Integrand size = 13, antiderivative size = 52 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {b \text {csch}(x)}{a^2}-\frac {\text {csch}^2(x)}{2 a}+\frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{a^3} \] Output:
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*s inh(x))/a^3
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.87 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {2 a b \text {csch}(x)-a^2 \text {csch}^2(x)+2 \left (a^2+b^2\right ) (\log (\sinh (x))-\log (a+b \sinh (x)))}{2 a^3} \] Input:
Integrate[Coth[x]^3/(a + b*Sinh[x]),x]
Output:
(2*a*b*Csch[x] - a^2*Csch[x]^2 + 2*(a^2 + b^2)*(Log[Sinh[x]] - Log[a + b*S inh[x]]))/(2*a^3)
Time = 0.31 (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 |
| \(\displaystyle \frac {b \text {csch}(x)}{a^2}+\frac {\left (a^2+b^2\right ) \log (b \sinh (x))}{a^3}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{a^3}-\frac {\text {csch}^2(x)}{2 a}\) |
Input:
Int[Coth[x]^3/(a + b*Sinh[x]),x]
Output:
(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
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 = 1.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.88
| method | result | size |
| risch | \(-\frac {2 \,{\mathrm e}^{x} \left (-b \,{\mathrm e}^{2 x}+{\mathrm e}^{x} a +b \right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} a^{2}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{a}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) b^{2}}{a^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{a}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) b^{2}}{a^{3}}\) | \(98\) |
| default | \(-\frac {\frac {\tanh \left (\frac {x}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {x}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (4 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {x}{2}\right )}+\frac {\left (-4 a^{2}-4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}{4 a^{3}}\) | \(104\) |
Input:
int(coth(x)^3/(a+b*sinh(x)),x,method=_RETURNVERBOSE)
Output:
-2*exp(x)*(-b*exp(2*x)+exp(x)*a+b)/(exp(2*x)-1)^2/a^2-1/a*ln(exp(2*x)+2/b* a*exp(x)-1)-1/a^3*ln(exp(2*x)+2/b*a*exp(x)-1)*b^2+1/a*ln(exp(2*x)-1)+1/a^3 *ln(exp(2*x)-1)*b^2
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (50) = 100\).
Time = 0.10 (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 )} \] Input:
integrate(coth(x)^3/(a+b*sinh(x)),x, algorithm="fricas")
Output:
(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))
\[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \] Input:
integrate(coth(x)**3/(a+b*sinh(x)),x)
Output:
Integral(coth(x)**3/(a + b*sinh(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (50) = 100\).
Time = 0.05 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.23 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=-\frac {2 \, {\left (b e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - b e^{\left (-3 \, x\right )}\right )}}{2 \, a^{2} e^{\left (-2 \, x\right )} - a^{2} e^{\left (-4 \, x\right )} - a^{2}} - \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{3}} \] Input:
integrate(coth(x)^3/(a+b*sinh(x)),x, algorithm="maxima")
Output:
-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 = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.40 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a^{3}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{3} b} - \frac {3 \, a^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 3 \, b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4 \, a b {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, a^{2}}{2 \, a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \] Input:
integrate(coth(x)^3/(a+b*sinh(x)),x, algorithm="giac")
Output:
(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 = 2.13 (sec) , antiderivative size = 1163, normalized size of antiderivative = 22.37 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx =\text {Too large to display} \] Input:
int(coth(x)^3/(a + b*sinh(x)),x)
Output:
((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^...
Time = 0.16 (sec) , antiderivative size = 374, normalized size of antiderivative = 7.19 \[ \int \frac {\coth ^3(x)}{a+b \sinh (x)} \, dx=\frac {e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a^{2}+e^{4 x} \mathrm {log}\left (e^{x}-1\right ) b^{2}+e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a^{2}+e^{4 x} \mathrm {log}\left (e^{x}+1\right ) b^{2}-e^{4 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) a^{2}-e^{4 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) b^{2}-e^{4 x} a^{2}+2 e^{3 x} a b -2 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{2}-2 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) b^{2}-2 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{2}-2 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) b^{2}+2 e^{2 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) a^{2}+2 e^{2 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) b^{2}-2 e^{x} a b +\mathrm {log}\left (e^{x}-1\right ) a^{2}+\mathrm {log}\left (e^{x}-1\right ) b^{2}+\mathrm {log}\left (e^{x}+1\right ) a^{2}+\mathrm {log}\left (e^{x}+1\right ) b^{2}-\mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) a^{2}-\mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) b^{2}-a^{2}}{a^{3} \left (e^{4 x}-2 e^{2 x}+1\right )} \] Input:
int(coth(x)^3/(a+b*sinh(x)),x)
Output:
(e**(4*x)*log(e**x - 1)*a**2 + e**(4*x)*log(e**x - 1)*b**2 + e**(4*x)*log( e**x + 1)*a**2 + e**(4*x)*log(e**x + 1)*b**2 - e**(4*x)*log(e**(2*x)*b + 2 *e**x*a - b)*a**2 - e**(4*x)*log(e**(2*x)*b + 2*e**x*a - b)*b**2 - e**(4*x )*a**2 + 2*e**(3*x)*a*b - 2*e**(2*x)*log(e**x - 1)*a**2 - 2*e**(2*x)*log(e **x - 1)*b**2 - 2*e**(2*x)*log(e**x + 1)*a**2 - 2*e**(2*x)*log(e**x + 1)*b **2 + 2*e**(2*x)*log(e**(2*x)*b + 2*e**x*a - b)*a**2 + 2*e**(2*x)*log(e**( 2*x)*b + 2*e**x*a - b)*b**2 - 2*e**x*a*b + log(e**x - 1)*a**2 + log(e**x - 1)*b**2 + log(e**x + 1)*a**2 + log(e**x + 1)*b**2 - log(e**(2*x)*b + 2*e* *x*a - b)*a**2 - log(e**(2*x)*b + 2*e**x*a - b)*b**2 - a**2)/(a**3*(e**(4* x) - 2*e**(2*x) + 1))