\(\int (b \coth ^4(c+d x))^{4/3} \, dx\) [44]
Optimal result
Integrand size = 14, antiderivative size = 353 \[
\int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {b \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{7 d}-\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}-\frac {b \sqrt [3]{b \coth ^4(c+d x)} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {b \sqrt [3]{b \coth ^4(c+d x)} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 b \sqrt [3]{b \coth ^4(c+d x)} \tanh (c+d x)}{d}
\]
[Out]
b*arctanh(coth(d*x+c)^(1/3))*(b*coth(d*x+c)^4)^(1/3)/d/coth(d*x+c)^(4/3)-3/7*b*coth(d*x+c)*(b*coth(d*x+c)^4)^(
1/3)/d-3/13*b*coth(d*x+c)^3*(b*coth(d*x+c)^4)^(1/3)/d-1/4*b*(b*coth(d*x+c)^4)^(1/3)*ln(1-coth(d*x+c)^(1/3)+cot
h(d*x+c)^(2/3))/d/coth(d*x+c)^(4/3)+1/4*b*(b*coth(d*x+c)^4)^(1/3)*ln(1+coth(d*x+c)^(1/3)+coth(d*x+c)^(2/3))/d/
coth(d*x+c)^(4/3)-1/2*b*arctan(1/3*(1-2*coth(d*x+c)^(1/3))*3^(1/2))*(b*coth(d*x+c)^4)^(1/3)*3^(1/2)/d/coth(d*x
+c)^(4/3)+1/2*b*arctan(1/3*(1+2*coth(d*x+c)^(1/3))*3^(1/2))*(b*coth(d*x+c)^4)^(1/3)*3^(1/2)/d/coth(d*x+c)^(4/3
)-3*b*(b*coth(d*x+c)^4)^(1/3)*tanh(d*x+c)/d
Rubi [A] (verified)
Time = 0.16 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.00, number of
steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3739, 3554, 3557, 335, 216,
648, 632, 210, 642, 212} \[
\int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\sqrt {3} b \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {b \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{7 d}-\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}-\frac {3 b \tanh (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{d}-\frac {b \sqrt [3]{b \coth ^4(c+d x)} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {b \sqrt [3]{b \coth ^4(c+d x)} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}
\]
[In]
Int[(b*Coth[c + d*x]^4)^(4/3),x]
[Out]
-1/2*(Sqrt[3]*b*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^4)^(1/3))/(d*Coth[c + d*x]^(4/3))
+ (Sqrt[3]*b*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^4)^(1/3))/(2*d*Coth[c + d*x]^(4/3))
+ (b*ArcTanh[Coth[c + d*x]^(1/3)]*(b*Coth[c + d*x]^4)^(1/3))/(d*Coth[c + d*x]^(4/3)) - (3*b*Coth[c + d*x]*(b*
Coth[c + d*x]^4)^(1/3))/(7*d) - (3*b*Coth[c + d*x]^3*(b*Coth[c + d*x]^4)^(1/3))/(13*d) - (b*(b*Coth[c + d*x]^4
)^(1/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(4/3)) + (b*(b*Coth[c + d*x]^4)
^(1/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(4/3)) - (3*b*(b*Coth[c + d*x]^4
)^(1/3)*Tanh[c + d*x])/d
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 212
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] && (GtQ[a, 0] || LtQ[b, 0])
Rule 216
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n
]], k, u}, Simp[u = Int[(r - s*Cos[(2*k*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*C
os[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n))*Int[1/(r^2 - s^2*x^2), x] + Dis
t[2*(r/(a*n)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
Rule 335
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 632
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 642
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 648
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 3554
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Rule 3557
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Rule 3739
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])
Rubi steps \begin{align*}
\text {integral}& = \frac {\left (b \sqrt [3]{b \coth ^4(c+d x)}\right ) \int \coth ^{\frac {16}{3}}(c+d x) \, dx}{\coth ^{\frac {4}{3}}(c+d x)} \\ & = -\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}+\frac {\left (b \sqrt [3]{b \coth ^4(c+d x)}\right ) \int \coth ^{\frac {10}{3}}(c+d x) \, dx}{\coth ^{\frac {4}{3}}(c+d x)} \\ & = -\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{7 d}-\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}+\frac {\left (b \sqrt [3]{b \coth ^4(c+d x)}\right ) \int \coth ^{\frac {4}{3}}(c+d x) \, dx}{\coth ^{\frac {4}{3}}(c+d x)} \\ & = -\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{7 d}-\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}-\frac {3 b \sqrt [3]{b \coth ^4(c+d x)} \tanh (c+d x)}{d}+\frac {\left (b \sqrt [3]{b \coth ^4(c+d x)}\right ) \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)} \, dx}{\coth ^{\frac {4}{3}}(c+d x)} \\ & = -\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{7 d}-\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}-\frac {3 b \sqrt [3]{b \coth ^4(c+d x)} \tanh (c+d x)}{d}-\frac {\left (b \sqrt [3]{b \coth ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{x^{2/3} \left (-1+x^2\right )} \, dx,x,\coth (c+d x)\right )}{d \coth ^{\frac {4}{3}}(c+d x)} \\ & = -\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{7 d}-\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}-\frac {3 b \sqrt [3]{b \coth ^4(c+d x)} \tanh (c+d x)}{d}-\frac {\left (3 b \sqrt [3]{b \coth ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {4}{3}}(c+d x)} \\ & = -\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{7 d}-\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}-\frac {3 b \sqrt [3]{b \coth ^4(c+d x)} \tanh (c+d x)}{d}+\frac {\left (b \sqrt [3]{b \coth ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \sqrt [3]{b \coth ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \sqrt [3]{b \coth ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {4}{3}}(c+d x)} \\ & = \frac {b \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{7 d}-\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}-\frac {3 b \sqrt [3]{b \coth ^4(c+d x)} \tanh (c+d x)}{d}-\frac {\left (b \sqrt [3]{b \coth ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \sqrt [3]{b \coth ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (3 b \sqrt [3]{b \coth ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (3 b \sqrt [3]{b \coth ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)} \\ & = \frac {b \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{7 d}-\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}-\frac {b \sqrt [3]{b \coth ^4(c+d x)} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {b \sqrt [3]{b \coth ^4(c+d x)} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 b \sqrt [3]{b \coth ^4(c+d x)} \tanh (c+d x)}{d}-\frac {\left (3 b \sqrt [3]{b \coth ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \coth ^{\frac {4}{3}}(c+d x)}-\frac {\left (3 b \sqrt [3]{b \coth ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \coth ^{\frac {4}{3}}(c+d x)} \\ & = -\frac {\sqrt {3} b \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {b \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{7 d}-\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}-\frac {b \sqrt [3]{b \coth ^4(c+d x)} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {b \sqrt [3]{b \coth ^4(c+d x)} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 b \sqrt [3]{b \coth ^4(c+d x)} \tanh (c+d x)}{d} \\
\end{align*}
Mathematica [A] (verified)
Time = 2.08 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.69
\[
\int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=-\frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (42 \coth ^4(c+d x) \sqrt [6]{\coth ^2(c+d x)}+13 \left (42 \sqrt [6]{\coth ^2(c+d x)}+6 \coth ^2(c+d x)^{7/6}+7 \log \left (1-\sqrt [6]{\coth ^2(c+d x)}\right )-7 \log \left (1+\sqrt [6]{\coth ^2(c+d x)}\right )-7 (-1)^{2/3} \log \left (1-\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )+7 (-1)^{2/3} \log \left (1+\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )-7 \sqrt [3]{-1} \log \left (1-(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )+7 \sqrt [3]{-1} \log \left (1+(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )\right )\right ) \tanh (c+d x)}{182 d \sqrt [6]{\coth ^2(c+d x)}}
\]
[In]
Integrate[(b*Coth[c + d*x]^4)^(4/3),x]
[Out]
-1/182*(b*(b*Coth[c + d*x]^4)^(1/3)*(42*Coth[c + d*x]^4*(Coth[c + d*x]^2)^(1/6) + 13*(42*(Coth[c + d*x]^2)^(1/
6) + 6*(Coth[c + d*x]^2)^(7/6) + 7*Log[1 - (Coth[c + d*x]^2)^(1/6)] - 7*Log[1 + (Coth[c + d*x]^2)^(1/6)] - 7*(
-1)^(2/3)*Log[1 - (-1)^(1/3)*(Coth[c + d*x]^2)^(1/6)] + 7*(-1)^(2/3)*Log[1 + (-1)^(1/3)*(Coth[c + d*x]^2)^(1/6
)] - 7*(-1)^(1/3)*Log[1 - (-1)^(2/3)*(Coth[c + d*x]^2)^(1/6)] + 7*(-1)^(1/3)*Log[1 + (-1)^(2/3)*(Coth[c + d*x]
^2)^(1/6)]))*Tanh[c + d*x])/(d*(Coth[c + d*x]^2)^(1/6))
Maple [F]
\[\int \left (b \coth \left (d x +c \right )^{4}\right )^{\frac {4}{3}}d x\]
[In]
int((b*coth(d*x+c)^4)^(4/3),x)
[Out]
int((b*coth(d*x+c)^4)^(4/3),x)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2864 vs. \(2 (295) = 590\).
Time = 0.32 (sec) , antiderivative size = 2864, normalized size of antiderivative = 8.11
\[
\int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=\text {Too large to display}
\]
[In]
integrate((b*coth(d*x+c)^4)^(4/3),x, algorithm="fricas")
[Out]
-1/364*(182*(sqrt(3)*b*cosh(d*x + c)^8 + 8*sqrt(3)*b*cosh(d*x + c)*sinh(d*x + c)^7 + sqrt(3)*b*sinh(d*x + c)^8
- 4*sqrt(3)*b*cosh(d*x + c)^6 + 4*(7*sqrt(3)*b*cosh(d*x + c)^2 - sqrt(3)*b)*sinh(d*x + c)^6 + 8*(7*sqrt(3)*b*
cosh(d*x + c)^3 - 3*sqrt(3)*b*cosh(d*x + c))*sinh(d*x + c)^5 + 6*sqrt(3)*b*cosh(d*x + c)^4 + 2*(35*sqrt(3)*b*c
osh(d*x + c)^4 - 30*sqrt(3)*b*cosh(d*x + c)^2 + 3*sqrt(3)*b)*sinh(d*x + c)^4 + 8*(7*sqrt(3)*b*cosh(d*x + c)^5
- 10*sqrt(3)*b*cosh(d*x + c)^3 + 3*sqrt(3)*b*cosh(d*x + c))*sinh(d*x + c)^3 - 4*sqrt(3)*b*cosh(d*x + c)^2 + 4*
(7*sqrt(3)*b*cosh(d*x + c)^6 - 15*sqrt(3)*b*cosh(d*x + c)^4 + 9*sqrt(3)*b*cosh(d*x + c)^2 - sqrt(3)*b)*sinh(d*
x + c)^2 + 8*(sqrt(3)*b*cosh(d*x + c)^7 - 3*sqrt(3)*b*cosh(d*x + c)^5 + 3*sqrt(3)*b*cosh(d*x + c)^3 - sqrt(3)*
b*cosh(d*x + c))*sinh(d*x + c) + sqrt(3)*b)*(-b)^(1/3)*arctan(1/3*(sqrt(3)*b + 2*sqrt(3)*(-b)^(2/3)*(b*cosh(d*
x + c)/sinh(d*x + c))^(1/3))/b) - 182*(sqrt(3)*b*cosh(d*x + c)^8 + 8*sqrt(3)*b*cosh(d*x + c)*sinh(d*x + c)^7 +
sqrt(3)*b*sinh(d*x + c)^8 - 4*sqrt(3)*b*cosh(d*x + c)^6 + 4*(7*sqrt(3)*b*cosh(d*x + c)^2 - sqrt(3)*b)*sinh(d*
x + c)^6 + 8*(7*sqrt(3)*b*cosh(d*x + c)^3 - 3*sqrt(3)*b*cosh(d*x + c))*sinh(d*x + c)^5 + 6*sqrt(3)*b*cosh(d*x
+ c)^4 + 2*(35*sqrt(3)*b*cosh(d*x + c)^4 - 30*sqrt(3)*b*cosh(d*x + c)^2 + 3*sqrt(3)*b)*sinh(d*x + c)^4 + 8*(7*
sqrt(3)*b*cosh(d*x + c)^5 - 10*sqrt(3)*b*cosh(d*x + c)^3 + 3*sqrt(3)*b*cosh(d*x + c))*sinh(d*x + c)^3 - 4*sqrt
(3)*b*cosh(d*x + c)^2 + 4*(7*sqrt(3)*b*cosh(d*x + c)^6 - 15*sqrt(3)*b*cosh(d*x + c)^4 + 9*sqrt(3)*b*cosh(d*x +
c)^2 - sqrt(3)*b)*sinh(d*x + c)^2 + 8*(sqrt(3)*b*cosh(d*x + c)^7 - 3*sqrt(3)*b*cosh(d*x + c)^5 + 3*sqrt(3)*b*
cosh(d*x + c)^3 - sqrt(3)*b*cosh(d*x + c))*sinh(d*x + c) + sqrt(3)*b)*b^(1/3)*arctan(-1/3*(sqrt(3)*b - 2*sqrt(
3)*b^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3))/b) + 91*(b*cosh(d*x + c)^8 + 8*b*cosh(d*x + c)*sinh(d*x + c)
^7 + b*sinh(d*x + c)^8 - 4*b*cosh(d*x + c)^6 + 4*(7*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^6 + 8*(7*b*cosh(d*x +
c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^5 + 6*b*cosh(d*x + c)^4 + 2*(35*b*cosh(d*x + c)^4 - 30*b*cosh(d*x + c
)^2 + 3*b)*sinh(d*x + c)^4 + 8*(7*b*cosh(d*x + c)^5 - 10*b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c))*sinh(d*x + c)^
3 - 4*b*cosh(d*x + c)^2 + 4*(7*b*cosh(d*x + c)^6 - 15*b*cosh(d*x + c)^4 + 9*b*cosh(d*x + c)^2 - b)*sinh(d*x +
c)^2 + 8*(b*cosh(d*x + c)^7 - 3*b*cosh(d*x + c)^5 + 3*b*cosh(d*x + c)^3 - b*cosh(d*x + c))*sinh(d*x + c) + b)*
(-b)^(1/3)*log((-b)^(2/3) - (-b)^(1/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))
^(2/3)) + 91*(b*cosh(d*x + c)^8 + 8*b*cosh(d*x + c)*sinh(d*x + c)^7 + b*sinh(d*x + c)^8 - 4*b*cosh(d*x + c)^6
+ 4*(7*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^6 + 8*(7*b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^5 +
6*b*cosh(d*x + c)^4 + 2*(35*b*cosh(d*x + c)^4 - 30*b*cosh(d*x + c)^2 + 3*b)*sinh(d*x + c)^4 + 8*(7*b*cosh(d*x
+ c)^5 - 10*b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c))*sinh(d*x + c)^3 - 4*b*cosh(d*x + c)^2 + 4*(7*b*cosh(d*x + c
)^6 - 15*b*cosh(d*x + c)^4 + 9*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^2 + 8*(b*cosh(d*x + c)^7 - 3*b*cosh(d*x +
c)^5 + 3*b*cosh(d*x + c)^3 - b*cosh(d*x + c))*sinh(d*x + c) + b)*b^(1/3)*log(b^(2/3) - b^(1/3)*(b*cosh(d*x + c
)/sinh(d*x + c))^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(2/3)) - 182*(b*cosh(d*x + c)^8 + 8*b*cosh(d*x + c)*s
inh(d*x + c)^7 + b*sinh(d*x + c)^8 - 4*b*cosh(d*x + c)^6 + 4*(7*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^6 + 8*(7*
b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^5 + 6*b*cosh(d*x + c)^4 + 2*(35*b*cosh(d*x + c)^4 - 30*b*
cosh(d*x + c)^2 + 3*b)*sinh(d*x + c)^4 + 8*(7*b*cosh(d*x + c)^5 - 10*b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c))*si
nh(d*x + c)^3 - 4*b*cosh(d*x + c)^2 + 4*(7*b*cosh(d*x + c)^6 - 15*b*cosh(d*x + c)^4 + 9*b*cosh(d*x + c)^2 - b)
*sinh(d*x + c)^2 + 8*(b*cosh(d*x + c)^7 - 3*b*cosh(d*x + c)^5 + 3*b*cosh(d*x + c)^3 - b*cosh(d*x + c))*sinh(d*
x + c) + b)*(-b)^(1/3)*log((-b)^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) - 182*(b*cosh(d*x + c)^8 + 8*b*
cosh(d*x + c)*sinh(d*x + c)^7 + b*sinh(d*x + c)^8 - 4*b*cosh(d*x + c)^6 + 4*(7*b*cosh(d*x + c)^2 - b)*sinh(d*x
+ c)^6 + 8*(7*b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^5 + 6*b*cosh(d*x + c)^4 + 2*(35*b*cosh(d*x
+ c)^4 - 30*b*cosh(d*x + c)^2 + 3*b)*sinh(d*x + c)^4 + 8*(7*b*cosh(d*x + c)^5 - 10*b*cosh(d*x + c)^3 + 3*b*co
sh(d*x + c))*sinh(d*x + c)^3 - 4*b*cosh(d*x + c)^2 + 4*(7*b*cosh(d*x + c)^6 - 15*b*cosh(d*x + c)^4 + 9*b*cosh(
d*x + c)^2 - b)*sinh(d*x + c)^2 + 8*(b*cosh(d*x + c)^7 - 3*b*cosh(d*x + c)^5 + 3*b*cosh(d*x + c)^3 - b*cosh(d*
x + c))*sinh(d*x + c) + b)*b^(1/3)*log(b^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) + 12*(111*b*cosh(d*x +
c)^8 + 888*b*cosh(d*x + c)*sinh(d*x + c)^7 + 111*b*sinh(d*x + c)^8 - 336*b*cosh(d*x + c)^6 + 84*(37*b*cosh(d*
x + c)^2 - 4*b)*sinh(d*x + c)^6 + 168*(37*b*cosh(d*x + c)^3 - 12*b*cosh(d*x + c))*sinh(d*x + c)^5 + 562*b*cosh
(d*x + c)^4 + 2*(3885*b*cosh(d*x + c)^4 - 2520*b*cosh(d*x + c)^2 + 281*b)*sinh(d*x + c)^4 + 8*(777*b*cosh(d*x
+ c)^5 - 840*b*cosh(d*x + c)^3 + 281*b*cosh(d*x + c))*sinh(d*x + c)^3 - 336*b*cosh(d*x + c)^2 + 12*(259*b*cosh
(d*x + c)^6 - 420*b*cosh(d*x + c)^4 + 281*b*cosh(d*x + c)^2 - 28*b)*sinh(d*x + c)^2 + 8*(111*b*cosh(d*x + c)^7
- 252*b*cosh(d*x + c)^5 + 281*b*cosh(d*x + c)^3 - 84*b*cosh(d*x + c))*sinh(d*x + c) + 111*b)*(b*cosh(d*x + c)
/sinh(d*x + c))^(1/3))/(d*cosh(d*x + c)^8 + 8*d*cosh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 - 4*d*cosh(d
*x + c)^6 + 4*(7*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^6 + 8*(7*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x
+ c)^5 + 6*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 - 30*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^4 + 8*(7*d
*cosh(d*x + c)^5 - 10*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^3 - 4*d*cosh(d*x + c)^2 + 4*(7*d*co
sh(d*x + c)^6 - 15*d*cosh(d*x + c)^4 + 9*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 - 3*d*c
osh(d*x + c)^5 + 3*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c) + d)
Sympy [F]
\[
\int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=\int \left (b \coth ^{4}{\left (c + d x \right )}\right )^{\frac {4}{3}}\, dx
\]
[In]
integrate((b*coth(d*x+c)**4)**(4/3),x)
[Out]
Integral((b*coth(c + d*x)**4)**(4/3), x)
Maxima [F]
\[
\int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=\int { \left (b \coth \left (d x + c\right )^{4}\right )^{\frac {4}{3}} \,d x }
\]
[In]
integrate((b*coth(d*x+c)^4)^(4/3),x, algorithm="maxima")
[Out]
integrate((b*coth(d*x + c)^4)^(4/3), x)
Giac [F]
\[
\int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=\int { \left (b \coth \left (d x + c\right )^{4}\right )^{\frac {4}{3}} \,d x }
\]
[In]
integrate((b*coth(d*x+c)^4)^(4/3),x, algorithm="giac")
[Out]
integrate((b*coth(d*x + c)^4)^(4/3), x)
Mupad [F(-1)]
Timed out. \[
\int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=\int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^4\right )}^{4/3} \,d x
\]
[In]
int((b*coth(c + d*x)^4)^(4/3),x)
[Out]
int((b*coth(c + d*x)^4)^(4/3), x)