Integrand size = 13, antiderivative size = 194 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {b^5 \arctan (\sinh (x))}{\left (a^2+b^2\right )^3}-\frac {b^3 \arctan (\sinh (x))}{2 \left (a^2+b^2\right )^2}-\frac {3 b \arctan (\sinh (x))}{8 \left (a^2+b^2\right )}+\frac {b^6 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^3}+\frac {\log (\sinh (x))}{a}-\frac {a \left (a^4+3 a^2 b^2+3 b^4\right ) \log (\tanh (x))}{\left (a^2+b^2\right )^3}+\frac {3 b \text {sech}(x) \tanh (x)}{8 \left (a^2+b^2\right )}-\frac {\left (a \left (a^2+2 b^2\right )-b^3 \text {csch}(x)\right ) \tanh ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac {(a-b \text {csch}(x)) \tanh ^4(x)}{4 \left (a^2+b^2\right )} \] Output:
-b^5*arctan(sinh(x))/(a^2+b^2)^3-1/2*b^3*arctan(sinh(x))/(a^2+b^2)^2-3*b*a rctan(sinh(x))/(8*a^2+8*b^2)+b^6*ln(a+b*csch(x))/a/(a^2+b^2)^3+ln(sinh(x)) /a-a*(a^4+3*a^2*b^2+3*b^4)*ln(tanh(x))/(a^2+b^2)^3+3*b*sech(x)*tanh(x)/(8* a^2+8*b^2)-1/2*(a*(a^2+2*b^2)-b^3*csch(x))*tanh(x)^2/(a^2+b^2)^2-(a-b*csch (x))*tanh(x)^4/(4*a^2+4*b^2)
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.30 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {a b \left (5 a^4+14 a^2 b^2+9 b^4\right ) \arctan (\sinh (x))+4 a \left (a^5+i a^4 b+3 a^3 b^2+3 i a^2 b^3+3 a b^4+3 i b^5\right ) \log (i-\sinh (x))+4 a \left (a^5-i a^4 b+3 a^3 b^2-3 i a^2 b^3+3 a b^4-3 i b^5\right ) \log (i+\sinh (x))+8 b^6 \log (b+a \sinh (x))+4 a^2 \left (2 a^4+5 a^2 b^2+3 b^4\right ) \text {sech}^2(x)-2 a^2 \left (a^2+b^2\right )^2 \text {sech}^4(x)+a b \left (5 a^4+14 a^2 b^2+9 b^4\right ) \text {sech}(x) \tanh (x)-2 a b \left (a^2+b^2\right )^2 \text {sech}^3(x) \tanh (x)}{8 a \left (a^2+b^2\right )^3} \] Input:
Integrate[Tanh[x]^5/(a + b*Csch[x]),x]
Output:
(a*b*(5*a^4 + 14*a^2*b^2 + 9*b^4)*ArcTan[Sinh[x]] + 4*a*(a^5 + I*a^4*b + 3 *a^3*b^2 + (3*I)*a^2*b^3 + 3*a*b^4 + (3*I)*b^5)*Log[I - Sinh[x]] + 4*a*(a^ 5 - I*a^4*b + 3*a^3*b^2 - (3*I)*a^2*b^3 + 3*a*b^4 - (3*I)*b^5)*Log[I + Sin h[x]] + 8*b^6*Log[b + a*Sinh[x]] + 4*a^2*(2*a^4 + 5*a^2*b^2 + 3*b^4)*Sech[ x]^2 - 2*a^2*(a^2 + b^2)^2*Sech[x]^4 + a*b*(5*a^4 + 14*a^2*b^2 + 9*b^4)*Se ch[x]*Tanh[x] - 2*a*b*(a^2 + b^2)^2*Sech[x]^3*Tanh[x])/(8*a*(a^2 + b^2)^3)
Time = 0.56 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.32, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 26, 4373, 25, 615, 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.
| \(\displaystyle \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\cot (i x)^5 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\cot (i x)^5 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 4373 |
| \(\displaystyle b^6 \int -\frac {\sinh (x)}{b (a+b \text {csch}(x)) \left (\text {csch}^2(x) b^2+b^2\right )^3}d(b \text {csch}(x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -b^6 \int \frac {\sinh (x)}{b (a+b \text {csch}(x)) \left (\text {csch}^2(x) b^2+b^2\right )^3}d(b \text {csch}(x))\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle -b^6 \int \left (\frac {-b^2-a \text {csch}(x) b}{b^2 \left (a^2+b^2\right ) \left (\text {csch}^2(x) b^2+b^2\right )^3}+\frac {\sinh (x)}{a b^7}-\frac {1}{a \left (a^2+b^2\right )^3 (a+b \text {csch}(x))}+\frac {-b^6-a \left (a^4+3 b^2 a^2+3 b^4\right ) \text {csch}(x) b}{b^6 \left (a^2+b^2\right )^3 \left (\text {csch}^2(x) b^2+b^2\right )}+\frac {-b^4-a \left (a^2+2 b^2\right ) \text {csch}(x) b}{b^4 \left (a^2+b^2\right )^2 \left (\text {csch}^2(x) b^2+b^2\right )^2}\right )d(b \text {csch}(x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b^6 \left (\frac {\arctan (\text {csch}(x))}{b \left (a^2+b^2\right )^3}+\frac {3 \arctan (\text {csch}(x))}{8 b^5 \left (a^2+b^2\right )}+\frac {\arctan (\text {csch}(x))}{2 b^3 \left (a^2+b^2\right )^2}-\frac {a-b \text {csch}(x)}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \text {csch}^2(x)+b^2\right )^2}+\frac {\log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^3}+\frac {3 \text {csch}(x)}{8 b^3 \left (a^2+b^2\right ) \left (b^2 \text {csch}^2(x)+b^2\right )}-\frac {a \left (a^2+2 b^2\right )-b^3 \text {csch}(x)}{2 b^4 \left (a^2+b^2\right )^2 \left (b^2 \text {csch}^2(x)+b^2\right )}+\frac {a \left (a^4+3 a^2 b^2+3 b^4\right ) \log \left (b^2 \text {csch}^2(x)+b^2\right )}{2 b^6 \left (a^2+b^2\right )^3}-\frac {\log (b \text {csch}(x))}{a b^6}\right )\) |
Input:
Int[Tanh[x]^5/(a + b*Csch[x]),x]
Output:
b^6*(ArcTan[Csch[x]]/(b*(a^2 + b^2)^3) + ArcTan[Csch[x]]/(2*b^3*(a^2 + b^2 )^2) + (3*ArcTan[Csch[x]])/(8*b^5*(a^2 + b^2)) - (a - b*Csch[x])/(4*b^2*(a ^2 + b^2)*(b^2 + b^2*Csch[x]^2)^2) + (3*Csch[x])/(8*b^3*(a^2 + b^2)*(b^2 + b^2*Csch[x]^2)) - (a*(a^2 + 2*b^2) - b^3*Csch[x])/(2*b^4*(a^2 + b^2)^2*(b ^2 + b^2*Csch[x]^2)) - Log[b*Csch[x]]/(a*b^6) + Log[a + b*Csch[x]]/(a*(a^2 + b^2)^3) + (a*(a^4 + 3*a^2*b^2 + 3*b^4)*Log[b^2 + b^2*Csch[x]^2])/(2*b^6 *(a^2 + b^2)^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] && ILtQ[p, 0]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1)) Subst[Int[(b^2 - x^ 2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 0.78 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.88
| method | result | size |
| default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\frac {2 \left (\left (-\frac {3}{8} a^{4} b -\frac {5}{4} a^{2} b^{3}-\frac {7}{8} b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{7}+\left (-a^{5}-3 a^{3} b^{2}-2 b^{4} a \right ) \tanh \left (\frac {x}{2}\right )^{6}+\left (-\frac {13}{4} a^{2} b^{3}-\frac {15}{8} b^{5}-\frac {11}{8} a^{4} b \right ) \tanh \left (\frac {x}{2}\right )^{5}+\left (-4 a^{5}-10 a^{3} b^{2}-6 b^{4} a \right ) \tanh \left (\frac {x}{2}\right )^{4}+\left (\frac {13}{4} a^{2} b^{3}+\frac {15}{8} b^{5}+\frac {11}{8} a^{4} b \right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a^{5}-3 a^{3} b^{2}-2 b^{4} a \right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (\frac {3}{8} a^{4} b +\frac {5}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )^{4}}+\frac {\left (8 a^{5}+24 a^{3} b^{2}+24 b^{4} a \right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )}{8}+\frac {\left (-3 a^{4} b -10 a^{2} b^{3}-15 b^{5}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}+\frac {b^{6} \ln \left (-b \tanh \left (\frac {x}{2}\right )^{2}+2 a \tanh \left (\frac {x}{2}\right )+b \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right ) a}\) | \(364\) |
| risch | \(\frac {x}{a}-\frac {2 x \,a^{5}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 x \,a^{3} b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 x \,b^{4} a}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 x \,b^{6}}{a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\left (5 a^{2} b \,{\mathrm e}^{6 x}+9 b^{3} {\mathrm e}^{6 x}+16 a^{3} {\mathrm e}^{5 x}+24 a \,b^{2} {\mathrm e}^{5 x}-3 a^{2} b \,{\mathrm e}^{4 x}+b^{3} {\mathrm e}^{4 x}+16 a^{3} {\mathrm e}^{3 x}+32 a \,b^{2} {\mathrm e}^{3 x}+3 a^{2} b \,{\mathrm e}^{2 x}-b^{3} {\mathrm e}^{2 x}+16 a^{3} {\mathrm e}^{x}+24 \,{\mathrm e}^{x} b^{2} a -5 a^{2} b -9 b^{3}\right ) {\mathrm e}^{x}}{4 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left ({\mathrm e}^{2 x}+1\right )^{4}}+\frac {5 i \ln \left ({\mathrm e}^{x}-i\right ) a^{2} b^{3}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 i \ln \left ({\mathrm e}^{x}+i\right ) a^{4} b}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {5 i \ln \left ({\mathrm e}^{x}+i\right ) a^{2} b^{3}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\ln \left ({\mathrm e}^{x}-i\right ) a^{5}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 \ln \left ({\mathrm e}^{x}-i\right ) a^{3} b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 \ln \left ({\mathrm e}^{x}-i\right ) b^{4} a}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {15 i \ln \left ({\mathrm e}^{x}+i\right ) b^{5}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 i \ln \left ({\mathrm e}^{x}-i\right ) a^{4} b}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {15 i \ln \left ({\mathrm e}^{x}-i\right ) b^{5}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\ln \left ({\mathrm e}^{x}+i\right ) a^{5}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 \ln \left ({\mathrm e}^{x}+i\right ) a^{3} b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 \ln \left ({\mathrm e}^{x}+i\right ) b^{4} a}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {b^{6} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(787\) |
Input:
int(tanh(x)^5/(a+b*csch(x)),x,method=_RETURNVERBOSE)
Output:
-1/a*ln(tanh(1/2*x)-1)+2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*(((-3/8*a^4*b-5/4*a ^2*b^3-7/8*b^5)*tanh(1/2*x)^7+(-a^5-3*a^3*b^2-2*a*b^4)*tanh(1/2*x)^6+(-13/ 4*a^2*b^3-15/8*b^5-11/8*a^4*b)*tanh(1/2*x)^5+(-4*a^5-10*a^3*b^2-6*a*b^4)*t anh(1/2*x)^4+(13/4*a^2*b^3+15/8*b^5+11/8*a^4*b)*tanh(1/2*x)^3+(-a^5-3*a^3* b^2-2*a*b^4)*tanh(1/2*x)^2+(3/8*a^4*b+5/4*a^2*b^3+7/8*b^5)*tanh(1/2*x))/(t anh(1/2*x)^2+1)^4+1/16*(8*a^5+24*a^3*b^2+24*a*b^4)*ln(tanh(1/2*x)^2+1)+1/8 *(-3*a^4*b-10*a^2*b^3-15*b^5)*arctan(tanh(1/2*x)))-1/a*ln(tanh(1/2*x)+1)+b ^6/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/a*ln(-b*tanh(1/2*x)^2+2*a*tanh(1/2*x)+b)
Leaf count of result is larger than twice the leaf count of optimal. 4025 vs. \(2 (185) = 370\).
Time = 0.20 (sec) , antiderivative size = 4025, normalized size of antiderivative = 20.75 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \] Input:
integrate(tanh(x)^5/(a+b*csch(x)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\tanh ^{5}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \] Input:
integrate(tanh(x)**5/(a+b*csch(x)),x)
Output:
Integral(tanh(x)**5/(a + b*csch(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (185) = 370\).
Time = 0.14 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.97 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{6} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}} + \frac {{\left (3 \, a^{4} b + 10 \, a^{2} b^{3} + 15 \, b^{5}\right )} \arctan \left (e^{\left (-x\right )}\right )}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (a^{5} + 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (5 \, a^{2} b + 9 \, b^{3}\right )} e^{\left (-x\right )} + 8 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} e^{\left (-2 \, x\right )} - {\left (3 \, a^{2} b - b^{3}\right )} e^{\left (-3 \, x\right )} + 16 \, {\left (a^{3} + 2 \, a b^{2}\right )} e^{\left (-4 \, x\right )} + {\left (3 \, a^{2} b - b^{3}\right )} e^{\left (-5 \, x\right )} + 8 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} e^{\left (-6 \, x\right )} - {\left (5 \, a^{2} b + 9 \, b^{3}\right )} e^{\left (-7 \, x\right )}}{4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} + \frac {x}{a} \] Input:
integrate(tanh(x)^5/(a+b*csch(x)),x, algorithm="maxima")
Output:
b^6*log(-2*b*e^(-x) + a*e^(-2*x) - a)/(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 ) + 1/4*(3*a^4*b + 10*a^2*b^3 + 15*b^5)*arctan(e^(-x))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (a^5 + 3*a^3*b^2 + 3*a*b^4)*log(e^(-2*x) + 1)/(a^6 + 3* a^4*b^2 + 3*a^2*b^4 + b^6) + 1/4*((5*a^2*b + 9*b^3)*e^(-x) + 8*(2*a^3 + 3* a*b^2)*e^(-2*x) - (3*a^2*b - b^3)*e^(-3*x) + 16*(a^3 + 2*a*b^2)*e^(-4*x) + (3*a^2*b - b^3)*e^(-5*x) + 8*(2*a^3 + 3*a*b^2)*e^(-6*x) - (5*a^2*b + 9*b^ 3)*e^(-7*x))/(a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*e^(-2*x) + 6*(a^4 + 2*a^2*b^2 + b^4)*e^(-4*x) + 4*(a^4 + 2*a^2*b^2 + b^4)*e^(-6*x) + (a^4 + 2*a^2*b^2 + b^4)*e^(-8*x)) + x/a
Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (185) = 370\).
Time = 0.13 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.23 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{6} \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}} - \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (3 \, a^{4} b + 10 \, a^{2} b^{3} + 15 \, b^{5}\right )}}{16 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (a^{5} + 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {3 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 9 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 9 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 5 \, a^{4} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 14 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 9 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 8 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 32 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 48 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 12 \, a^{4} b {\left (e^{\left (-x\right )} - e^{x}\right )} + 40 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} + 28 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )} + 16 \, a^{3} b^{2} + 64 \, a b^{4}}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} \] Input:
integrate(tanh(x)^5/(a+b*csch(x)),x, algorithm="giac")
Output:
b^6*log(abs(-a*(e^(-x) - e^x) + 2*b))/(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 ) - 1/16*(pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*(3*a^4*b + 10*a^2*b^3 + 15*b^5)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*(a^5 + 3*a^3*b^2 + 3*a* b^4)*log((e^(-x) - e^x)^2 + 4)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 1/4*( 3*a^5*(e^(-x) - e^x)^4 + 9*a^3*b^2*(e^(-x) - e^x)^4 + 9*a*b^4*(e^(-x) - e^ x)^4 + 5*a^4*b*(e^(-x) - e^x)^3 + 14*a^2*b^3*(e^(-x) - e^x)^3 + 9*b^5*(e^( -x) - e^x)^3 + 8*a^5*(e^(-x) - e^x)^2 + 32*a^3*b^2*(e^(-x) - e^x)^2 + 48*a *b^4*(e^(-x) - e^x)^2 + 12*a^4*b*(e^(-x) - e^x) + 40*a^2*b^3*(e^(-x) - e^x ) + 28*b^5*(e^(-x) - e^x) + 16*a^3*b^2 + 64*a*b^4)/((a^6 + 3*a^4*b^2 + 3*a ^2*b^4 + b^6)*((e^(-x) - e^x)^2 + 4)^2)
Time = 8.11 (sec) , antiderivative size = 611, normalized size of antiderivative = 3.15 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {\frac {6\,{\mathrm {e}}^x\,\left (a^2\,b+b^3\right )}{{\left (a^2+b^2\right )}^2}+\frac {8\,\left (a^4+a^2\,b^2\right )}{a\,{\left (a^2+b^2\right )}^2}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {4\,a}{a^2+b^2}+\frac {4\,b\,{\mathrm {e}}^x}{a^2+b^2}}{4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {x}{a}-\frac {\frac {{\mathrm {e}}^x\,\left (9\,a^2\,b+13\,b^3\right )}{2\,{\left (a^2+b^2\right )}^2}+\frac {2\,\left (4\,a^4+5\,a^2\,b^2\right )}{a\,{\left (a^2+b^2\right )}^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {\frac {{\mathrm {e}}^x\,\left (5\,a^4\,b+14\,a^2\,b^3+9\,b^5\right )}{4\,{\left (a^2+b^2\right )}^3}+\frac {2\,\left (2\,a^6+5\,a^4\,b^2+3\,a^2\,b^4\right )}{a\,{\left (a^2+b^2\right )}^3}}{{\mathrm {e}}^{2\,x}+1}+\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,\left (-8\,a^2+a\,b\,21{}\mathrm {i}+15\,b^2\right )}{8\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}+\frac {b^6\,\ln \left (64\,a^{13}\,{\mathrm {e}}^{2\,x}-64\,a\,b^{12}-64\,a^{13}+159\,a^3\,b^{10}-492\,a^5\,b^8-1214\,a^7\,b^6-1020\,a^9\,b^4-393\,a^{11}\,b^2+128\,b^{13}\,{\mathrm {e}}^x-159\,a^3\,b^{10}\,{\mathrm {e}}^{2\,x}+492\,a^5\,b^8\,{\mathrm {e}}^{2\,x}+1214\,a^7\,b^6\,{\mathrm {e}}^{2\,x}+1020\,a^9\,b^4\,{\mathrm {e}}^{2\,x}+393\,a^{11}\,b^2\,{\mathrm {e}}^{2\,x}+128\,a^{12}\,b\,{\mathrm {e}}^x+64\,a\,b^{12}\,{\mathrm {e}}^{2\,x}-318\,a^2\,b^{11}\,{\mathrm {e}}^x+984\,a^4\,b^9\,{\mathrm {e}}^x+2428\,a^6\,b^7\,{\mathrm {e}}^x+2040\,a^8\,b^5\,{\mathrm {e}}^x+786\,a^{10}\,b^3\,{\mathrm {e}}^x\right )}{a^7+3\,a^5\,b^2+3\,a^3\,b^4+a\,b^6}+\frac {\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,\left (-a^2\,8{}\mathrm {i}+21\,a\,b+b^2\,15{}\mathrm {i}\right )}{8\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )} \] Input:
int(tanh(x)^5/(a + b/sinh(x)),x)
Output:
((6*exp(x)*(a^2*b + b^3))/(a^2 + b^2)^2 + (8*(a^4 + a^2*b^2))/(a*(a^2 + b^ 2)^2))/(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1) - ((4*a)/(a^2 + b^2) + (4* b*exp(x))/(a^2 + b^2))/(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1) - x/a - ((exp(x)*(9*a^2*b + 13*b^3))/(2*(a^2 + b^2)^2) + (2*(4*a^4 + 5* a^2*b^2))/(a*(a^2 + b^2)^2))/(2*exp(2*x) + exp(4*x) + 1) + ((exp(x)*(5*a^4 *b + 9*b^5 + 14*a^2*b^3))/(4*(a^2 + b^2)^3) + (2*(2*a^6 + 3*a^2*b^4 + 5*a^ 4*b^2))/(a*(a^2 + b^2)^3))/(exp(2*x) + 1) + (log(exp(x)*1i + 1)*(a*b*21i - 8*a^2 + 15*b^2))/(8*(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)) + (b^6*log(64*a^ 13*exp(2*x) - 64*a*b^12 - 64*a^13 + 159*a^3*b^10 - 492*a^5*b^8 - 1214*a^7* b^6 - 1020*a^9*b^4 - 393*a^11*b^2 + 128*b^13*exp(x) - 159*a^3*b^10*exp(2*x ) + 492*a^5*b^8*exp(2*x) + 1214*a^7*b^6*exp(2*x) + 1020*a^9*b^4*exp(2*x) + 393*a^11*b^2*exp(2*x) + 128*a^12*b*exp(x) + 64*a*b^12*exp(2*x) - 318*a^2* b^11*exp(x) + 984*a^4*b^9*exp(x) + 2428*a^6*b^7*exp(x) + 2040*a^8*b^5*exp( x) + 786*a^10*b^3*exp(x)))/(a*b^6 + a^7 + 3*a^3*b^4 + 3*a^5*b^2) + (log(ex p(x) + 1i)*(21*a*b - a^2*8i + b^2*15i))/(8*(a*b^2*3i + 3*a^2*b - a^3*1i - b^3))
Time = 0.23 (sec) , antiderivative size = 1309, normalized size of antiderivative = 6.75 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx =\text {Too large to display} \] Input:
int(tanh(x)^5/(a+b*csch(x)),x)
Output:
( - 3*e**(8*x)*atan(e**x)*a**5*b - 10*e**(8*x)*atan(e**x)*a**3*b**3 - 15*e **(8*x)*atan(e**x)*a*b**5 - 12*e**(6*x)*atan(e**x)*a**5*b - 40*e**(6*x)*at an(e**x)*a**3*b**3 - 60*e**(6*x)*atan(e**x)*a*b**5 - 18*e**(4*x)*atan(e**x )*a**5*b - 60*e**(4*x)*atan(e**x)*a**3*b**3 - 90*e**(4*x)*atan(e**x)*a*b** 5 - 12*e**(2*x)*atan(e**x)*a**5*b - 40*e**(2*x)*atan(e**x)*a**3*b**3 - 60* e**(2*x)*atan(e**x)*a*b**5 - 3*atan(e**x)*a**5*b - 10*atan(e**x)*a**3*b**3 - 15*atan(e**x)*a*b**5 + 4*e**(8*x)*log(e**(2*x) + 1)*a**6 + 12*e**(8*x)* log(e**(2*x) + 1)*a**4*b**2 + 12*e**(8*x)*log(e**(2*x) + 1)*a**2*b**4 + 4* e**(8*x)*log(e**(2*x)*a + 2*e**x*b - a)*b**6 - 4*e**(8*x)*a**6*x - 4*e**(8 *x)*a**6 - 12*e**(8*x)*a**4*b**2*x - 10*e**(8*x)*a**4*b**2 - 12*e**(8*x)*a **2*b**4*x - 6*e**(8*x)*a**2*b**4 - 4*e**(8*x)*b**6*x + 5*e**(7*x)*a**5*b + 14*e**(7*x)*a**3*b**3 + 9*e**(7*x)*a*b**5 + 16*e**(6*x)*log(e**(2*x) + 1 )*a**6 + 48*e**(6*x)*log(e**(2*x) + 1)*a**4*b**2 + 48*e**(6*x)*log(e**(2*x ) + 1)*a**2*b**4 + 16*e**(6*x)*log(e**(2*x)*a + 2*e**x*b - a)*b**6 - 16*e* *(6*x)*a**6*x - 48*e**(6*x)*a**4*b**2*x - 48*e**(6*x)*a**2*b**4*x - 16*e** (6*x)*b**6*x - 3*e**(5*x)*a**5*b - 2*e**(5*x)*a**3*b**3 + e**(5*x)*a*b**5 + 24*e**(4*x)*log(e**(2*x) + 1)*a**6 + 72*e**(4*x)*log(e**(2*x) + 1)*a**4* b**2 + 72*e**(4*x)*log(e**(2*x) + 1)*a**2*b**4 + 24*e**(4*x)*log(e**(2*x)* a + 2*e**x*b - a)*b**6 - 24*e**(4*x)*a**6*x - 8*e**(4*x)*a**6 - 72*e**(4*x )*a**4*b**2*x - 12*e**(4*x)*a**4*b**2 - 72*e**(4*x)*a**2*b**4*x - 4*e**...