Integrand size = 13, antiderivative size = 52 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {b \text {csch}(x)}{a^2}-\frac {\text {csch}^2(x)}{2 a}+\frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{a^3} \]
Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.87 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {2 a b \text {csch}(x)-a^2 \text {csch}^2(x)+2 \left (a^2+b^2\right ) (\log (\sinh (x))-\log (a+b \sinh (x)))}{2 a^3} \]
Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 26, 3200, 25, 522, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\tan (i x)^3 (a-i b \sin (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{(a-i b \sin (i x)) \tan (i x)^3}dx\) |
| \(\Big \downarrow \) 3200 |
| \(\displaystyle -\int -\frac {\text {csch}^3(x) \left (\sinh ^2(x) b^2+b^2\right )}{b^3 (a+b \sinh (x))}d(b \sinh (x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\text {csch}^3(x) \left (b^2 \sinh ^2(x)+b^2\right )}{b^3 (a+b \sinh (x))}d(b \sinh (x))\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (-\frac {\text {csch}^2(x)}{a^2}+\frac {-a^2-b^2}{a^3 (a+b \sinh (x))}+\frac {\left (a^2+b^2\right ) \text {csch}(x)}{a^3 b}+\frac {\text {csch}^3(x)}{a b}\right )d(b \sinh (x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \text {csch}(x)}{a^2}+\frac {\left (a^2+b^2\right ) \log (b \sinh (x))}{a^3}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{a^3}-\frac {\text {csch}^2(x)}{2 a}\) |
(b*Csch[x])/a^2 - Csch[x]^2/(2*a) + ((a^2 + b^2)*Log[b*Sinh[x]])/a^3 - ((a ^2 + b^2)*Log[a + b*Sinh[x]])/a^3
3.3.34.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(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]
Time = 0.88 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.88
| method | result | size |
| risch | \(-\frac {2 \,{\mathrm e}^{x} \left (-b \,{\mathrm e}^{2 x}+{\mathrm e}^{x} a +b \right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} a^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{a}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) b^{2}}{a^{3}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{a}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) b^{2}}{a^{3}}\) | \(98\) |
| default | \(-\frac {\frac {\tanh \left (\frac {x}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {x}{2}\right )}{4 a^{2}}+\frac {\left (-4 a^{2}-4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}{4 a^{3}}-\frac {1}{8 a \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (4 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {x}{2}\right )}\) | \(104\) |
-2*exp(x)*(-b*exp(2*x)+exp(x)*a+b)/(exp(2*x)-1)^2/a^2+1/a*ln(exp(2*x)-1)+1 /a^3*ln(exp(2*x)-1)*b^2-1/a*ln(exp(2*x)+2*a/b*exp(x)-1)-1/a^3*ln(exp(2*x)+ 2*a/b*exp(x)-1)*b^2
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (50) = 100\).
Time = 0.29 (sec) , antiderivative size = 427, normalized size of antiderivative = 8.21 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {2 \, a b \cosh \left (x\right )^{3} + 2 \, a b \sinh \left (x\right )^{3} - 2 \, a^{2} \cosh \left (x\right )^{2} - 2 \, a b \cosh \left (x\right ) + 2 \, {\left (3 \, a b \cosh \left (x\right ) - a^{2}\right )} \sinh \left (x\right )^{2} - {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} + b^{2}\right )} \sinh \left (x\right )^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} - a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} + b^{2} + 4 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{3} - {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} + b^{2}\right )} \sinh \left (x\right )^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} - a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} + b^{2} + 4 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{3} - {\left (a^{2} + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (3 \, a b \cosh \left (x\right )^{2} - 2 \, a^{2} \cosh \left (x\right ) - a b\right )} \sinh \left (x\right )}{a^{3} \cosh \left (x\right )^{4} + 4 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{3} \sinh \left (x\right )^{4} - 2 \, a^{3} \cosh \left (x\right )^{2} + a^{3} + 2 \, {\left (3 \, a^{3} \cosh \left (x\right )^{2} - a^{3}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{3} \cosh \left (x\right )^{3} - a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )} \]
(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)*co sh(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)*c osh(x)^3 - (a^2 + b^2)*cosh(x))*sinh(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*cosh(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))
\[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (50) = 100\).
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.23 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=-\frac {2 \, {\left (b e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - b e^{\left (-3 \, x\right )}\right )}}{2 \, a^{2} e^{\left (-2 \, x\right )} - a^{2} e^{\left (-4 \, x\right )} - a^{2}} - \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{3}} \]
-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
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (50) = 100\).
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.40 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a^{3}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{3} b} - \frac {3 \, a^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 3 \, b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4 \, a b {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, a^{2}}{2 \, a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \]
(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)
Time = 1.97 (sec) , antiderivative size = 1163, normalized size of antiderivative = 22.37 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\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} \]
((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^...