Integrand size = 13, antiderivative size = 76 \[ \int \frac {\coth ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {2 b \text {csch}(x)}{a^3}-\frac {\text {csch}^2(x)}{2 a^2}+\frac {\left (a^2+3 b^2\right ) \log (\sinh (x))}{a^4}-\frac {\left (a^2+3 b^2\right ) \log (a+b \sinh (x))}{a^4}+\frac {a^2+b^2}{a^3 (a+b \sinh (x))} \] Output:
2*b*csch(x)/a^3-1/2*csch(x)^2/a^2+(a^2+3*b^2)*ln(sinh(x))/a^4-(a^2+3*b^2)* ln(a+b*sinh(x))/a^4+(a^2+b^2)/a^3/(a+b*sinh(x))
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \frac {\coth ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {4 a b \text {csch}(x)-a^2 \text {csch}^2(x)+2 \left (a^2+3 b^2\right ) \log (\sinh (x))-2 \left (a^2+3 b^2\right ) \log (a+b \sinh (x))+\frac {2 a \left (a^2+b^2\right )}{a+b \sinh (x)}}{2 a^4} \] Input:
Integrate[Coth[x]^3/(a + b*Sinh[x])^2,x]
Output:
(4*a*b*Csch[x] - a^2*Csch[x]^2 + 2*(a^2 + 3*b^2)*Log[Sinh[x]] - 2*(a^2 + 3 *b^2)*Log[a + b*Sinh[x]] + (2*a*(a^2 + b^2))/(a + b*Sinh[x]))/(2*a^4)
Time = 0.36 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03, 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.
\(\displaystyle \int \frac {\coth ^3(x)}{(a+b \sinh (x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i}{\tan (i x)^3 (a-i b \sin (i x))^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {1}{(a-i b \sin (i x))^2 \tan (i x)^3}dx\) |
\(\Big \downarrow \) 3200 |
\(\displaystyle -\int -\frac {\text {csch}^3(x) \left (\sinh ^2(x) b^2+b^2\right )}{b^3 (a+b \sinh (x))^2}d(b \sinh (x))\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\text {csch}^3(x) \left (b^2 \sinh ^2(x)+b^2\right )}{b^3 (a+b \sinh (x))^2}d(b \sinh (x))\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (-\frac {2 \text {csch}^2(x)}{a^3}+\frac {\text {csch}^3(x)}{a^2 b}+\frac {-a^2-3 b^2}{a^4 (a+b \sinh (x))}+\frac {\left (a^2+3 b^2\right ) \text {csch}(x)}{a^4 b}+\frac {-a^2-b^2}{a^3 (a+b \sinh (x))^2}\right )d(b \sinh (x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 b \text {csch}(x)}{a^3}-\frac {\text {csch}^2(x)}{2 a^2}+\frac {\left (a^2+3 b^2\right ) \log (b \sinh (x))}{a^4}-\frac {\left (a^2+3 b^2\right ) \log (a+b \sinh (x))}{a^4}+\frac {a^2+b^2}{a^3 (a+b \sinh (x))}\) |
Input:
Int[Coth[x]^3/(a + b*Sinh[x])^2,x]
Output:
(2*b*Csch[x])/a^3 - Csch[x]^2/(2*a^2) + ((a^2 + 3*b^2)*Log[b*Sinh[x]])/a^4 - ((a^2 + 3*b^2)*Log[a + b*Sinh[x]])/a^4 + (a^2 + b^2)/(a^3*(a + b*Sinh[x ]))
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 = 5.52 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.87
method | result | size |
default | \(-\frac {\frac {\tanh \left (\frac {x}{2}\right )^{2} a}{2}+4 b \tanh \left (\frac {x}{2}\right )}{4 a^{3}}-\frac {1}{8 a^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (4 a^{2}+12 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a^{4}}+\frac {b}{\tanh \left (\frac {x}{2}\right ) a^{3}}-\frac {2 \left (\frac {\left (-a^{2} b -b^{3}\right ) \tanh \left (\frac {x}{2}\right )}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}+\frac {\left (a^{2}+3 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}{2}\right )}{a^{4}}\) | \(142\) |
risch | \(\frac {2 \,{\mathrm e}^{x} \left ({\mathrm e}^{4 x} a^{2}+3 b^{2} {\mathrm e}^{4 x}+3 a b \,{\mathrm e}^{3 x}-4 \,{\mathrm e}^{2 x} a^{2}-6 \,{\mathrm e}^{2 x} b^{2}-3 b \,{\mathrm e}^{x} a +a^{2}+3 b^{2}\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} a^{3} \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{a^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 x}-1\right ) b^{2}}{a^{4}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{a^{2}}-\frac {3 \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) b^{2}}{a^{4}}\) | \(161\) |
Input:
int(coth(x)^3/(a+b*sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
-1/4/a^3*(1/2*tanh(1/2*x)^2*a+4*b*tanh(1/2*x))-1/8/a^2/tanh(1/2*x)^2+1/4/a ^4*(4*a^2+12*b^2)*ln(tanh(1/2*x))+b/tanh(1/2*x)/a^3-2/a^4*((-a^2*b-b^3)*ta nh(1/2*x)/(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)-a)+1/2*(a^2+3*b^2)*ln(tanh(1/2* x)^2*a-2*b*tanh(1/2*x)-a))
Leaf count of result is larger than twice the leaf count of optimal. 1463 vs. \(2 (74) = 148\).
Time = 0.10 (sec) , antiderivative size = 1463, normalized size of antiderivative = 19.25 \[ \int \frac {\coth ^3(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \] Input:
integrate(coth(x)^3/(a+b*sinh(x))^2,x, algorithm="fricas")
Output:
(6*a^2*b*cosh(x)^4 + 2*(a^3 + 3*a*b^2)*cosh(x)^5 + 2*(a^3 + 3*a*b^2)*sinh( x)^5 - 6*a^2*b*cosh(x)^2 + 2*(3*a^2*b + 5*(a^3 + 3*a*b^2)*cosh(x))*sinh(x) ^4 - 4*(2*a^3 + 3*a*b^2)*cosh(x)^3 + 4*(6*a^2*b*cosh(x) - 2*a^3 - 3*a*b^2 + 5*(a^3 + 3*a*b^2)*cosh(x)^2)*sinh(x)^3 + 2*(18*a^2*b*cosh(x)^2 + 10*(a^3 + 3*a*b^2)*cosh(x)^3 - 3*a^2*b - 6*(2*a^3 + 3*a*b^2)*cosh(x))*sinh(x)^2 + 2*(a^3 + 3*a*b^2)*cosh(x) - ((a^2*b + 3*b^3)*cosh(x)^6 + (a^2*b + 3*b^3)* sinh(x)^6 + 2*(a^3 + 3*a*b^2)*cosh(x)^5 + 2*(a^3 + 3*a*b^2 + 3*(a^2*b + 3* b^3)*cosh(x))*sinh(x)^5 - 3*(a^2*b + 3*b^3)*cosh(x)^4 - (3*a^2*b + 9*b^3 - 15*(a^2*b + 3*b^3)*cosh(x)^2 - 10*(a^3 + 3*a*b^2)*cosh(x))*sinh(x)^4 - 4* (a^3 + 3*a*b^2)*cosh(x)^3 + 4*(5*(a^2*b + 3*b^3)*cosh(x)^3 - a^3 - 3*a*b^2 + 5*(a^3 + 3*a*b^2)*cosh(x)^2 - 3*(a^2*b + 3*b^3)*cosh(x))*sinh(x)^3 - a^ 2*b - 3*b^3 + 3*(a^2*b + 3*b^3)*cosh(x)^2 + (15*(a^2*b + 3*b^3)*cosh(x)^4 + 20*(a^3 + 3*a*b^2)*cosh(x)^3 + 3*a^2*b + 9*b^3 - 18*(a^2*b + 3*b^3)*cosh (x)^2 - 12*(a^3 + 3*a*b^2)*cosh(x))*sinh(x)^2 + 2*(a^3 + 3*a*b^2)*cosh(x) + 2*(3*(a^2*b + 3*b^3)*cosh(x)^5 + 5*(a^3 + 3*a*b^2)*cosh(x)^4 - 6*(a^2*b + 3*b^3)*cosh(x)^3 + a^3 + 3*a*b^2 - 6*(a^3 + 3*a*b^2)*cosh(x)^2 + 3*(a^2* b + 3*b^3)*cosh(x))*sinh(x))*log(2*(b*sinh(x) + a)/(cosh(x) - sinh(x))) + ((a^2*b + 3*b^3)*cosh(x)^6 + (a^2*b + 3*b^3)*sinh(x)^6 + 2*(a^3 + 3*a*b^2) *cosh(x)^5 + 2*(a^3 + 3*a*b^2 + 3*(a^2*b + 3*b^3)*cosh(x))*sinh(x)^5 - 3*( a^2*b + 3*b^3)*cosh(x)^4 - (3*a^2*b + 9*b^3 - 15*(a^2*b + 3*b^3)*cosh(x...
\[ \int \frac {\coth ^3(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \] Input:
integrate(coth(x)**3/(a+b*sinh(x))**2,x)
Output:
Integral(coth(x)**3/(a + b*sinh(x))**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (74) = 148\).
Time = 0.05 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.66 \[ \int \frac {\coth ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {2 \, {\left (3 \, a b e^{\left (-2 \, x\right )} - 3 \, a b e^{\left (-4 \, x\right )} + {\left (a^{2} + 3 \, b^{2}\right )} e^{\left (-x\right )} - 2 \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} e^{\left (-3 \, x\right )} + {\left (a^{2} + 3 \, b^{2}\right )} e^{\left (-5 \, x\right )}\right )}}{2 \, a^{4} e^{\left (-x\right )} - 3 \, a^{3} b e^{\left (-2 \, x\right )} - 4 \, a^{4} e^{\left (-3 \, x\right )} + 3 \, a^{3} b e^{\left (-4 \, x\right )} + 2 \, a^{4} e^{\left (-5 \, x\right )} - a^{3} b e^{\left (-6 \, x\right )} + a^{3} b} - \frac {{\left (a^{2} + 3 \, b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{4}} + \frac {{\left (a^{2} + 3 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{4}} + \frac {{\left (a^{2} + 3 \, b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{4}} \] Input:
integrate(coth(x)^3/(a+b*sinh(x))^2,x, algorithm="maxima")
Output:
2*(3*a*b*e^(-2*x) - 3*a*b*e^(-4*x) + (a^2 + 3*b^2)*e^(-x) - 2*(2*a^2 + 3*b ^2)*e^(-3*x) + (a^2 + 3*b^2)*e^(-5*x))/(2*a^4*e^(-x) - 3*a^3*b*e^(-2*x) - 4*a^4*e^(-3*x) + 3*a^3*b*e^(-4*x) + 2*a^4*e^(-5*x) - a^3*b*e^(-6*x) + a^3* b) - (a^2 + 3*b^2)*log(-2*a*e^(-x) + b*e^(-2*x) - b)/a^4 + (a^2 + 3*b^2)*l og(e^(-x) + 1)/a^4 + (a^2 + 3*b^2)*log(e^(-x) - 1)/a^4
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (74) = 148\).
Time = 0.13 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.50 \[ \int \frac {\coth ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (a^{2} + 3 \, b^{2}\right )} \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a^{4}} - \frac {{\left (a^{2} b + 3 \, b^{3}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{4} b} + \frac {a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )} + 3 \, b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} - 4 \, a^{3} - 8 \, a b^{2}}{{\left (b {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )} a^{4}} - \frac {3 \, a^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 9 \, b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 8 \, a b {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, a^{2}}{2 \, a^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \] Input:
integrate(coth(x)^3/(a+b*sinh(x))^2,x, algorithm="giac")
Output:
(a^2 + 3*b^2)*log(abs(-e^(-x) + e^x))/a^4 - (a^2*b + 3*b^3)*log(abs(-b*(e^ (-x) - e^x) + 2*a))/(a^4*b) + (a^2*b*(e^(-x) - e^x) + 3*b^3*(e^(-x) - e^x) - 4*a^3 - 8*a*b^2)/((b*(e^(-x) - e^x) - 2*a)*a^4) - 1/2*(3*a^2*(e^(-x) - e^x)^2 + 9*b^2*(e^(-x) - e^x)^2 + 8*a*b*(e^(-x) - e^x) + 4*a^2)/(a^4*(e^(- x) - e^x)^2)
Time = 2.47 (sec) , antiderivative size = 1375, normalized size of antiderivative = 18.09 \[ \int \frac {\coth ^3(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \] Input:
int(coth(x)^3/(a + b*sinh(x))^2,x)
Output:
(2*exp(x)*(a*b^7 + 2*a^3*b^5 + a^5*b^3))/(a^4*(b^5 + a^2*b^3)*(2*a*exp(x) - b + b*exp(2*x))) - 2/(a^2*(exp(4*x) - 2*exp(2*x) + 1)) - ((2*atan((4*a^9 *b*((a^2 + 3*b^2)^2)^(1/2)*(-a^8)^(1/2) + 12*a^5*b^5*((a^2 + 3*b^2)^2)^(1/ 2)*(-a^8)^(1/2) + 16*a^7*b^3*((a^2 + 3*b^2)^2)^(1/2)*(-a^8)^(1/2))*(exp(x) *((a^2 + 2*b^2)^2/(16*a^10*b^2*(a^4 + 3*b^4 + 4*a^2*b^2)^2) - 1/(16*a^6*b^ 2*(a^2 + 3*b^2)^2*(a^2 + b^2)^2)) + (a^2 + 2*b^2)/(8*a^9*b*(a^4 + 3*b^4 + 4*a^2*b^2)^2) + 1/(8*a^7*b*(a^2 + 3*b^2)^2*(a^2 + b^2)^2))) - 2*atan((a^2* (-a^8)^(1/2)*(a^4 + 9*b^4 + 6*a^2*b^2)^(1/2) + 2*b^2*(-a^8)^(1/2)*(a^4 + 9 *b^4 + 6*a^2*b^2)^(1/2))/(2*a^4*(a^4 + 3*b^4 + 4*a^2*b^2)) + ((a^8 + 3*a^6 *b^2)*(-a^8)^(1/2))/(2*a^8*((a^2 + 3*b^2)^2)^(1/2)*(a^2 + b^2)) - (a^8*b^2 *exp(2*x)*(-a^8)^(1/2)*((4*(a^2 + 2*b^2)*(a^4 + 9*b^4 + 6*a^2*b^2))/(a^12* b^2*(a^4 + 3*b^4 + 4*a^2*b^2)) + (4*(a^2*(-a^8)^(1/2)*(a^4 + 9*b^4 + 6*a^2 *b^2)^(1/2) + 2*b^2*(-a^8)^(1/2)*(a^4 + 9*b^4 + 6*a^2*b^2)^(1/2))*(a^4 + 9 *b^4 + 6*a^2*b^2)^(1/2))/(a^12*b^2*(-a^8)^(1/2)*(a^4 + 3*b^4 + 4*a^2*b^2)) + (2*(2*a^7*b + 6*a^5*b^3)*(a^4 + 9*b^4 + 6*a^2*b^2)^(1/2))/(a^15*b^3*((a ^2 + 3*b^2)^2)^(1/2)*(a^2 + b^2)) + (4*(a^8 + 3*a^6*b^2)*(a^4 + 9*b^4 + 6* a^2*b^2)^(1/2))/(a^16*b^2*((a^2 + 3*b^2)^2)^(1/2)*(a^2 + b^2))))/(8*(a^4 + 9*b^4 + 6*a^2*b^2)^(1/2)) + (a^8*b^2*exp(3*x)*((2*(a^8 + 3*a^6*b^2)*(a^4 + 9*b^4 + 6*a^2*b^2)^(1/2))/(a^15*b^3*((a^2 + 3*b^2)^2)^(1/2)*(a^2 + b^2)) - (2*(a^2 + 2*b^2)*(a^2*(-a^8)^(1/2)*(a^4 + 9*b^4 + 6*a^2*b^2)^(1/2) +...
Time = 0.16 (sec) , antiderivative size = 955, normalized size of antiderivative = 12.57 \[ \int \frac {\coth ^3(x)}{(a+b \sinh (x))^2} \, dx =\text {Too large to display} \] Input:
int(coth(x)^3/(a+b*sinh(x))^2,x)
Output:
(e**(6*x)*log(e**x - 1)*a**2*b + 3*e**(6*x)*log(e**x - 1)*b**3 + e**(6*x)* log(e**x + 1)*a**2*b + 3*e**(6*x)*log(e**x + 1)*b**3 - e**(6*x)*log(e**(2* x)*b + 2*e**x*a - b)*a**2*b - 3*e**(6*x)*log(e**(2*x)*b + 2*e**x*a - b)*b* *3 - e**(6*x)*a**2*b - 3*e**(6*x)*b**3 + 2*e**(5*x)*log(e**x - 1)*a**3 + 6 *e**(5*x)*log(e**x - 1)*a*b**2 + 2*e**(5*x)*log(e**x + 1)*a**3 + 6*e**(5*x )*log(e**x + 1)*a*b**2 - 2*e**(5*x)*log(e**(2*x)*b + 2*e**x*a - b)*a**3 - 6*e**(5*x)*log(e**(2*x)*b + 2*e**x*a - b)*a*b**2 - 3*e**(4*x)*log(e**x - 1 )*a**2*b - 9*e**(4*x)*log(e**x - 1)*b**3 - 3*e**(4*x)*log(e**x + 1)*a**2*b - 9*e**(4*x)*log(e**x + 1)*b**3 + 3*e**(4*x)*log(e**(2*x)*b + 2*e**x*a - b)*a**2*b + 9*e**(4*x)*log(e**(2*x)*b + 2*e**x*a - b)*b**3 + 9*e**(4*x)*a* *2*b + 9*e**(4*x)*b**3 - 4*e**(3*x)*log(e**x - 1)*a**3 - 12*e**(3*x)*log(e **x - 1)*a*b**2 - 4*e**(3*x)*log(e**x + 1)*a**3 - 12*e**(3*x)*log(e**x + 1 )*a*b**2 + 4*e**(3*x)*log(e**(2*x)*b + 2*e**x*a - b)*a**3 + 12*e**(3*x)*lo g(e**(2*x)*b + 2*e**x*a - b)*a*b**2 - 4*e**(3*x)*a**3 + 3*e**(2*x)*log(e** x - 1)*a**2*b + 9*e**(2*x)*log(e**x - 1)*b**3 + 3*e**(2*x)*log(e**x + 1)*a **2*b + 9*e**(2*x)*log(e**x + 1)*b**3 - 3*e**(2*x)*log(e**(2*x)*b + 2*e**x *a - b)*a**2*b - 9*e**(2*x)*log(e**(2*x)*b + 2*e**x*a - b)*b**3 - 9*e**(2* x)*a**2*b - 9*e**(2*x)*b**3 + 2*e**x*log(e**x - 1)*a**3 + 6*e**x*log(e**x - 1)*a*b**2 + 2*e**x*log(e**x + 1)*a**3 + 6*e**x*log(e**x + 1)*a*b**2 - 2* e**x*log(e**(2*x)*b + 2*e**x*a - b)*a**3 - 6*e**x*log(e**(2*x)*b + 2*e*...