Optimal. Leaf size=111 \[ \frac {b \coth (c+d x)}{a^2 d}+\frac {2 b \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]
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Rubi [A] time = 0.57, 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 = {2889, 3056, 3055, 3001, 3770, 2660, 618, 204} \[ \frac {2 b \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2889
Rule 3001
Rule 3055
Rule 3056
Rule 3770
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 ) \operatorname {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 ) \operatorname {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 = 1.22, size = 145, normalized size = 1.31 \[ \frac {4 \left (a^2+2 b^2\right ) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+16 b \sqrt {-a^2-b^2} \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )-a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+4 a b \tanh \left (\frac {1}{2} (c+d x)\right )+4 a b \coth \left (\frac {1}{2} (c+d x)\right )}{8 a^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 892, normalized size = 8.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.85, size = 221, normalized size = 1.99 \[ -\frac {\frac {{\left (a^{2} e^{c} + 2 \, b^{2} e^{c}\right )} e^{\left (-c\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{3}} - \frac {{\left (a^{2} e^{c} + 2 \, b^{2} e^{c}\right )} e^{\left (-c\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{3}} + \frac {2 \, {\left (a^{2} b e^{c} + b^{3} e^{c}\right )} e^{\left (-c\right )} \log \left (\frac {{\left | 2 \, b e^{\left (d x + 2 \, c\right )} + 2 \, a e^{c} - 2 \, \sqrt {a^{2} + b^{2}} e^{c} \right |}}{{\left | 2 \, b e^{\left (d x + 2 \, c\right )} + 2 \, a e^{c} + 2 \, \sqrt {a^{2} + b^{2}} e^{c} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} + \frac {2 \, {\left (a e^{\left (3 \, d x + 3 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a e^{\left (d x + c\right )} + 2 \, b\right )}}{a^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 162, normalized size = 1.46 \[ \frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{2 d \,a^{2}}-\frac {2 b \sqrt {a^{2}+b^{2}}\, \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,a^{3}}-\frac {1}{8 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{d \,a^{3}}+\frac {b}{2 d \,a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 217, normalized size = 1.95 \[ \frac {a e^{\left (-d x - c\right )} + 2 \, b e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-3 \, d x - 3 \, c\right )} - 2 \, b}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} - \frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{2 \, a^{3} d} + \frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{2 \, a^{3} d} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{3} d} \]
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 \[ \frac {{\mathrm {e}}^{c+d\,x}}{a\,d-a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d-2\,a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+a\,d\,{\mathrm {e}}^{4\,c+4\,d\,x}}-\frac {2\,b}{a^2\,d-a^2\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}+\frac {\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{2\,a\,d}-\frac {\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{2\,a\,d}+\frac {b^2\,\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^3\,d}-\frac {b^2\,\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^3\,d}-\frac {b\,\ln \left (32\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,a\,b^3-16\,a^3\,b-8\,b^3\,\sqrt {a^2+b^2}+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,a^2\,b\,\sqrt {a^2+b^2}+40\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}+24\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^3\,d}+\frac {b\,\ln \left (8\,b^3\,\sqrt {a^2+b^2}-16\,a\,b^3-16\,a^3\,b+32\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^2\,b\,\sqrt {a^2+b^2}+40\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-32\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}-24\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^3\,d} \]
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 \[ \int \frac {\coth ^{2}{\left (c + d x \right )} \operatorname {csch}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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