3.113 \(\int \frac{\tanh ^5(x)}{a+b \text{csch}(x)} \, dx\)

Optimal. Leaf size=194 \[ -\frac{a \left (3 a^2 b^2+a^4+3 b^4\right ) \log (\tanh (x))}{\left (a^2+b^2\right )^3}-\frac{b^5 \tan ^{-1}(\sinh (x))}{\left (a^2+b^2\right )^3}-\frac{b^3 \tan ^{-1}(\sinh (x))}{2 \left (a^2+b^2\right )^2}-\frac{3 b \tan ^{-1}(\sinh (x))}{8 \left (a^2+b^2\right )}+\frac{b^6 \log (a+b \text{csch}(x))}{a \left (a^2+b^2\right )^3}-\frac{\tanh ^4(x) (a-b \text{csch}(x))}{4 \left (a^2+b^2\right )}-\frac{\tanh ^2(x) \left (a \left (a^2+2 b^2\right )-b^3 \text{csch}(x)\right )}{2 \left (a^2+b^2\right )^2}+\frac{3 b \tanh (x) \text{sech}(x)}{8 \left (a^2+b^2\right )}+\frac{\log (\sinh (x))}{a} \]

[Out]

-((b^5*ArcTan[Sinh[x]])/(a^2 + b^2)^3) - (b^3*ArcTan[Sinh[x]])/(2*(a^2 + b^2)^2) - (3*b*ArcTan[Sinh[x]])/(8*(a
^2 + b^2)) + (b^6*Log[a + b*Csch[x]])/(a*(a^2 + b^2)^3) + Log[Sinh[x]]/a - (a*(a^4 + 3*a^2*b^2 + 3*b^4)*Log[Ta
nh[x]])/(a^2 + b^2)^3 + (3*b*Sech[x]*Tanh[x])/(8*(a^2 + b^2)) - ((a*(a^2 + 2*b^2) - b^3*Csch[x])*Tanh[x]^2)/(2
*(a^2 + b^2)^2) - ((a - b*Csch[x])*Tanh[x]^4)/(4*(a^2 + b^2))

________________________________________________________________________________________

Rubi [A]  time = 0.254877, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {3885, 894, 639, 199, 203, 635, 260} \[ -\frac{a \left (3 a^2 b^2+a^4+3 b^4\right ) \log (\tanh (x))}{\left (a^2+b^2\right )^3}-\frac{b^5 \tan ^{-1}(\sinh (x))}{\left (a^2+b^2\right )^3}-\frac{b^3 \tan ^{-1}(\sinh (x))}{2 \left (a^2+b^2\right )^2}-\frac{3 b \tan ^{-1}(\sinh (x))}{8 \left (a^2+b^2\right )}+\frac{b^6 \log (a+b \text{csch}(x))}{a \left (a^2+b^2\right )^3}-\frac{\tanh ^4(x) (a-b \text{csch}(x))}{4 \left (a^2+b^2\right )}-\frac{\tanh ^2(x) \left (a \left (a^2+2 b^2\right )-b^3 \text{csch}(x)\right )}{2 \left (a^2+b^2\right )^2}+\frac{3 b \tanh (x) \text{sech}(x)}{8 \left (a^2+b^2\right )}+\frac{\log (\sinh (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^5/(a + b*Csch[x]),x]

[Out]

-((b^5*ArcTan[Sinh[x]])/(a^2 + b^2)^3) - (b^3*ArcTan[Sinh[x]])/(2*(a^2 + b^2)^2) - (3*b*ArcTan[Sinh[x]])/(8*(a
^2 + b^2)) + (b^6*Log[a + b*Csch[x]])/(a*(a^2 + b^2)^3) + Log[Sinh[x]]/a - (a*(a^4 + 3*a^2*b^2 + 3*b^4)*Log[Ta
nh[x]])/(a^2 + b^2)^3 + (3*b*Sech[x]*Tanh[x])/(8*(a^2 + b^2)) - ((a*(a^2 + 2*b^2) - b^3*Csch[x])*Tanh[x]^2)/(2
*(a^2 + b^2)^2) - ((a - b*Csch[x])*Tanh[x]^4)/(4*(a^2 + b^2))

Rule 3885

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Dist[(-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]

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{\tanh ^5(x)}{a+b \text{csch}(x)} \, dx &=b^6 \operatorname{Subst}\left (\int \frac{1}{x (a+x) \left (-b^2-x^2\right )^3} \, dx,x,b \text{csch}(x)\right )\\ &=b^6 \operatorname{Subst}\left (\int \left (-\frac{1}{a b^6 x}+\frac{1}{a \left (a^2+b^2\right )^3 (a+x)}+\frac{b^2+a x}{b^2 \left (a^2+b^2\right ) \left (b^2+x^2\right )^3}+\frac{b^4+a \left (a^2+2 b^2\right ) x}{b^4 \left (a^2+b^2\right )^2 \left (b^2+x^2\right )^2}+\frac{b^6+a \left (a^4+3 a^2 b^2+3 b^4\right ) x}{b^6 \left (a^2+b^2\right )^3 \left (b^2+x^2\right )}\right ) \, dx,x,b \text{csch}(x)\right )\\ &=\frac{b^6 \log (a+b \text{csch}(x))}{a \left (a^2+b^2\right )^3}+\frac{\log (\sinh (x))}{a}+\frac{\operatorname{Subst}\left (\int \frac{b^6+a \left (a^4+3 a^2 b^2+3 b^4\right ) x}{b^2+x^2} \, dx,x,b \text{csch}(x)\right )}{\left (a^2+b^2\right )^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{b^4+a \left (a^2+2 b^2\right ) x}{\left (b^2+x^2\right )^2} \, dx,x,b \text{csch}(x)\right )}{\left (a^2+b^2\right )^2}+\frac{b^4 \operatorname{Subst}\left (\int \frac{b^2+a x}{\left (b^2+x^2\right )^3} \, dx,x,b \text{csch}(x)\right )}{a^2+b^2}\\ &=\frac{b^6 \log (a+b \text{csch}(x))}{a \left (a^2+b^2\right )^3}+\frac{\log (\sinh (x))}{a}-\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 )}+\frac{b^6 \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \text{csch}(x)\right )}{\left (a^2+b^2\right )^3}+\frac{b^4 \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \text{csch}(x)\right )}{2 \left (a^2+b^2\right )^2}+\frac{\left (3 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{\left (b^2+x^2\right )^2} \, dx,x,b \text{csch}(x)\right )}{4 \left (a^2+b^2\right )}+\frac{\left (a \left (a^4+3 a^2 b^2+3 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \text{csch}(x)\right )}{\left (a^2+b^2\right )^3}\\ &=-\frac{b^5 \tan ^{-1}(\sinh (x))}{\left (a^2+b^2\right )^3}-\frac{b^3 \tan ^{-1}(\sinh (x))}{2 \left (a^2+b^2\right )^2}+\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 )}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \text{csch}(x)\right )}{8 \left (a^2+b^2\right )}\\ &=-\frac{b^5 \tan ^{-1}(\sinh (x))}{\left (a^2+b^2\right )^3}-\frac{b^3 \tan ^{-1}(\sinh (x))}{2 \left (a^2+b^2\right )^2}-\frac{3 b \tan ^{-1}(\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 )}\\ \end{align*}

Mathematica [C]  time = 0.492813, size = 253, normalized size = 1.3 \[ \frac{-2 a^2 \left (a^2+b^2\right )^2 \text{sech}^4(x)+4 a^2 \left (5 a^2 b^2+2 a^4+3 b^4\right ) \text{sech}^2(x)+4 a \left (3 a^3 b^2+3 i a^2 b^3+i a^4 b+a^5+3 a b^4+3 i b^5\right ) \log (-\sinh (x)+i)+4 a \left (3 a^3 b^2-3 i a^2 b^3-i a^4 b+a^5+3 a b^4-3 i b^5\right ) \log (\sinh (x)+i)+a b \left (14 a^2 b^2+5 a^4+9 b^4\right ) \tan ^{-1}(\sinh (x))-2 a b \left (a^2+b^2\right )^2 \tanh (x) \text{sech}^3(x)+a b \left (14 a^2 b^2+5 a^4+9 b^4\right ) \tanh (x) \text{sech}(x)+8 b^6 \log (a \sinh (x)+b)}{8 a \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]^5/(a + b*Csch[x]),x]

[Out]

(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 + S
inh[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)*Sech[x]*Tanh[x] - 2*a*b*(a^2 + b^2)^2*Sech[x]^3*Tanh[x])/(8*a*(a^2 + b^2
)^3)

________________________________________________________________________________________

Maple [B]  time = 0.063, size = 1323, normalized size = 6.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^5/(a+b*csch(x)),x)

[Out]

-5/2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^7*a^2*b^3-20/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/
(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^4*a^3*b^2-6/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^2*a^
3*b^2-6/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^6*a^3*b^2-4/(a^4+2*a^2*b^2+b^4)/(a^2+b^2
)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^6*a*b^4-13/2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^5
*a^2*b^3+3/4/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)*a^4*b-12/(a^4+2*a^2*b^2+b^4)/(a^2+b
^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^4*a*b^4+13/2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)
^3*a^2*b^3+11/4/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^3*a^4*b-11/4/(a^4+2*a^2*b^2+b^4)
/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^5*a^4*b-3/4/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(
1/2*x)^7*a^4*b+5/2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)*a^2*b^3-4/(a^4+2*a^2*b^2+b^4)
/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^2*a*b^4-1/a*ln(tanh(1/2*x)+1)-1/a*ln(tanh(1/2*x)-1)-15/4/(a^4+2*a^2
*b^2+b^4)/(a^2+b^2)*arctan(tanh(1/2*x))*b^5+1/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*ln(tanh(1/2*x)^2+1)*a^5+15/4/(a^4+
2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^3*b^5-2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+
1)^4*tanh(1/2*x)^2*a^5+7/4/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)*b^5+3/(a^4+2*a^2*b^2+
b^4)/(a^2+b^2)*ln(tanh(1/2*x)^2+1)*a^3*b^2+3/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*ln(tanh(1/2*x)^2+1)*a*b^4-7/4/(a^4+
2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^7*b^5+b^6/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/a*ln(tanh(1/2
*x)^2*b-2*a*tanh(1/2*x)-b)-5/2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*arctan(tanh(1/2*x))*a^2*b^3-15/4/(a^4+2*a^2*b^2+b
^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^5*b^5-8/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1
/2*x)^4*a^5-2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^6*a^5-3/4/(a^4+2*a^2*b^2+b^4)/(a^2
+b^2)*arctan(tanh(1/2*x))*a^4*b

________________________________________________________________________________________

Maxima [B]  time = 1.60776, size = 517, normalized size = 2.66 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^5/(a+b*csch(x)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.64532, size = 9482, normalized size = 48.88 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^5/(a+b*csch(x)),x, algorithm="fricas")

[Out]

-1/4*(4*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x)^8 + 4*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*sinh(x)^8 -
(5*a^5*b + 14*a^3*b^3 + 9*a*b^5)*cosh(x)^7 - (5*a^5*b + 14*a^3*b^3 + 9*a*b^5 - 32*(a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*x*cosh(x))*sinh(x)^7 - 8*(2*a^6 + 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x)*cos
h(x)^6 - (16*a^6 + 40*a^4*b^2 + 24*a^2*b^4 - 112*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x)^2 - 16*(a^6 + 3
*a^4*b^2 + 3*a^2*b^4 + b^6)*x + 7*(5*a^5*b + 14*a^3*b^3 + 9*a*b^5)*cosh(x))*sinh(x)^6 + (3*a^5*b + 2*a^3*b^3 -
a*b^5)*cosh(x)^5 + (3*a^5*b + 2*a^3*b^3 - a*b^5 + 224*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x)^3 - 21*(5
*a^5*b + 14*a^3*b^3 + 9*a*b^5)*cosh(x)^2 - 48*(2*a^6 + 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*x)*cosh(x))*sinh(x)^5 - 8*(2*a^6 + 6*a^4*b^2 + 4*a^2*b^4 - 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x)*cos
h(x)^4 - (16*a^6 + 48*a^4*b^2 + 32*a^2*b^4 - 280*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x)^4 + 35*(5*a^5*b
+ 14*a^3*b^3 + 9*a*b^5)*cosh(x)^3 + 120*(2*a^6 + 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
)*x)*cosh(x)^2 - 24*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x - 5*(3*a^5*b + 2*a^3*b^3 - a*b^5)*cosh(x))*sinh(x)^4
- (3*a^5*b + 2*a^3*b^3 - a*b^5)*cosh(x)^3 + (224*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x)^5 - 3*a^5*b -
2*a^3*b^3 + a*b^5 - 35*(5*a^5*b + 14*a^3*b^3 + 9*a*b^5)*cosh(x)^4 - 160*(2*a^6 + 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^
6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x)*cosh(x)^3 + 10*(3*a^5*b + 2*a^3*b^3 - a*b^5)*cosh(x)^2 - 32*(2*a^6 + 6*a^4
*b^2 + 4*a^2*b^4 - 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x)*cosh(x))*sinh(x)^3 - 8*(2*a^6 + 5*a^4*b^2 + 3*a^2*
b^4 - 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x)*cosh(x)^2 + (112*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x)^
6 - 16*a^6 - 40*a^4*b^2 - 24*a^2*b^4 - 21*(5*a^5*b + 14*a^3*b^3 + 9*a*b^5)*cosh(x)^5 - 120*(2*a^6 + 5*a^4*b^2
+ 3*a^2*b^4 - 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x)*cosh(x)^4 + 10*(3*a^5*b + 2*a^3*b^3 - a*b^5)*cosh(x)^3
- 48*(2*a^6 + 6*a^4*b^2 + 4*a^2*b^4 - 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x)*cosh(x)^2 + 16*(a^6 + 3*a^4*b^2
+ 3*a^2*b^4 + b^6)*x - 3*(3*a^5*b + 2*a^3*b^3 - a*b^5)*cosh(x))*sinh(x)^2 + 4*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*x + ((3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)^8 + 8*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)*sinh(x)^7
+ (3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*sinh(x)^8 + 4*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)^6 + 4*(3*a^5*b + 1
0*a^3*b^3 + 15*a*b^5 + 7*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)^2)*sinh(x)^6 + 3*a^5*b + 10*a^3*b^3 + 15*a*
b^5 + 8*(7*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)^3 + 3*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x))*sinh(x)^
5 + 6*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)^4 + 2*(9*a^5*b + 30*a^3*b^3 + 45*a*b^5 + 35*(3*a^5*b + 10*a^3*
b^3 + 15*a*b^5)*cosh(x)^4 + 30*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(3*a^5*b + 10*a^3
*b^3 + 15*a*b^5)*cosh(x)^5 + 10*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)^3 + 3*(3*a^5*b + 10*a^3*b^3 + 15*a*b
^5)*cosh(x))*sinh(x)^3 + 4*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)^2 + 4*(7*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5
)*cosh(x)^6 + 3*a^5*b + 10*a^3*b^3 + 15*a*b^5 + 15*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)^4 + 9*(3*a^5*b +
10*a^3*b^3 + 15*a*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)^7 + 3*(3*a^5*b + 10
*a^3*b^3 + 15*a*b^5)*cosh(x)^5 + 3*(3*a^5*b + 10*a^3*b^3 + 15*a*b^5)*cosh(x)^3 + (3*a^5*b + 10*a^3*b^3 + 15*a*
b^5)*cosh(x))*sinh(x))*arctan(cosh(x) + sinh(x)) + (5*a^5*b + 14*a^3*b^3 + 9*a*b^5)*cosh(x) - 4*(b^6*cosh(x)^8
+ 8*b^6*cosh(x)*sinh(x)^7 + b^6*sinh(x)^8 + 4*b^6*cosh(x)^6 + 6*b^6*cosh(x)^4 + 4*b^6*cosh(x)^2 + 4*(7*b^6*co
sh(x)^2 + b^6)*sinh(x)^6 + b^6 + 8*(7*b^6*cosh(x)^3 + 3*b^6*cosh(x))*sinh(x)^5 + 2*(35*b^6*cosh(x)^4 + 30*b^6*
cosh(x)^2 + 3*b^6)*sinh(x)^4 + 8*(7*b^6*cosh(x)^5 + 10*b^6*cosh(x)^3 + 3*b^6*cosh(x))*sinh(x)^3 + 4*(7*b^6*cos
h(x)^6 + 15*b^6*cosh(x)^4 + 9*b^6*cosh(x)^2 + b^6)*sinh(x)^2 + 8*(b^6*cosh(x)^7 + 3*b^6*cosh(x)^5 + 3*b^6*cosh
(x)^3 + b^6*cosh(x))*sinh(x))*log(2*(a*sinh(x) + b)/(cosh(x) - sinh(x))) - 4*((a^6 + 3*a^4*b^2 + 3*a^2*b^4)*co
sh(x)^8 + 8*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)*sinh(x)^7 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4)*sinh(x)^8 + 4*(a^6
+ 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^6 + 4*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + 7*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)
^2)*sinh(x)^6 + a^6 + 3*a^4*b^2 + 3*a^2*b^4 + 8*(7*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^3 + 3*(a^6 + 3*a^4*b^
2 + 3*a^2*b^4)*cosh(x))*sinh(x)^5 + 6*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^4 + 2*(3*a^6 + 9*a^4*b^2 + 9*a^2*b
^4 + 35*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^4 + 30*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^2)*sinh(x)^4 + 8*(7
*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^5 + 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^3 + 3*(a^6 + 3*a^4*b^2 + 3
*a^2*b^4)*cosh(x))*sinh(x)^3 + 4*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^2 + 4*(7*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*
cosh(x)^6 + a^6 + 3*a^4*b^2 + 3*a^2*b^4 + 15*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^4 + 9*(a^6 + 3*a^4*b^2 + 3*
a^2*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^7 + 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*c
osh(x)^5 + 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^3 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x))*sinh(x))*log(2*c
osh(x)/(cosh(x) - sinh(x))) + (32*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x*cosh(x)^7 - 7*(5*a^5*b + 14*a^3*b^3 +
9*a*b^5)*cosh(x)^6 + 5*a^5*b + 14*a^3*b^3 + 9*a*b^5 - 48*(2*a^6 + 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)*x)*cosh(x)^5 + 5*(3*a^5*b + 2*a^3*b^3 - a*b^5)*cosh(x)^4 - 32*(2*a^6 + 6*a^4*b^2 + 4*a^2*b^4
- 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x)*cosh(x)^3 - 3*(3*a^5*b + 2*a^3*b^3 - a*b^5)*cosh(x)^2 - 16*(2*a^6
+ 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x)*cosh(x))*sinh(x))/((a^7 + 3*a^5*b^2 + 3*a^3
*b^4 + a*b^6)*cosh(x)^8 + 8*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)*sinh(x)^7 + (a^7 + 3*a^5*b^2 + 3*a^3
*b^4 + a*b^6)*sinh(x)^8 + a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + 4*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x
)^6 + 4*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + 7*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^2)*sinh(x)^6 +
8*(7*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^3 + 3*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x))*sinh(x
)^5 + 6*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^4 + 2*(3*a^7 + 9*a^5*b^2 + 9*a^3*b^4 + 3*a*b^6 + 35*(a^7
+ 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^4 + 30*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^2)*sinh(x)^4 +
8*(7*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^5 + 10*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^3 + 3*
(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x))*sinh(x)^3 + 4*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^2 +
4*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + 7*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^6 + 15*(a^7 + 3*a^5*
b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^4 + 9*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^2)*sinh(x)^2 + 8*((a^7 +
3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^7 + 3*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^5 + 3*(a^7 + 3*a^5*
b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^3 + (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x))*sinh(x))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{5}{\left (x \right )}}{a + b \operatorname{csch}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)**5/(a+b*csch(x)),x)

[Out]

Integral(tanh(x)**5/(a + b*csch(x)), x)

________________________________________________________________________________________

Giac [B]  time = 1.21098, size = 583, normalized size = 3.01 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^5/(a+b*csch(x)),x, algorithm="giac")

[Out]

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)