Optimal. Leaf size=77 \[ \frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\coth (c+d x)}{a d} \]
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Rubi [A]
time = 0.19, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2802, 3135,
3080, 3855, 2739, 632, 210} \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^2 d}+\frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {\coth (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2802
Rule 3080
Rule 3135
Rule 3855
Rubi steps
\begin {align*} \int \frac {\coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\int \frac {\text {csch}^2(c+d x) \left (1+\sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx\\ &=-\frac {\coth (c+d x)}{a d}+\frac {i \int \frac {\text {csch}(c+d x) (i b-i a \sinh (c+d x))}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {\coth (c+d x)}{a d}-\frac {b \int \text {csch}(c+d x) \, dx}{a^2}+\frac {\left (a^2+b^2\right ) \int \frac {1}{a+b \sinh (c+d x)} \, dx}{a^2}\\ &=\frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {\coth (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^2 d}\\ &=\frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {\coth (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^2 d}\\ &=\frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 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^2 d}-\frac {\coth (c+d x)}{a d}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 98, normalized size = 1.27 \begin {gather*} -\frac {4 \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 )+a \coth \left (\frac {1}{2} (c+d x)\right )+2 b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+a \tanh \left (\frac {1}{2} (c+d x)\right )}{2 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.42, size = 105, normalized size = 1.36
method | result | size |
derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {\left (-4 a^{2}-4 b^{2}\right ) \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 a^{2} \sqrt {a^{2}+b^{2}}}}{d}\) | \(105\) |
default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {\left (-4 a^{2}-4 b^{2}\right ) \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 a^{2} \sqrt {a^{2}+b^{2}}}}{d}\) | \(105\) |
risch | \(-\frac {2}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}+\frac {b \ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}-\frac {b \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}+\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,a^{2}}-\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,a^{2}}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 134, normalized size = 1.74 \begin {gather*} \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} 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^{2} d} + \frac {2}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} 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. 360 vs.
\(2 (74) = 148\).
time = 0.37, size = 360, normalized size = 4.68 \begin {gather*} \frac {\sqrt {a^{2} + b^{2}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \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 ) + {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) - 2 \, a}{a^{2} d \cosh \left (d x + c\right )^{2} + 2 \, a^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} d \sinh \left (d x + c\right )^{2} - a^{2} 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 {\coth ^{2}{\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 = 120, normalized size = 1.56 \begin {gather*} \frac {\frac {b \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{2}} - \frac {b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{2}} + \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^{2}} - \frac {2}{a {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.64, size = 380, normalized size = 4.94 \begin {gather*} \frac {2}{a\,d-a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {b\,\ln \left (32\,a^2+32\,b^2-32\,a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-32\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^2\,d}+\frac {b\,\ln \left (32\,a^2+32\,b^2+32\,a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^2\,d}+\frac {\ln \left (128\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-64\,a\,b^3-64\,a^3\,b-32\,b^3\,\sqrt {a^2+b^2}+32\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-64\,a^2\,b\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+128\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}+96\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2\,d}-\frac {\ln \left (32\,b^3\,\sqrt {a^2+b^2}-64\,a\,b^3-64\,a^3\,b+128\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+64\,a^2\,b\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-128\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}-96\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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