3.234 \(\int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx\)

Optimal. Leaf size=52 \[ \frac {b \text {csch}(x)}{a^2}+\frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{a^3}-\frac {\text {csch}^2(x)}{2 a} \]

[Out]

b*csch(x)/a^2-1/2*csch(x)^2/a+(a^2+b^2)*ln(sinh(x))/a^3-(a^2+b^2)*ln(a+b*sinh(x))/a^3

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Rubi [A]  time = 0.09, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2721, 894} \[ \frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{a^3}+\frac {b \text {csch}(x)}{a^2}-\frac {\text {csch}^2(x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^3/(a + b*Sinh[x]),x]

[Out]

(b*Csch[x])/a^2 - Csch[x]^2/(2*a) + ((a^2 + b^2)*Log[Sinh[x]])/a^3 - ((a^2 + b^2)*Log[a + b*Sinh[x]])/a^3

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2721

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rubi steps

\begin {align*} \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {-b^2-x^2}{x^3 (a+x)} \, dx,x,b \sinh (x)\right )\\ &=-\operatorname {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 (x)\right )\\ &=\frac {b \text {csch}(x)}{a^2}-\frac {\text {csch}^2(x)}{2 a}+\frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{a^3}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 45, normalized size = 0.87 \[ \frac {2 \left (a^2+b^2\right ) (\log (\sinh (x))-\log (a+b \sinh (x)))-a^2 \text {csch}^2(x)+2 a b \text {csch}(x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^3/(a + b*Sinh[x]),x]

[Out]

(2*a*b*Csch[x] - a^2*Csch[x]^2 + 2*(a^2 + b^2)*(Log[Sinh[x]] - Log[a + b*Sinh[x]]))/(2*a^3)

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fricas [B]  time = 0.75, size = 427, normalized size = 8.21 \[ \frac {2 \, a b \cosh \relax (x)^{3} + 2 \, a b \sinh \relax (x)^{3} - 2 \, a^{2} \cosh \relax (x)^{2} - 2 \, a b \cosh \relax (x) + 2 \, {\left (3 \, a b \cosh \relax (x) - a^{2}\right )} \sinh \relax (x)^{2} - {\left ({\left (a^{2} + b^{2}\right )} \cosh \relax (x)^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (a^{2} + b^{2}\right )} \sinh \relax (x)^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \relax (x)^{2} - a^{2} - b^{2}\right )} \sinh \relax (x)^{2} + a^{2} + b^{2} + 4 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \relax (x)^{3} - {\left (a^{2} + b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, {\left (b \sinh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + {\left ({\left (a^{2} + b^{2}\right )} \cosh \relax (x)^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (a^{2} + b^{2}\right )} \sinh \relax (x)^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \relax (x)^{2} - a^{2} - b^{2}\right )} \sinh \relax (x)^{2} + a^{2} + b^{2} + 4 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \relax (x)^{3} - {\left (a^{2} + b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (3 \, a b \cosh \relax (x)^{2} - 2 \, a^{2} \cosh \relax (x) - a b\right )} \sinh \relax (x)}{a^{3} \cosh \relax (x)^{4} + 4 \, a^{3} \cosh \relax (x) \sinh \relax (x)^{3} + a^{3} \sinh \relax (x)^{4} - 2 \, a^{3} \cosh \relax (x)^{2} + a^{3} + 2 \, {\left (3 \, a^{3} \cosh \relax (x)^{2} - a^{3}\right )} \sinh \relax (x)^{2} + 4 \, {\left (a^{3} \cosh \relax (x)^{3} - a^{3} \cosh \relax (x)\right )} \sinh \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*sinh(x)),x, algorithm="fricas")

[Out]

(2*a*b*cosh(x)^3 + 2*a*b*sinh(x)^3 - 2*a^2*cosh(x)^2 - 2*a*b*cosh(x) + 2*(3*a*b*cosh(x) - a^2)*sinh(x)^2 - ((a
^2 + b^2)*cosh(x)^4 + 4*(a^2 + b^2)*cosh(x)*sinh(x)^3 + (a^2 + b^2)*sinh(x)^4 - 2*(a^2 + b^2)*cosh(x)^2 + 2*(3
*(a^2 + b^2)*cosh(x)^2 - a^2 - b^2)*sinh(x)^2 + a^2 + b^2 + 4*((a^2 + b^2)*cosh(x)^3 - (a^2 + b^2)*cosh(x))*si
nh(x))*log(2*(b*sinh(x) + a)/(cosh(x) - sinh(x))) + ((a^2 + b^2)*cosh(x)^4 + 4*(a^2 + b^2)*cosh(x)*sinh(x)^3 +
(a^2 + b^2)*sinh(x)^4 - 2*(a^2 + b^2)*cosh(x)^2 + 2*(3*(a^2 + b^2)*cosh(x)^2 - a^2 - b^2)*sinh(x)^2 + a^2 + b
^2 + 4*((a^2 + b^2)*cosh(x)^3 - (a^2 + b^2)*cosh(x))*sinh(x))*log(2*sinh(x)/(cosh(x) - sinh(x))) + 2*(3*a*b*co
sh(x)^2 - 2*a^2*cosh(x) - a*b)*sinh(x))/(a^3*cosh(x)^4 + 4*a^3*cosh(x)*sinh(x)^3 + a^3*sinh(x)^4 - 2*a^3*cosh(
x)^2 + a^3 + 2*(3*a^3*cosh(x)^2 - a^3)*sinh(x)^2 + 4*(a^3*cosh(x)^3 - a^3*cosh(x))*sinh(x))

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giac [B]  time = 0.27, size = 125, normalized size = 2.40 \[ \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a^{3}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{3} b} - \frac {3 \, a^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 3 \, b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4 \, a b {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, a^{2}}{2 \, a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*sinh(x)),x, algorithm="giac")

[Out]

(a^2 + b^2)*log(abs(-e^(-x) + e^x))/a^3 - (a^2*b + b^3)*log(abs(-b*(e^(-x) - e^x) + 2*a))/(a^3*b) - 1/2*(3*a^2
*(e^(-x) - e^x)^2 + 3*b^2*(e^(-x) - e^x)^2 + 4*a*b*(e^(-x) - e^x) + 4*a^2)/(a^3*(e^(-x) - e^x)^2)

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maple [B]  time = 0.07, size = 120, normalized size = 2.31 \[ -\frac {\tanh ^{2}\left (\frac {x}{2}\right )}{8 a}-\frac {\tanh \left (\frac {x}{2}\right ) b}{2 a^{2}}-\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}{a}-\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right ) b^{2}}{a^{3}}-\frac {1}{8 a \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right ) b^{2}}{a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {x}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3/(a+b*sinh(x)),x)

[Out]

-1/8/a*tanh(1/2*x)^2-1/2/a^2*tanh(1/2*x)*b-1/a*ln(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)-1/a^3*ln(a*tanh(1/2*x)^2-
2*tanh(1/2*x)*b-a)*b^2-1/8/a/tanh(1/2*x)^2+1/a*ln(tanh(1/2*x))+1/a^3*ln(tanh(1/2*x))*b^2+1/2*b/a^2/tanh(1/2*x)

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maxima [B]  time = 0.35, size = 116, normalized size = 2.23 \[ -\frac {2 \, {\left (b e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - b e^{\left (-3 \, x\right )}\right )}}{2 \, a^{2} e^{\left (-2 \, x\right )} - a^{2} e^{\left (-4 \, x\right )} - a^{2}} - \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*sinh(x)),x, algorithm="maxima")

[Out]

-2*(b*e^(-x) - a*e^(-2*x) - b*e^(-3*x))/(2*a^2*e^(-2*x) - a^2*e^(-4*x) - a^2) - (a^2 + b^2)*log(-2*a*e^(-x) +
b*e^(-2*x) - b)/a^3 + (a^2 + b^2)*log(e^(-x) + 1)/a^3 + (a^2 + b^2)*log(e^(-x) - 1)/a^3

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mupad [B]  time = 1.24, size = 1163, normalized size = 22.37 \[ \frac {\left (2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {-a^6}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6}\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{2\,a^3\,{\left (a^2+b^2\right )}^2}+\frac {\left (a^7+a^5\,b^2\right )\,\sqrt {-a^6}}{2\,a^6\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}-\frac {a^6\,b^2\,{\mathrm {e}}^x\,\sqrt {-a^6}\,\left (\frac {8\,\left (a^4+2\,a^2\,b^2+b^4\right )}{a^8\,b\,{\left (a^2+b^2\right )}^2}-\frac {4\,\left (2\,a^6\,b+2\,a^4\,b^3\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{12}\,b^2\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}+\frac {2\,\left (a^7+a^5\,b^2\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,\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}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6}\,\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\,\sqrt {-a^6}\,{\left (a^2+b^2\right )}^2}\right )}{8\,\sqrt {a^4+2\,a^2\,b^2+b^4}}-\frac {a^6\,b^2\,{\mathrm {e}}^{2\,x}\,\sqrt {-a^6}\,\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\,{\left (a^2+b^2\right )}^2}+\frac {4\,\left (a^2\,\sqrt {-a^6}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6}\,\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\,\sqrt {-a^6}\,{\left (a^2+b^2\right )}^2}+\frac {2\,\left (2\,a^6\,b+2\,a^4\,b^3\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,\sqrt {{\left (a^2+b^2\right )}^2}\,\left (a^2+b^2\right )}+\frac {4\,\left (a^7+a^5\,b^2\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{12}\,b^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\,x}\,\left (\frac {2\,\left (a^7+a^5\,b^2\right )\,\sqrt {a^4+2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,\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}\,\sqrt {a^4+2\,a^2\,b^2+b^4}+2\,b^2\,\sqrt {-a^6}\,\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\,\sqrt {-a^6}\,{\left (a^2+b^2\right )}^2}\right )\,\sqrt {-a^6}}{8\,\sqrt {a^4+2\,a^2\,b^2+b^4}}\right )-2\,\mathrm {atan}\left (\left (4\,a^6\,b\,\sqrt {-a^6}\,{\left (a^2+b^2\right )}^2+4\,a^4\,b^3\,\sqrt {-a^6}\,{\left (a^2+b^2\right )}^2\right )\,\left (\frac {1}{8\,a^5\,b\,\sqrt {{\left (a^2+b^2\right )}^2}\,{\left (a^2+b^2\right )}^3}-{\mathrm {e}}^x\,\left (\frac {1}{16\,a^4\,b^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\,\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\,\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}}-\frac {2}{a\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {\frac {2}{a}-\frac {2\,b\,{\mathrm {e}}^x}{a^2}}{{\mathrm {e}}^{2\,x}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3/(a + b*sinh(x)),x)

[Out]

((2*atan((a^2*(-a^6)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2) + 2*b^2*(-a^6)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(
2*a^3*(a^2 + b^2)^2) + ((a^7 + a^5*b^2)*(-a^6)^(1/2))/(2*a^6*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)) - (a^6*b^2*exp
(x)*(-a^6)^(1/2)*((8*(a^4 + b^4 + 2*a^2*b^2))/(a^8*b*(a^2 + b^2)^2) - (4*(2*a^6*b + 2*a^4*b^3)*(a^4 + b^4 + 2*
a^2*b^2)^(1/2))/(a^12*b^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)) + (2*(a^7 + a^5*b^2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2
))/(a^11*b^3*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)) - (2*(a^2 + 2*b^2)*(a^2*(-a^6)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(
1/2) + 2*b^2*(-a^6)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^10*b^3*(-a^6)^(1/2)
*(a^2 + b^2)^2)))/(8*(a^4 + b^4 + 2*a^2*b^2)^(1/2)) - (a^6*b^2*exp(2*x)*(-a^6)^(1/2)*((4*(a^2 + 2*b^2)*(a^4 +
b^4 + 2*a^2*b^2))/(a^9*b^2*(a^2 + b^2)^2) + (4*(a^2*(-a^6)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2) + 2*b^2*(-a^6)^
(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^9*b^2*(-a^6)^(1/2)*(a^2 + b^2)^2) + (2*
(2*a^6*b + 2*a^4*b^3)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^11*b^3*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)) + (4*(a^7 +
a^5*b^2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^12*b^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2))))/(8*(a^4 + b^4 + 2*a^2*b
^2)^(1/2)) + (a^6*b^2*exp(3*x)*((2*(a^7 + a^5*b^2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^11*b^3*((a^2 + b^2)^2)^(1
/2)*(a^2 + b^2)) - (2*(a^2 + 2*b^2)*(a^2*(-a^6)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2) + 2*b^2*(-a^6)^(1/2)*(a^4
+ b^4 + 2*a^2*b^2)^(1/2))*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^10*b^3*(-a^6)^(1/2)*(a^2 + b^2)^2))*(-a^6)^(1/2))/
(8*(a^4 + b^4 + 2*a^2*b^2)^(1/2))) - 2*atan((4*a^6*b*(-a^6)^(1/2)*(a^2 + b^2)^2 + 4*a^4*b^3*(-a^6)^(1/2)*(a^2
+ b^2)^2)*(1/(8*a^5*b*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)^3) - exp(x)*(1/(16*a^4*b^2*((a^2 + b^2)^2)^(1/2)*(a^2
+ b^2)^3) - (a^2 + 2*b^2)^2/(16*a^8*b^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)^3)) + (a^2 + 2*b^2)/(8*a^7*b*((a^2 +
b^2)^2)^(1/2)*(a^2 + b^2)^3))))*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(-a^6)^(1/2) - 2/(a*(exp(4*x) - 2*exp(2*x) + 1
)) - (2/a - (2*b*exp(x))/a^2)/(exp(2*x) - 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{3}{\relax (x )}}{a + b \sinh {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**3/(a+b*sinh(x)),x)

[Out]

Integral(coth(x)**3/(a + b*sinh(x)), x)

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