3.14 \(\int \frac{1}{(b \coth (c+d x))^{4/3}} \, dx\)
Optimal. Leaf size=238 \[ -\frac{\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac{\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{2 b^{4/3} d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}+1}{\sqrt{3}}\right )}{2 b^{4/3} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}-\frac{3}{b d \sqrt [3]{b \coth (c+d x)}} \]
[Out]
(Sqrt[3]*ArcTan[(1 - (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*b^(4/3)*d) - (Sqrt[3]*ArcTan[(1 + (2*(b
*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*b^(4/3)*d) + ArcTanh[(b*Coth[c + d*x])^(1/3)/b^(1/3)]/(b^(4/3)*d)
- 3/(b*d*(b*Coth[c + d*x])^(1/3)) - Log[b^(2/3) - b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)]/
(4*b^(4/3)*d) + Log[b^(2/3) + b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)]/(4*b^(4/3)*d)
________________________________________________________________________________________
Rubi [A] time = 0.31572, antiderivative size = 238, normalized size of antiderivative = 1.,
number of steps used = 13, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used =
{3474, 3476, 329, 296, 634, 618, 204, 628, 206} \[ -\frac{\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac{\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{2 b^{4/3} d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}+1}{\sqrt{3}}\right )}{2 b^{4/3} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}-\frac{3}{b d \sqrt [3]{b \coth (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Int[(b*Coth[c + d*x])^(-4/3),x]
[Out]
(Sqrt[3]*ArcTan[(1 - (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*b^(4/3)*d) - (Sqrt[3]*ArcTan[(1 + (2*(b
*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*b^(4/3)*d) + ArcTanh[(b*Coth[c + d*x])^(1/3)/b^(1/3)]/(b^(4/3)*d)
- 3/(b*d*(b*Coth[c + d*x])^(1/3)) - Log[b^(2/3) - b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)]/
(4*b^(4/3)*d) + Log[b^(2/3) + b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)]/(4*b^(4/3)*d)
Rule 3474
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)), x] - Dist[
1/b^2, Int[(b*Tan[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
Rule 3476
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 329
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 296
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt
[-(a/b), n]], k, u}, Simp[u = Int[(r*Cos[(2*k*m*Pi)/n] - s*Cos[(2*k*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi
)/n]*x + s^2*x^2), x] + Int[(r*Cos[(2*k*m*Pi)/n] + s*Cos[(2*k*(m + 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x
+ s^2*x^2), x]; (2*r^(m + 2)*Int[1/(r^2 - s^2*x^2), x])/(a*n*s^m) + Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k,
1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]
Rule 634
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 618
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 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 628
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{1}{(b \coth (c+d x))^{4/3}} \, dx &=-\frac{3}{b d \sqrt [3]{b \coth (c+d x)}}+\frac{\int (b \coth (c+d x))^{2/3} \, dx}{b^2}\\ &=-\frac{3}{b d \sqrt [3]{b \coth (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{x^{2/3}}{-b^2+x^2} \, dx,x,b \coth (c+d x)\right )}{b d}\\ &=-\frac{3}{b d \sqrt [3]{b \coth (c+d x)}}-\frac{3 \operatorname{Subst}\left (\int \frac{x^4}{-b^2+x^6} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{b d}\\ &=-\frac{3}{b d \sqrt [3]{b \coth (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{\sqrt [3]{b}}{2}-\frac{x}{2}}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{b^{4/3} d}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{\sqrt [3]{b}}{2}+\frac{x}{2}}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{b^{4/3} d}+\frac{\operatorname{Subst}\left (\int \frac{1}{b^{2/3}-x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{b d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}-\frac{3}{b d \sqrt [3]{b \coth (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{-\sqrt [3]{b}+2 x}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 b^{4/3} d}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{b}+2 x}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 b^{4/3} d}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 b d}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 b d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}-\frac{3}{b d \sqrt [3]{b \coth (c+d x)}}-\frac{\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac{\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{4/3} d}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{4/3} d}\\ &=\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{2 b^{4/3} d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{2 b^{4/3} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}-\frac{3}{b d \sqrt [3]{b \coth (c+d x)}}-\frac{\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac{\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{4/3} d}\\ \end{align*}
Mathematica [C] time = 0.0619372, size = 36, normalized size = 0.15 \[ -\frac{3 \, _2F_1\left (-\frac{1}{6},1;\frac{5}{6};\coth ^2(c+d x)\right )}{b d \sqrt [3]{b \coth (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(b*Coth[c + d*x])^(-4/3),x]
[Out]
(-3*Hypergeometric2F1[-1/6, 1, 5/6, Coth[c + d*x]^2])/(b*d*(b*Coth[c + d*x])^(1/3))
________________________________________________________________________________________
Maple [A] time = 0.016, size = 211, normalized size = 0.9 \begin{align*} -3\,{\frac{1}{bd\sqrt [3]{b{\rm coth} \left (dx+c\right )}}}+{\frac{1}{2\,d}\ln \left ( \sqrt [3]{b{\rm coth} \left (dx+c\right )}+\sqrt [3]{b} \right ){b}^{-{\frac{4}{3}}}}-{\frac{1}{4\,d}\ln \left ({b}^{{\frac{2}{3}}}-\sqrt [3]{b}\sqrt [3]{b{\rm coth} \left (dx+c\right )}+ \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{4}{3}}}}-{\frac{\sqrt{3}}{2\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{b{\rm coth} \left (dx+c\right )}}{\sqrt [3]{b}}}-1 \right ) } \right ){b}^{-{\frac{4}{3}}}}-{\frac{1}{2\,d}\ln \left ( \sqrt [3]{b{\rm coth} \left (dx+c\right )}-\sqrt [3]{b} \right ){b}^{-{\frac{4}{3}}}}+{\frac{1}{4\,d}\ln \left ({b}^{{\frac{2}{3}}}+\sqrt [3]{b}\sqrt [3]{b{\rm coth} \left (dx+c\right )}+ \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{4}{3}}}}-{\frac{\sqrt{3}}{2\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{\frac{\sqrt [3]{b{\rm coth} \left (dx+c\right )}}{\sqrt [3]{b}}} \right ) } \right ){b}^{-{\frac{4}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*coth(d*x+c))^(4/3),x)
[Out]
-3/b/d/(b*coth(d*x+c))^(1/3)+1/2/b^(4/3)/d*ln((b*coth(d*x+c))^(1/3)+b^(1/3))-1/4*ln(b^(2/3)-b^(1/3)*(b*coth(d*
x+c))^(1/3)+(b*coth(d*x+c))^(2/3))/b^(4/3)/d-1/2/b^(4/3)/d*3^(1/2)*arctan(1/3*3^(1/2)*(2*(b*coth(d*x+c))^(1/3)
/b^(1/3)-1))-1/2/b^(4/3)/d*ln((b*coth(d*x+c))^(1/3)-b^(1/3))+1/4*ln(b^(2/3)+b^(1/3)*(b*coth(d*x+c))^(1/3)+(b*c
oth(d*x+c))^(2/3))/b^(4/3)/d-1/2*arctan(1/3*(1+2*(b*coth(d*x+c))^(1/3)/b^(1/3))*3^(1/2))*3^(1/2)/b^(4/3)/d
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \coth \left (d x + c\right )\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*coth(d*x+c))^(4/3),x, algorithm="maxima")
[Out]
integrate((b*coth(d*x + c))^(-4/3), x)
________________________________________________________________________________________
Fricas [B] time = 2.44882, size = 9677, normalized size = 40.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*coth(d*x+c))^(4/3),x, algorithm="fricas")
[Out]
[1/4*(sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)*sqrt((-b)^(1/3)/b)
*log(3*b*cosh(d*x + c)^2 + 6*b*cosh(d*x + c)*sinh(d*x + c) + 3*b*sinh(d*x + c)^2 - 3*(cosh(d*x + c)^2 + 2*cosh
(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*(-b)^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) - sqrt(3)*(2*(
cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*(-b)^(2/3)*(b*cosh(d*x + c)/sinh(d*x +
c))^(2/3) + (b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - b)*(-b)^(1/3) - (b*cosh
(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - b)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3))*
sqrt((-b)^(1/3)/b) + b) + sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b
)*sqrt(-1/b^(2/3))*log(-(2*sqrt(3)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*b^(
2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3)*sqrt(-1/b^(2/3)) - b*cosh(d*x + c)^2 - 2*b*cosh(d*x + c)*sinh(d*x +
c) - b*sinh(d*x + c)^2 - sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - b
)*b^(1/3)*sqrt(-1/b^(2/3)) + (sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2
- b)*sqrt(-1/b^(2/3)) + 3*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*b^(2/3))*(b
*cosh(d*x + c)/sinh(d*x + c))^(1/3) - 3*b)/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)
) + (cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*(-b)^(2/3)*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)) - (cosh(d*x + c)^2 + 2*cos
h(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*b^(2/3)*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)) - 2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x
+ c)^2 + 1)*(-b)^(2/3)*log((-b)^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) + 2*(cosh(d*x + c)^2 + 2*cosh(
d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*b^(2/3)*log(b^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) - 1
2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3
))/(b^2*d*cosh(d*x + c)^2 + 2*b^2*d*cosh(d*x + c)*sinh(d*x + c) + b^2*d*sinh(d*x + c)^2 + b^2*d), -1/4*(2*sqrt
(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)*sqrt(-(-b)^(1/3)/b)*arctan(-
1/3*sqrt(3)*(-b)^(1/3)*sqrt(-(-b)^(1/3)/b) + 2/3*sqrt(3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3)*sqrt(-(-b)^(1/3
)/b)) - sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)*sqrt(-1/b^(2/3))
*log(-(2*sqrt(3)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*b^(2/3)*(b*cosh(d*x +
c)/sinh(d*x + c))^(2/3)*sqrt(-1/b^(2/3)) - b*cosh(d*x + c)^2 - 2*b*cosh(d*x + c)*sinh(d*x + c) - b*sinh(d*x +
c)^2 - sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - b)*b^(1/3)*sqrt(-1/
b^(2/3)) + (sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - b)*sqrt(-1/b^(2
/3)) + 3*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*b^(2/3))*(b*cosh(d*x + c)/sin
h(d*x + c))^(1/3) - 3*b)/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - (cosh(d*x + c)
^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*(-b)^(2/3)*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)) + (cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*
x + c) + sinh(d*x + c)^2 + 1)*b^(2/3)*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)) + 2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*(-b)^
(2/3)*log((-b)^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) - 2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x
+ c) + sinh(d*x + c)^2 + 1)*b^(2/3)*log(b^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) + 12*(cosh(d*x + c)^2
+ 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3))/(b^2*d*cosh(d*x
+ c)^2 + 2*b^2*d*cosh(d*x + c)*sinh(d*x + c) + b^2*d*sinh(d*x + c)^2 + b^2*d), 1/4*(sqrt(3)*(b*cosh(d*x + c)^
2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)*sqrt((-b)^(1/3)/b)*log(3*b*cosh(d*x + c)^2 + 6*b*
cosh(d*x + c)*sinh(d*x + c) + 3*b*sinh(d*x + c)^2 - 3*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(
d*x + c)^2 - 1)*(-b)^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) - sqrt(3)*(2*(cosh(d*x + c)^2 + 2*cosh(d*x +
c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*(-b)^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3) + (b*cosh(d*x + c)^2
+ 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - b)*(-b)^(1/3) - (b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)
*sinh(d*x + c) + b*sinh(d*x + c)^2 - b)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3))*sqrt((-b)^(1/3)/b) + b) + (cosh
(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*(-b)^(2/3)*log((-b)^(2/3) - (-b)^(1/3)*(b*c
osh(d*x + c)/sinh(d*x + c))^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(2/3)) - (cosh(d*x + c)^2 + 2*cosh(d*x + c
)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*b^(2/3)*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)) - 2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 +
1)*(-b)^(2/3)*log((-b)^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) + 2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*
sinh(d*x + c) + sinh(d*x + c)^2 + 1)*b^(2/3)*log(b^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) - 2*sqrt(3)*
(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)*arctan(-1/3*sqrt(3)*(b^(1/3) - 2
*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3))/b^(1/3))/b^(1/3) - 12*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c)
+ sinh(d*x + c)^2 - 1)*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3))/(b^2*d*cosh(d*x + c)^2 + 2*b^2*d*cosh(d*x + c)*
sinh(d*x + c) + b^2*d*sinh(d*x + c)^2 + b^2*d), -1/4*(2*sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*
x + c) + b*sinh(d*x + c)^2 + b)*sqrt(-(-b)^(1/3)/b)*arctan(-1/3*sqrt(3)*(-b)^(1/3)*sqrt(-(-b)^(1/3)/b) + 2/3*s
qrt(3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3)*sqrt(-(-b)^(1/3)/b)) - (cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*
x + c) + sinh(d*x + c)^2 + 1)*(-b)^(2/3)*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)) + (cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1
)*b^(2/3)*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))
+ 2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*(-b)^(2/3)*log((-b)^(1/3) + (b*co
sh(d*x + c)/sinh(d*x + c))^(1/3)) - 2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*
b^(2/3)*log(b^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) + 2*sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c
)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)*arctan(-1/3*sqrt(3)*(b^(1/3) - 2*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3
))/b^(1/3))/b^(1/3) + 12*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*(b*cosh(d*x +
c)/sinh(d*x + c))^(2/3))/(b^2*d*cosh(d*x + c)^2 + 2*b^2*d*cosh(d*x + c)*sinh(d*x + c) + b^2*d*sinh(d*x + c)^2
+ b^2*d)]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \coth{\left (c + d x \right )}\right )^{\frac{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*coth(d*x+c))**(4/3),x)
[Out]
Integral((b*coth(c + d*x))**(-4/3), x)
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*coth(d*x+c))^(4/3),x, algorithm="giac")
[Out]
Exception raised: TypeError