Integrand size = 13, antiderivative size = 134 \[ \int \frac {\sinh ^3(x)}{a+b \coth (x)} \, dx=-\frac {b^4 \text {arctanh}\left (\frac {(b+a \coth (x)) \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {a b^2 \cosh (x)}{\left (a^2-b^2\right )^2}-\frac {a \cosh (x)}{a^2-b^2}+\frac {a \cosh ^3(x)}{3 \left (a^2-b^2\right )}-\frac {b^3 \sinh (x)}{\left (a^2-b^2\right )^2}-\frac {b \sinh ^3(x)}{3 \left (a^2-b^2\right )} \] Output:
-b^4*arctanh((b+a*coth(x))*sinh(x)/(a^2-b^2)^(1/2))/(a^2-b^2)^(5/2)+a*b^2* cosh(x)/(a^2-b^2)^2-a*cosh(x)/(a^2-b^2)+a*cosh(x)^3/(3*a^2-3*b^2)-b^3*sinh (x)/(a^2-b^2)^2-b*sinh(x)^3/(3*a^2-3*b^2)
Time = 0.74 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.28 \[ \int \frac {\sinh ^3(x)}{a+b \coth (x)} \, dx=\frac {24 b^4 \sqrt {a+b} \arctan \left (\frac {a+b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a+b} \sqrt {a+b}}\right )-3 a \sqrt {-a+b} \left (3 a^3+3 a^2 b-7 a b^2-7 b^3\right ) \cosh (x)-a (-a+b)^{3/2} (a+b)^2 \cosh (3 x)+3 b \sqrt {-a+b} \left (a^3+a^2 b-5 a b^2-5 b^3\right ) \sinh (x)+b (-a+b)^{3/2} (a+b)^2 \sinh (3 x)}{12 (-a+b)^{5/2} (a+b)^3} \] Input:
Integrate[Sinh[x]^3/(a + b*Coth[x]),x]
Output:
(24*b^4*Sqrt[a + b]*ArcTan[(a + b*Tanh[x/2])/(Sqrt[-a + b]*Sqrt[a + b])] - 3*a*Sqrt[-a + b]*(3*a^3 + 3*a^2*b - 7*a*b^2 - 7*b^3)*Cosh[x] - a*(-a + b) ^(3/2)*(a + b)^2*Cosh[3*x] + 3*b*Sqrt[-a + b]*(a^3 + a^2*b - 5*a*b^2 - 5*b ^3)*Sinh[x] + b*(-a + b)^(3/2)*(a + b)^2*Sinh[3*x])/(12*(-a + b)^(5/2)*(a + b)^3)
Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.99, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.769, Rules used = {3042, 26, 3990, 26, 3042, 26, 3967, 26, 3042, 26, 3113, 2009, 3990, 26, 3042, 26, 3967, 26, 3042, 26, 3118, 3988, 219}
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.
| \(\displaystyle \int \frac {\sinh ^3(x)}{a+b \coth (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sec \left (-\frac {\pi }{2}+i x\right )^3 \left (a-i b \tan \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\sec \left (i x-\frac {\pi }{2}\right )^3 \left (a-i b \tan \left (i x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 3990 |
| \(\displaystyle i \left (\frac {\int -i (a-b \coth (x)) \sinh ^3(x)dx}{a^2-b^2}-\frac {b^2 \int \frac {i \sinh (x)}{a+b \coth (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i \int (a-b \coth (x)) \sinh ^3(x)dx}{a^2-b^2}-\frac {i b^2 \int \frac {\sinh (x)}{a+b \coth (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i b^2 \int -\frac {i}{\sec \left (i x-\frac {\pi }{2}\right ) \left (a-i b \tan \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}-\frac {i \int \frac {i \left (a+i b \tan \left (i x-\frac {\pi }{2}\right )\right )}{\sec \left (i x-\frac {\pi }{2}\right )^3}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {\int \frac {a+i b \tan \left (i x-\frac {\pi }{2}\right )}{\sec \left (i x-\frac {\pi }{2}\right )^3}dx}{a^2-b^2}-\frac {b^2 \int \frac {1}{\sec \left (i x-\frac {\pi }{2}\right ) \left (a-i b \tan \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle i \left (\frac {a \int -i \sinh ^3(x)dx+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \int \frac {1}{\sec \left (i x-\frac {\pi }{2}\right ) \left (a-i b \tan \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {\frac {1}{3} i b \sinh ^3(x)-i a \int \sinh ^3(x)dx}{a^2-b^2}-\frac {b^2 \int \frac {1}{\sec \left (i x-\frac {\pi }{2}\right ) \left (a-i b \tan \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {\frac {1}{3} i b \sinh ^3(x)-i a \int i \sin (i x)^3dx}{a^2-b^2}-\frac {b^2 \int \frac {1}{\sec \left (i x-\frac {\pi }{2}\right ) \left (a-i b \tan \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {a \int \sin (i x)^3dx+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \int \frac {1}{\sec \left (i x-\frac {\pi }{2}\right ) \left (a-i b \tan \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle i \left (\frac {i a \int \left (1-\cosh ^2(x)\right )d\cosh (x)+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \int \frac {1}{\sec \left (i x-\frac {\pi }{2}\right ) \left (a-i b \tan \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \int \frac {1}{\sec \left (i x-\frac {\pi }{2}\right ) \left (a-i b \tan \left (i x-\frac {\pi }{2}\right )\right )}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3990 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \left (\frac {\int i (a-b \coth (x)) \sinh (x)dx}{a^2-b^2}-\frac {b^2 \int -\frac {i \text {csch}(x)}{a+b \coth (x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \left (\frac {i \int (a-b \coth (x)) \sinh (x)dx}{a^2-b^2}+\frac {i b^2 \int \frac {\text {csch}(x)}{a+b \coth (x)}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \left (\frac {i b^2 \int \frac {i \sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{a^2-b^2}+\frac {i \int -\frac {i \left (a+i b \tan \left (i x-\frac {\pi }{2}\right )\right )}{\sec \left (i x-\frac {\pi }{2}\right )}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \left (\frac {\int \frac {a+i b \tan \left (i x-\frac {\pi }{2}\right )}{\sec \left (i x-\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {b^2 \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \left (\frac {a \int i \sinh (x)dx-i b \sinh (x)}{a^2-b^2}-\frac {b^2 \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \left (\frac {i a \int \sinh (x)dx-i b \sinh (x)}{a^2-b^2}-\frac {b^2 \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \left (\frac {i a \int -i \sin (i x)dx-i b \sinh (x)}{a^2-b^2}-\frac {b^2 \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \left (\frac {a \int \sin (i x)dx-i b \sinh (x)}{a^2-b^2}-\frac {b^2 \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \left (\frac {i a \cosh (x)-i b \sinh (x)}{a^2-b^2}-\frac {b^2 \int \frac {\sec \left (i x-\frac {\pi }{2}\right )}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3988 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \left (\frac {i a \cosh (x)-i b \sinh (x)}{a^2-b^2}-\frac {i b^2 \int \frac {1}{a^2-b^2-(b+a \coth (x))^2 \sinh ^2(x)}d((b+a \coth (x)) \sinh (x))}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle i \left (\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )+\frac {1}{3} i b \sinh ^3(x)}{a^2-b^2}-\frac {b^2 \left (\frac {i a \cosh (x)-i b \sinh (x)}{a^2-b^2}-\frac {i b^2 \text {arctanh}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}\right )}{a^2-b^2}\right )\) |
Input:
Int[Sinh[x]^3/(a + b*Coth[x]),x]
Output:
I*((I*a*(Cosh[x] - Cosh[x]^3/3) + (I/3)*b*Sinh[x]^3)/(a^2 - b^2) - (b^2*(( (-I)*b^2*ArcTanh[((b + a*Coth[x])*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^( 3/2) + (I*a*Cosh[x] - I*b*Sinh[x])/(a^2 - b^2)))/(a^2 - b^2))
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
Int[sec[(e_.) + (f_.)*(x_)]/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbo l] :> Simp[-f^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, (b - a*Tan[e + f *x])/Sec[e + f*x]], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_ Symbol] :> Simp[1/(a^2 + b^2) Int[Sec[e + f*x]^m*(a - b*Tan[e + f*x]), x] , x] + Simp[b^2/(a^2 + b^2) Int[Sec[e + f*x]^(m + 2)/(a + b*Tan[e + f*x]) , x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[(m - 1)/2, 0]
Time = 1.94 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.28
| method | result | size |
| risch | \(\frac {{\mathrm e}^{3 x}}{24 a +24 b}-\frac {3 \,{\mathrm e}^{x} a}{8 \left (a +b \right )^{2}}-\frac {5 \,{\mathrm e}^{x} b}{8 \left (a +b \right )^{2}}-\frac {3 \,{\mathrm e}^{-x} a}{8 \left (a -b \right )^{2}}+\frac {5 \,{\mathrm e}^{-x} b}{8 \left (a -b \right )^{2}}+\frac {{\mathrm e}^{-3 x}}{24 a -24 b}+\frac {b^{4} \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(172\) |
| default | \(-\frac {16}{\left (32 a -32 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {32}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3} \left (32 a -32 b \right )}-\frac {a -2 b}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {2 b^{4} \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {-a^{2}+b^{2}}}-\frac {32}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3} \left (32 a +32 b \right )}-\frac {16}{\left (32 a +32 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {-a -2 b}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(175\) |
Input:
int(sinh(x)^3/(a+b*coth(x)),x,method=_RETURNVERBOSE)
Output:
1/24/(a+b)*exp(x)^3-3/8/(a+b)^2*exp(x)*a-5/8/(a+b)^2*exp(x)*b-3/8/(a-b)^2/ exp(x)*a+5/8/(a-b)^2/exp(x)*b+1/24/(a-b)/exp(x)^3+1/(a^2-b^2)^(1/2)*b^4/(a +b)^2/(a-b)^2*ln(exp(x)-(a-b)/(a^2-b^2)^(1/2))-1/(a^2-b^2)^(1/2)*b^4/(a+b) ^2/(a-b)^2*ln(exp(x)+(a-b)/(a^2-b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 902 vs. \(2 (126) = 252\).
Time = 0.14 (sec) , antiderivative size = 1859, normalized size of antiderivative = 13.87 \[ \int \frac {\sinh ^3(x)}{a+b \coth (x)} \, dx=\text {Too large to display} \] Input:
integrate(sinh(x)^3/(a+b*coth(x)),x, algorithm="fricas")
Output:
[1/24*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^6 + 6*( a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)*sinh(x)^5 + (a^ 5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*sinh(x)^6 + a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 - 3*(3*a^5 - a^4*b - 10*a^3*b^2 + 6*a ^2*b^3 + 7*a*b^4 - 5*b^5)*cosh(x)^4 - 3*(3*a^5 - a^4*b - 10*a^3*b^2 + 6*a^ 2*b^3 + 7*a*b^4 - 5*b^5 - 5*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a *b^4 - b^5)*cosh(x)^3 - 3*(3*a^5 - a^4*b - 10*a^3*b^2 + 6*a^2*b^3 + 7*a*b^ 4 - 5*b^5)*cosh(x))*sinh(x)^3 - 3*(3*a^5 + a^4*b - 10*a^3*b^2 - 6*a^2*b^3 + 7*a*b^4 + 5*b^5)*cosh(x)^2 - 3*(3*a^5 + a^4*b - 10*a^3*b^2 - 6*a^2*b^3 + 7*a*b^4 + 5*b^5 - 5*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*c osh(x)^4 + 6*(3*a^5 - a^4*b - 10*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 - 5*b^5)*co sh(x)^2)*sinh(x)^2 + 24*(b^4*cosh(x)^3 + 3*b^4*cosh(x)^2*sinh(x) + 3*b^4*c osh(x)*sinh(x)^2 + b^4*sinh(x)^3)*sqrt(a^2 - b^2)*log(((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 - 2*sqrt(a^2 - b^2)*(cosh(x ) + sinh(x)) + a - b)/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 - a + b)) + 6*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^5 - 2*(3*a^5 - a^4*b - 10*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 - 5*b^5)*cosh(x)^3 - (3*a^5 + a^4*b - 10*a^3*b^2 - 6*a^2*b^3 + 7*a*b^4 + 5*b ^5)*cosh(x))*sinh(x))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + ...
\[ \int \frac {\sinh ^3(x)}{a+b \coth (x)} \, dx=\int \frac {\sinh ^{3}{\left (x \right )}}{a + b \coth {\left (x \right )}}\, dx \] Input:
integrate(sinh(x)**3/(a+b*coth(x)),x)
Output:
Integral(sinh(x)**3/(a + b*coth(x)), x)
Exception generated. \[ \int \frac {\sinh ^3(x)}{a+b \coth (x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(sinh(x)^3/(a+b*coth(x)),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.14 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh ^3(x)}{a+b \coth (x)} \, dx=-\frac {2 \, b^{4} \arctan \left (-\frac {a e^{x} + b e^{x}}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {{\left (9 \, a e^{\left (2 \, x\right )} - 15 \, b e^{\left (2 \, x\right )} - a + b\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {a^{2} e^{\left (3 \, x\right )} + 2 \, a b e^{\left (3 \, x\right )} + b^{2} e^{\left (3 \, x\right )} - 9 \, a^{2} e^{x} - 24 \, a b e^{x} - 15 \, b^{2} e^{x}}{24 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} \] Input:
integrate(sinh(x)^3/(a+b*coth(x)),x, algorithm="giac")
Output:
-2*b^4*arctan(-(a*e^x + b*e^x)/sqrt(-a^2 + b^2))/((a^4 - 2*a^2*b^2 + b^4)* sqrt(-a^2 + b^2)) - 1/24*(9*a*e^(2*x) - 15*b*e^(2*x) - a + b)*e^(-3*x)/(a^ 2 - 2*a*b + b^2) + 1/24*(a^2*e^(3*x) + 2*a*b*e^(3*x) + b^2*e^(3*x) - 9*a^2 *e^x - 24*a*b*e^x - 15*b^2*e^x)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3)
Time = 3.15 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.28 \[ \int \frac {\sinh ^3(x)}{a+b \coth (x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24\,a-24\,b}+\frac {{\mathrm {e}}^{3\,x}}{24\,a+24\,b}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a-5\,b\right )}{8\,{\left (a-b\right )}^2}-\frac {{\mathrm {e}}^x\,\left (3\,a+5\,b\right )}{8\,{\left (a+b\right )}^2}-\frac {b^4\,\ln \left (2\,a^3\,b-2\,a\,b^3+a^4-b^4+{\mathrm {e}}^x\,{\left (a+b\right )}^{7/2}\,\sqrt {a-b}\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}+\frac {b^4\,\ln \left (2\,a\,b^3-2\,a^3\,b-a^4+b^4+{\mathrm {e}}^x\,{\left (a+b\right )}^{7/2}\,\sqrt {a-b}\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}} \] Input:
int(sinh(x)^3/(a + b*coth(x)),x)
Output:
exp(-3*x)/(24*a - 24*b) + exp(3*x)/(24*a + 24*b) - (exp(-x)*(3*a - 5*b))/( 8*(a - b)^2) - (exp(x)*(3*a + 5*b))/(8*(a + b)^2) - (b^4*log(2*a^3*b - 2*a *b^3 + a^4 - b^4 + exp(x)*(a + b)^(7/2)*(a - b)^(1/2)))/((a + b)^(5/2)*(a - b)^(5/2)) + (b^4*log(2*a*b^3 - 2*a^3*b - a^4 + b^4 + exp(x)*(a + b)^(7/2 )*(a - b)^(1/2)))/((a + b)^(5/2)*(a - b)^(5/2))
Time = 0.23 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.36 \[ \int \frac {\sinh ^3(x)}{a+b \coth (x)} \, dx=\frac {-e^{6 x} a^{4} b -2 e^{6 x} a^{3} b^{2}+2 e^{6 x} a^{2} b^{3}+e^{6 x} a \,b^{4}-3 e^{2 x} a^{4} b +30 e^{2 x} a^{3} b^{2}+18 e^{2 x} a^{2} b^{3}-21 e^{2 x} a \,b^{4}+3 e^{4 x} a^{4} b +30 e^{4 x} a^{3} b^{2}-18 e^{4 x} a^{2} b^{3}-21 e^{4 x} a \,b^{4}-48 e^{3 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {-a^{2}+b^{2}}}\right ) b^{4}+a^{5}-2 a^{3} b^{2}+b^{5}-9 e^{4 x} a^{5}+15 e^{4 x} b^{5}-2 a^{2} b^{3}+a^{4} b +e^{6 x} a^{5}-e^{6 x} b^{5}-9 e^{2 x} a^{5}-15 e^{2 x} b^{5}+a \,b^{4}}{24 e^{3 x} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )} \] Input:
int(sinh(x)^3/(a+b*coth(x)),x)
Output:
( - 48*e**(3*x)*sqrt( - a**2 + b**2)*atan((e**x*a + e**x*b)/sqrt( - a**2 + b**2))*b**4 + e**(6*x)*a**5 - e**(6*x)*a**4*b - 2*e**(6*x)*a**3*b**2 + 2* e**(6*x)*a**2*b**3 + e**(6*x)*a*b**4 - e**(6*x)*b**5 - 9*e**(4*x)*a**5 + 3 *e**(4*x)*a**4*b + 30*e**(4*x)*a**3*b**2 - 18*e**(4*x)*a**2*b**3 - 21*e**( 4*x)*a*b**4 + 15*e**(4*x)*b**5 - 9*e**(2*x)*a**5 - 3*e**(2*x)*a**4*b + 30* e**(2*x)*a**3*b**2 + 18*e**(2*x)*a**2*b**3 - 21*e**(2*x)*a*b**4 - 15*e**(2 *x)*b**5 + a**5 + a**4*b - 2*a**3*b**2 - 2*a**2*b**3 + a*b**4 + b**5)/(24* e**(3*x)*(a**6 - 3*a**4*b**2 + 3*a**2*b**4 - b**6))