Integrand size = 13, antiderivative size = 107 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac {2 b^4 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \sqrt {a^2+b^2}}-\frac {\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac {b \cosh (x) \sinh (x)}{2 a^2}+\frac {\cosh (x) \sinh ^2(x)}{3 a} \]
1/2*b*(a^2-2*b^2)*x/a^4-1/3*(2*a^2-3*b^2)*cosh(x)/a^3-1/2*b*cosh(x)*sinh(x )/a^2+1/3*cosh(x)*sinh(x)^2/a-2*b^4*arctanh((a-b*tanh(1/2*x))/(a^2+b^2)^(1 /2))/a^4/(a^2+b^2)^(1/2)
Time = 1.52 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {\left (-9 a^3+12 a b^2\right ) \cosh (x)+a^3 \cosh (3 x)+3 b \left (2 a^2 x-4 b^2 x+\frac {8 b^3 \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-a^2 \sinh (2 x)\right )}{12 a^4} \]
((-9*a^3 + 12*a*b^2)*Cosh[x] + a^3*Cosh[3*x] + 3*b*(2*a^2*x - 4*b^2*x + (8 *b^3*ArcTan[(a - b*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - a^2*Si nh[2*x]))/(12*a^4)
Result contains complex when optimal does not.
Time = 1.05 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.36, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.769, Rules used = {3042, 26, 4340, 26, 3042, 25, 4592, 26, 26, 3042, 26, 4592, 27, 3042, 4407, 26, 3042, 26, 4318, 3042, 3139, 1083, 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.
| \(\displaystyle \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\csc (i x)^3 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\csc (i x)^3 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 4340 |
| \(\displaystyle i \left (\frac {\int \frac {i \left (2 b \text {csch}^2(x)+2 a \text {csch}(x)+3 b\right ) \sinh ^2(x)}{a+b \text {csch}(x)}dx}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i \int \frac {\left (2 b \text {csch}^2(x)+2 a \text {csch}(x)+3 b\right ) \sinh ^2(x)}{a+b \text {csch}(x)}dx}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i \int -\frac {-2 b \csc (i x)^2+2 i a \csc (i x)+3 b}{\csc (i x)^2 (a+i b \csc (i x))}dx}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\frac {i \int \frac {-2 b \csc (i x)^2+2 i a \csc (i x)+3 b}{\csc (i x)^2 (a+i b \csc (i x))}dx}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\int -\frac {i \left (-3 i b^2 \text {csch}^2(x)+i a b \text {csch}(x)+2 \left (2 i a^2-3 i b^2\right )\right ) \sinh (x)}{a+b \text {csch}(x)}dx}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \int \frac {i \left (-3 b^2 \text {csch}^2(x)+a b \text {csch}(x)+2 \left (2 a^2-3 b^2\right )\right ) \sinh (x)}{a+b \text {csch}(x)}dx}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {\int \frac {\left (-3 b^2 \text {csch}^2(x)+a b \text {csch}(x)+2 \left (2 a^2-3 b^2\right )\right ) \sinh (x)}{a+b \text {csch}(x)}dx}{2 a}-\frac {3 b \sinh (x) \cosh (x)}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}-\frac {\int -\frac {i \left (3 b^2 \csc (i x)^2+i a b \csc (i x)+2 \left (2 a^2-3 b^2\right )\right )}{\csc (i x) (a+i b \csc (i x))}dx}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \int \frac {3 b^2 \csc (i x)^2+i a b \csc (i x)+2 \left (2 a^2-3 b^2\right )}{\csc (i x) (a+i b \csc (i x))}dx}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {\int \frac {3 i \left (a \text {csch}(x) b^2+\left (a^2-2 b^2\right ) b\right )}{a+b \text {csch}(x)}dx}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {3 i \int \frac {a \text {csch}(x) b^2+\left (a^2-2 b^2\right ) b}{a+b \text {csch}(x)}dx}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {3 i \int \frac {i a \csc (i x) b^2+\left (a^2-2 b^2\right ) b}{a+i b \csc (i x)}dx}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {3 i \left (\frac {b x \left (a^2-2 b^2\right )}{a}+\frac {2 i b^4 \int -\frac {i \text {csch}(x)}{a+b \text {csch}(x)}dx}{a}\right )}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {3 i \left (\frac {2 b^4 \int \frac {\text {csch}(x)}{a+b \text {csch}(x)}dx}{a}+\frac {b x \left (a^2-2 b^2\right )}{a}\right )}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {3 i \left (\frac {b x \left (a^2-2 b^2\right )}{a}+\frac {2 b^4 \int \frac {i \csc (i x)}{a+i b \csc (i x)}dx}{a}\right )}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {3 i \left (\frac {b x \left (a^2-2 b^2\right )}{a}+\frac {2 i b^4 \int \frac {\csc (i x)}{a+i b \csc (i x)}dx}{a}\right )}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {3 i \left (\frac {2 b^3 \int \frac {1}{\frac {a \sinh (x)}{b}+1}dx}{a}+\frac {b x \left (a^2-2 b^2\right )}{a}\right )}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {3 i \left (\frac {b x \left (a^2-2 b^2\right )}{a}+\frac {2 b^3 \int \frac {1}{1-\frac {i a \sin (i x)}{b}}dx}{a}\right )}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {3 i \left (\frac {4 b^3 \int \frac {1}{-\tanh ^2\left (\frac {x}{2}\right )+\frac {2 a \tanh \left (\frac {x}{2}\right )}{b}+1}d\tanh \left (\frac {x}{2}\right )}{a}+\frac {b x \left (a^2-2 b^2\right )}{a}\right )}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {3 i \left (\frac {b x \left (a^2-2 b^2\right )}{a}-\frac {8 b^3 \int \frac {1}{4 \left (\frac {a^2}{b^2}+1\right )-\left (\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )^2}d\left (\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )}{a}\right )}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {3 b \sinh (x) \cosh (x)}{2 a}+\frac {i \left (\frac {2 i \left (2 a^2-3 b^2\right ) \cosh (x)}{a}-\frac {3 i \left (\frac {b x \left (a^2-2 b^2\right )}{a}-\frac {4 b^4 \text {arctanh}\left (\frac {b \left (\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\right )}{a}\right )}{2 a}\right )}{3 a}-\frac {i \sinh ^2(x) \cosh (x)}{3 a}\right )\) |
I*(((-1/3*I)*Cosh[x]*Sinh[x]^2)/a - ((I/3)*(((I/2)*(((-3*I)*((b*(a^2 - 2*b ^2)*x)/a - (4*b^4*ArcTanh[(b*((2*a)/b - 2*Tanh[x/2]))/(2*Sqrt[a^2 + b^2])] )/(a*Sqrt[a^2 + b^2])))/a + ((2*I)*(2*a^2 - 3*b^2)*Cosh[x])/a))/a - (3*b*C osh[x]*Sinh[x])/(2*a)))/a)
3.1.77.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[Cot[e + f*x]*((d*Csc[e + f*x])^n/(a*f*n)), x] - Sim p[1/(a*d*n) Int[((d*Csc[e + f*x])^(n + 1)/(a + b*Csc[e + f*x]))*Simp[b*n - a*(n + 1)*Csc[e + f*x] - b*(n + 1)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(197\) vs. \(2(93)=186\).
Time = 0.42 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.85
| method | result | size |
| default | \(-\frac {2 b^{4} \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{4} \sqrt {a^{2}+b^{2}}}+\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {a -b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {a^{2}+a b -2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b \left (a^{2}-2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a^{4}}-\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a +b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {-a^{2}+a b +2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {b \left (a^{2}-2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a^{4}}\) | \(198\) |
| risch | \(\frac {b x}{2 a^{2}}-\frac {b^{3} x}{a^{4}}+\frac {{\mathrm e}^{3 x}}{24 a}-\frac {b \,{\mathrm e}^{2 x}}{8 a^{2}}-\frac {3 \,{\mathrm e}^{x}}{8 a}+\frac {{\mathrm e}^{x} b^{2}}{2 a^{3}}-\frac {3 \,{\mathrm e}^{-x}}{8 a}+\frac {{\mathrm e}^{-x} b^{2}}{2 a^{3}}+\frac {b \,{\mathrm e}^{-2 x}}{8 a^{2}}+\frac {{\mathrm e}^{-3 x}}{24 a}+\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, a^{4}}-\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, a^{4}}\) | \(201\) |
-2*b^4/a^4/(a^2+b^2)^(1/2)*arctanh(1/2*(-2*b*tanh(1/2*x)+2*a)/(a^2+b^2)^(1 /2))+1/3/a/(tanh(1/2*x)+1)^3-1/2*(a-b)/a^2/(tanh(1/2*x)+1)^2-1/2*(a^2+a*b- 2*b^2)/a^3/(tanh(1/2*x)+1)+1/2*b*(a^2-2*b^2)/a^4*ln(tanh(1/2*x)+1)-1/3/a/( tanh(1/2*x)-1)^3-1/2*(a+b)/a^2/(tanh(1/2*x)-1)^2-1/2*(-a^2+a*b+2*b^2)/a^3/ (tanh(1/2*x)-1)-1/2*b*(a^2-2*b^2)/a^4*ln(tanh(1/2*x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (95) = 190\).
Time = 0.29 (sec) , antiderivative size = 807, normalized size of antiderivative = 7.54 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \]
1/24*((a^5 + a^3*b^2)*cosh(x)^6 + (a^5 + a^3*b^2)*sinh(x)^6 - 3*(a^4*b + a ^2*b^3)*cosh(x)^5 - 3*(a^4*b + a^2*b^3 - 2*(a^5 + a^3*b^2)*cosh(x))*sinh(x )^5 + a^5 + a^3*b^2 + 12*(a^4*b - a^2*b^3 - 2*b^5)*x*cosh(x)^3 - 3*(3*a^5 - a^3*b^2 - 4*a*b^4)*cosh(x)^4 - 3*(3*a^5 - a^3*b^2 - 4*a*b^4 - 5*(a^5 + a ^3*b^2)*cosh(x)^2 + 5*(a^4*b + a^2*b^3)*cosh(x))*sinh(x)^4 + 2*(10*(a^5 + a^3*b^2)*cosh(x)^3 - 15*(a^4*b + a^2*b^3)*cosh(x)^2 + 6*(a^4*b - a^2*b^3 - 2*b^5)*x - 6*(3*a^5 - a^3*b^2 - 4*a*b^4)*cosh(x))*sinh(x)^3 - 3*(3*a^5 - a^3*b^2 - 4*a*b^4)*cosh(x)^2 - 3*(3*a^5 - a^3*b^2 - 4*a*b^4 - 5*(a^5 + a^3 *b^2)*cosh(x)^4 + 10*(a^4*b + a^2*b^3)*cosh(x)^3 - 12*(a^4*b - a^2*b^3 - 2 *b^5)*x*cosh(x) + 6*(3*a^5 - a^3*b^2 - 4*a*b^4)*cosh(x)^2)*sinh(x)^2 + 24* (b^4*cosh(x)^3 + 3*b^4*cosh(x)^2*sinh(x) + 3*b^4*cosh(x)*sinh(x)^2 + b^4*s inh(x)^3)*sqrt(a^2 + b^2)*log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b*cosh( x) + a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2 + b^2)*(a*co sh(x) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*co sh(x) + b)*sinh(x) - a)) + 3*(a^4*b + a^2*b^3)*cosh(x) + 3*(2*(a^5 + a^3*b ^2)*cosh(x)^5 + a^4*b + a^2*b^3 - 5*(a^4*b + a^2*b^3)*cosh(x)^4 + 12*(a^4* b - a^2*b^3 - 2*b^5)*x*cosh(x)^2 - 4*(3*a^5 - a^3*b^2 - 4*a*b^4)*cosh(x)^3 - 2*(3*a^5 - a^3*b^2 - 4*a*b^4)*cosh(x))*sinh(x))/((a^6 + a^4*b^2)*cosh(x )^3 + 3*(a^6 + a^4*b^2)*cosh(x)^2*sinh(x) + 3*(a^6 + a^4*b^2)*cosh(x)*sinh (x)^2 + (a^6 + a^4*b^2)*sinh(x)^3)
\[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\sinh ^{3}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.47 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{4} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} - \frac {{\left (3 \, a b e^{\left (-x\right )} - a^{2} + 3 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )} e^{\left (3 \, x\right )}}{24 \, a^{3}} + \frac {3 \, a b e^{\left (-2 \, x\right )} + a^{2} e^{\left (-3 \, x\right )} - 3 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-x\right )}}{24 \, a^{3}} + \frac {{\left (a^{2} b - 2 \, b^{3}\right )} x}{2 \, a^{4}} \]
b^4*log((a*e^(-x) - b - sqrt(a^2 + b^2))/(a*e^(-x) - b + sqrt(a^2 + b^2))) /(sqrt(a^2 + b^2)*a^4) - 1/24*(3*a*b*e^(-x) - a^2 + 3*(3*a^2 - 4*b^2)*e^(- 2*x))*e^(3*x)/a^3 + 1/24*(3*a*b*e^(-2*x) + a^2*e^(-3*x) - 3*(3*a^2 - 4*b^2 )*e^(-x))/a^3 + 1/2*(a^2*b - 2*b^3)*x/a^4
Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.45 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{4} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} + \frac {a^{2} e^{\left (3 \, x\right )} - 3 \, a b e^{\left (2 \, x\right )} - 9 \, a^{2} e^{x} + 12 \, b^{2} e^{x}}{24 \, a^{3}} + \frac {{\left (a^{2} b - 2 \, b^{3}\right )} x}{2 \, a^{4}} + \frac {{\left (3 \, a^{2} b e^{x} + a^{3} - 3 \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-3 \, x\right )}}{24 \, a^{4}} \]
b^4*log(abs(2*a*e^x + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^x + 2*b + 2*sqrt( a^2 + b^2)))/(sqrt(a^2 + b^2)*a^4) + 1/24*(a^2*e^(3*x) - 3*a*b*e^(2*x) - 9 *a^2*e^x + 12*b^2*e^x)/a^3 + 1/2*(a^2*b - 2*b^3)*x/a^4 + 1/24*(3*a^2*b*e^x + a^3 - 3*(3*a^3 - 4*a*b^2)*e^(2*x))*e^(-3*x)/a^4
Time = 2.53 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.86 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {{\mathrm {e}}^{3\,x}}{24\,a}+\frac {x\,\left (a^2\,b-2\,b^3\right )}{2\,a^4}-\frac {{\mathrm {e}}^x\,\left (3\,a^2-4\,b^2\right )}{8\,a^3}+\frac {b\,{\mathrm {e}}^{-2\,x}}{8\,a^2}-\frac {b\,{\mathrm {e}}^{2\,x}}{8\,a^2}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a^2-4\,b^2\right )}{8\,a^3}-\frac {b^4\,\ln \left (-\frac {2\,b^4\,{\mathrm {e}}^x}{a^5}-\frac {2\,b^4\,\left (a-b\,{\mathrm {e}}^x\right )}{a^5\,\sqrt {a^2+b^2}}\right )}{a^4\,\sqrt {a^2+b^2}}+\frac {b^4\,\ln \left (\frac {2\,b^4\,\left (a-b\,{\mathrm {e}}^x\right )}{a^5\,\sqrt {a^2+b^2}}-\frac {2\,b^4\,{\mathrm {e}}^x}{a^5}\right )}{a^4\,\sqrt {a^2+b^2}} \]
exp(-3*x)/(24*a) + exp(3*x)/(24*a) + (x*(a^2*b - 2*b^3))/(2*a^4) - (exp(x) *(3*a^2 - 4*b^2))/(8*a^3) + (b*exp(-2*x))/(8*a^2) - (b*exp(2*x))/(8*a^2) - (exp(-x)*(3*a^2 - 4*b^2))/(8*a^3) - (b^4*log(- (2*b^4*exp(x))/a^5 - (2*b^ 4*(a - b*exp(x)))/(a^5*(a^2 + b^2)^(1/2))))/(a^4*(a^2 + b^2)^(1/2)) + (b^4 *log((2*b^4*(a - b*exp(x)))/(a^5*(a^2 + b^2)^(1/2)) - (2*b^4*exp(x))/a^5)) /(a^4*(a^2 + b^2)^(1/2))