Optimal. Leaf size=71 \[ \frac {x}{b}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a b d} \]
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Rubi [A]
time = 0.16, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2968, 3137,
2739, 632, 210, 3855} \begin {gather*} \frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a b d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {x}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2968
Rule 3137
Rule 3855
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx &=\int \frac {\text {csch}(c+d x) \left (1+\sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx\\ &=\frac {x}{b}+\frac {\int \text {csch}(c+d x) \, dx}{a}-\frac {\left (a^2+b^2\right ) \int \frac {1}{a+b \sinh (c+d x)} \, dx}{a b}\\ &=\frac {x}{b}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {\left (2 i \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a b d}\\ &=\frac {x}{b}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}-\frac {\left (4 i \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a b d}\\ &=\frac {x}{b}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a b d}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 80, normalized size = 1.13 \begin {gather*} \frac {a (c+d x)+2 \sqrt {-a^2-b^2} \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )+b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.54, size = 109, normalized size = 1.54
method | result | size |
derivativedivides | \(\frac {-\frac {\left (2 a^{2}+2 b^{2}\right ) \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a b \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}}{d}\) | \(109\) |
default | \(\frac {-\frac {\left (2 a^{2}+2 b^{2}\right ) \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a b \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}}{d}\) | \(109\) |
risch | \(\frac {x}{b}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}+\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d b a}-\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d b a}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 126, normalized size = 1.77 \begin {gather*} \frac {d x + c}{b d} - \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{a b 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. 209 vs.
\(2 (68) = 136\).
time = 0.41, size = 209, normalized size = 2.94 \begin {gather*} \frac {a d x - b \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + b \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right )}{a b d} \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 {\cosh {\left (c + d x \right )} \coth {\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 [A]
time = 0.47, size = 113, normalized size = 1.59 \begin {gather*} \frac {\frac {d x + c}{b} - \frac {\log \left (e^{\left (d x + c\right )} + 1\right )}{a} + \frac {\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{a b}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.47, size = 384, normalized size = 5.41 \begin {gather*} \frac {x}{b}+\frac {\ln \left (32\,a\,b^2+32\,a^3-32\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-32\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a\,d}-\frac {\ln \left (32\,a\,b^2+32\,a^3+32\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a\,d}-\frac {\ln \left (128\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-64\,a^2\,b^3-64\,a^4\,b-32\,a\,b^3\,\sqrt {a^2+b^2}-64\,a^3\,b\,\sqrt {a^2+b^2}+160\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+128\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}+32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+96\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a\,b\,d}+\frac {\ln \left (128\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-64\,a^2\,b^3-64\,a^4\,b+32\,a\,b^3\,\sqrt {a^2+b^2}+64\,a^3\,b\,\sqrt {a^2+b^2}+160\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-128\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}+32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-96\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a\,b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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