Optimal. Leaf size=111 \[ -\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}+\frac {2 b \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]
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Rubi [A]
time = 0.37, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2968, 3135,
3134, 3080, 3855, 2739, 632, 210} \begin {gather*} \frac {b \coth (c+d x)}{a^2 d}+\frac {2 b \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^3 d}-\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2968
Rule 3080
Rule 3134
Rule 3135
Rule 3855
Rubi steps
\begin {align*} \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\int \frac {\text {csch}^3(c+d x) \left (1+\sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx\\ &=-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {i \int \frac {\text {csch}^2(c+d x) \left (2 i b-i a \sinh (c+d x)+i b \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{2 a}\\ &=\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\int \frac {\text {csch}(c+d x) \left (-a^2-2 b^2+a b \sinh (c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{2 a^2}\\ &=\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {1}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac {\left (a^2+2 b^2\right ) \int \text {csch}(c+d x) \, dx}{2 a^3}\\ &=-\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {\left (2 i b \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^3 d}\\ &=-\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\left (4 i b \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^3 d}\\ &=-\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}+\frac {2 b \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^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A]
time = 1.02, size = 145, normalized size = 1.31 \begin {gather*} \frac {16 b \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 )+4 a b \coth \left (\frac {1}{2} (c+d x)\right )-a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+4 \left (a^2+2 b^2\right ) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+4 a b \tanh \left (\frac {1}{2} (c+d x)\right )}{8 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.62, size = 140, normalized size = 1.26
method | result | size |
derivativedivides | \(\frac {\frac {\frac {a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {2 b \sqrt {a^{2}+b^{2}}\, \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (2 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(140\) |
default | \(\frac {\frac {\frac {a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {2 b \sqrt {a^{2}+b^{2}}\, \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (2 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(140\) |
risch | \(-\frac {a \,{\mathrm e}^{3 d x +3 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a \,{\mathrm e}^{d x +c}+2 b}{a^{2} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\sqrt {a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}-\frac {\sqrt {a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d a}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{a^{3} d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d a}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{a^{3} d}\) | \(215\) |
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. 217 vs.
\(2 (104) = 208\).
time = 0.48, size = 217, normalized size = 1.95 \begin {gather*} \frac {a e^{\left (-d x - c\right )} + 2 \, b e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-3 \, d x - 3 \, c\right )} - 2 \, b}{{\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} + 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{2 \, a^{3} d} + \frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{2 \, a^{3} d} - \frac {{\left (a^{2} b + b^{3}\right )} \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 )}{\sqrt {a^{2} + b^{2}} 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. 892 vs.
\(2 (104) = 208\).
time = 0.38, size = 892, normalized size = 8.04 \begin {gather*} -\frac {2 \, a^{2} \cosh \left (d x + c\right )^{3} + 2 \, a^{2} \sinh \left (d x + c\right )^{3} - 4 \, a b \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) + 2 \, {\left (3 \, a^{2} \cosh \left (d x + c\right ) - 2 \, a b\right )} \sinh \left (d x + c\right )^{2} - 2 \, {\left (b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} - 2 \, b \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} - b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b\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 ) + 4 \, a b + {\left ({\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 2 \, b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - {\left ({\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 2 \, b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} - 4 \, a b \cosh \left (d x + c\right ) + a^{2}\right )} \sinh \left (d x + c\right )}{2 \, {\left (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 )\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 ^{2}{\left (c + d x \right )} \operatorname {csch}{\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 = 0.45, size = 182, normalized size = 1.64 \begin {gather*} -\frac {\frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{3}} - \frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{3}} + \frac {2 \, {\left (a^{2} b + b^{3}\right )} \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 )}{\sqrt {a^{2} + b^{2}} a^{3}} + \frac {2 \, {\left (a e^{\left (3 \, d x + 3 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a e^{\left (d x + c\right )} + 2 \, b\right )}}{a^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.69, size = 628, normalized size = 5.66 \begin {gather*} \frac {{\mathrm {e}}^{c+d\,x}}{a\,d-a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d-2\,a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+a\,d\,{\mathrm {e}}^{4\,c+4\,d\,x}}-\frac {2\,b}{a^2\,d-a^2\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}+\frac {\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{2\,a\,d}-\frac {\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{2\,a\,d}+\frac {b^2\,\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^3\,d}-\frac {b^2\,\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^3\,d}-\frac {b\,\ln \left (32\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,a\,b^3-16\,a^3\,b-8\,b^3\,\sqrt {a^2+b^2}+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,a^2\,b\,\sqrt {a^2+b^2}+40\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}+24\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^3\,d}+\frac {b\,\ln \left (8\,b^3\,\sqrt {a^2+b^2}-16\,a\,b^3-16\,a^3\,b+32\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^2\,b\,\sqrt {a^2+b^2}+40\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-32\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}-24\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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