (d x + c\right )^{3} - 2 \, a^{2} \cosh \left (d x + c\right )^{2} - 2 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (3 \, a b \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )^{2} - {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + b^{2} + 4 \, {\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 )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + b^{2} + 4 \, {\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 )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (3 \, a b \cosh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )}{a^{3} d \cosh \left (d x + c\right )^{4} + 4 \, a^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} d \sinh \left (d x + c\right )^{4} - 2 \, a^{3} d \cosh \left (d x + c\right )^{2} + a^{3} d + 2 \, {\left (3 \, a^{3} d \cosh \left (d x + c\right )^{2} - a^{3} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} d \cosh \left (d x + c\right )^{3} - a^{3} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\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 \frac {\coth ^{3}{\left (c + d x \right )}}{a + b \sinh {\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. 184 vs.
\(2 (78) = 156\).
time = 0.48, size = 184, normalized size = 2.30 \begin {gather*} \frac {\frac {2 \, {\left (a^{2} + b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{3}} - \frac {2 \, {\left (a^{2} b + b^{3}\right )} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{3} b} - \frac {3 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 3 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 4 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 4 \, a^{2}}{a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.01, size = 1329, normalized size = 16.61 \begin {gather*} \frac {\left (2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {-a^6\,d^2}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6\,d^2}\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{2\,a^3\,d\,{\left (a^2+b^2\right )}^2}+\frac {\left (d\,a^7+d\,a^5\,b^2\right )\,\sqrt {-a^6\,d^2}}{2\,a^6\,d^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}-\frac {a^6\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-a^6\,d^2}\,\left (\frac {4\,\left (a^2+2\,b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}{a^9\,b^2\,d\,{\left (a^2+b^2\right )}^2}+\frac {2\,\left (2\,d\,a^6\,b+2\,d\,a^4\,b^3\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,d^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}+\frac {4\,\left (a^2\,\sqrt {-a^6\,d^2}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6\,d^2}\,\sqrt {a^4+2\,a^2\,b^2+b^4}\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^9\,b^2\,d\,{\left (a^2+b^2\right )}^2\,\sqrt {-a^6\,d^2}}+\frac {4\,\left (d\,a^7+d\,a^5\,b^2\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{12}\,b^2\,d^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}\right )}{8\,\sqrt {a^4+2\,a^2\,b^2+b^4}}+\frac {a^6\,b^2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (\frac {2\,\left (d\,a^7+d\,a^5\,b^2\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,d^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}-\frac {2\,\left (a^2+2\,b^2\right )\,\left (a^2\,\sqrt {-a^6\,d^2}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6\,d^2}\,\sqrt {a^4+2\,a^2\,b^2+b^4}\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{10}\,b^3\,d\,{\left (a^2+b^2\right )}^2\,\sqrt {-a^6\,d^2}}\right )\,\sqrt {-a^6\,d^2}}{8\,\sqrt {a^4+2\,a^2\,b^2+b^4}}-\frac {a^6\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-a^6\,d^2}\,\left (\frac {8\,\left (a^4+2\,a^2\,b^2+b^4\right )}{a^8\,b\,d\,{\left (a^2+b^2\right )}^2}-\frac {4\,\left (2\,d\,a^6\,b+2\,d\,a^4\,b^3\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{12}\,b^2\,d^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}+\frac {2\,\left (d\,a^7+d\,a^5\,b^2\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,d^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}-\frac {2\,\left (a^2+2\,b^2\right )\,\left (a^2\,\sqrt {-a^6\,d^2}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6\,d^2}\,\sqrt {a^4+2\,a^2\,b^2+b^4}\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{10}\,b^3\,d\,{\left (a^2+b^2\right )}^2\,\sqrt {-a^6\,d^2}}\right )}{8\,\sqrt {a^4+2\,a^2\,b^2+b^4}}\right )-2\,\mathrm {atan}\left (\left (4\,a^6\,b\,d\,{\left (a^2+b^2\right )}^2\,\sqrt {-a^6\,d^2}+4\,a^4\,b^3\,d\,{\left (a^2+b^2\right )}^2\,\sqrt {-a^6\,d^2}\right )\,\left (\frac {1}{8\,a^5\,b\,d^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,{\left (a^2+b^2\right )}^3}-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {1}{16\,a^4\,b^2\,d^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,{\left (a^2+b^2\right )}^3}-\frac {{\left (a^2+2\,b^2\right )}^2}{16\,a^8\,b^2\,d^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,{\left (a^2+b^2\right )}^3}\right )+\frac {a^2+2\,b^2}{8\,a^7\,b\,d^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,{\left (a^2+b^2\right )}^3}\right )\right )\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{\sqrt {-a^6\,d^2}}-\frac {\frac {2}{a\,d}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{a^2\,d}}{{\mathrm {e}}^{2\,c+2\,d\,x}-1}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\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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