\(\int (1+\coth (x))^{5/2} \, dx\) [71]

Optimal result

Integrand size = 8, antiderivative size = 45 \[ \int (1+\coth (x))^{5/2} \, dx=4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2} \]

[Out]

-2/3*(1+coth(x))^(3/2)+4*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))*2^(1/2)-4*(1+coth(x))^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3559, 3561, 212} \[ \int (1+\coth (x))^{5/2} \, dx=4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )-\frac {2}{3} (\coth (x)+1)^{3/2}-4 \sqrt {\coth (x)+1} \]

[In]

Int[(1 + Coth[x])^(5/2),x]

[Out]

4*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - 4*Sqrt[1 + Coth[x]] - (2*(1 + Coth[x])^(3/2))/3

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 3559

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))
), x] + Dist[2*a, Int[(a + b*Tan[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && G
tQ[n, 1]

Rule 3561

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3} (1+\coth (x))^{3/2}+2 \int (1+\coth (x))^{3/2} \, dx \\ & = -4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2}+4 \int \sqrt {1+\coth (x)} \, dx \\ & = -4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2}+8 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right ) \\ & = 4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int (1+\coth (x))^{5/2} \, dx=4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-\frac {2}{3} \sqrt {1+\coth (x)} (7+\coth (x)) \]

[In]

Integrate[(1 + Coth[x])^(5/2),x]

[Out]

4*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - (2*Sqrt[1 + Coth[x]]*(7 + Coth[x]))/3

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(-\frac {2 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}{3}+4 \,\operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-4 \sqrt {1+\coth \left (x \right )}\)	\(35\)
default	\(-\frac {2 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}{3}+4 \,\operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-4 \sqrt {1+\coth \left (x \right )}\)	\(35\)

[In]

int((1+coth(x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*(1+coth(x))^(3/2)+4*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))*2^(1/2)-4*(1+coth(x))^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (34) = 68\).

Time = 0.27 (sec) , antiderivative size = 259, normalized size of antiderivative = 5.76 \[ \int (1+\coth (x))^{5/2} \, dx=-\frac {2 \, {\left (2 \, \sqrt {2} {\left (4 \, \sqrt {2} \cosh \left (x\right )^{3} + 12 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + 4 \, \sqrt {2} \sinh \left (x\right )^{3} + 3 \, {\left (4 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right ) - 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \, {\left (\sqrt {2} \cosh \left (x\right )^{4} + 4 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {2} \cosh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} - \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - 1\right )\right )}}{3 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]

[In]

integrate((1+coth(x))^(5/2),x, algorithm="fricas")

[Out]

-2/3*(2*sqrt(2)*(4*sqrt(2)*cosh(x)^3 + 12*sqrt(2)*cosh(x)*sinh(x)^2 + 4*sqrt(2)*sinh(x)^3 + 3*(4*sqrt(2)*cosh(
x)^2 - sqrt(2))*sinh(x) - 3*sqrt(2)*cosh(x))*sqrt(sinh(x)/(cosh(x) - sinh(x))) - 3*(sqrt(2)*cosh(x)^4 + 4*sqrt
(2)*cosh(x)*sinh(x)^3 + sqrt(2)*sinh(x)^4 + 2*(3*sqrt(2)*cosh(x)^2 - sqrt(2))*sinh(x)^2 - 2*sqrt(2)*cosh(x)^2
+ 4*(sqrt(2)*cosh(x)^3 - sqrt(2)*cosh(x))*sinh(x) + sqrt(2))*log(2*sqrt(2)*sqrt(sinh(x)/(cosh(x) - sinh(x)))*(
cosh(x) + sinh(x)) + 2*cosh(x)^2 + 4*cosh(x)*sinh(x) + 2*sinh(x)^2 - 1))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + si
nh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)

Sympy [F]

\[ \int (1+\coth (x))^{5/2} \, dx=\int \left (\coth {\left (x \right )} + 1\right )^{\frac {5}{2}}\, dx \]

[In]

integrate((1+coth(x))**(5/2),x)

[Out]

Integral((coth(x) + 1)**(5/2), x)

Maxima [F]

\[ \int (1+\coth (x))^{5/2} \, dx=\int { {\left (\coth \left (x\right ) + 1\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((1+coth(x))^(5/2),x, algorithm="maxima")

[Out]

integrate((coth(x) + 1)^(5/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (34) = 68\).

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.49 \[ \int (1+\coth (x))^{5/2} \, dx=-\frac {2}{3} \, \sqrt {2} {\left (\frac {2 \, {\left (6 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} + 9 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 9 \, e^{\left (2 \, x\right )} + 4\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1\right )}^{3}} + 3 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]

[In]

integrate((1+coth(x))^(5/2),x, algorithm="giac")

[Out]

-2/3*sqrt(2)*(2*(6*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^2 + 9*sqrt(e^(4*x) - e^(2*x)) - 9*e^(2*x) + 4)/(sqrt(e^
(4*x) - e^(2*x)) - e^(2*x) + 1)^3 + 3*log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1)))*sgn(e^(2*x) - 1)

Mupad [B] (verification not implemented)

Time = 1.88 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int (1+\coth (x))^{5/2} \, dx=\sqrt {8}\,\ln \left (-2\,\sqrt {8}\,\sqrt {\mathrm {coth}\left (x\right )+1}-8\right )-\frac {2\,{\left (\mathrm {coth}\left (x\right )+1\right )}^{3/2}}{3}-2\,\sqrt {2}\,\ln \left (4\,\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}-8\right )-4\,\sqrt {\mathrm {coth}\left (x\right )+1} \]

[In]

int((coth(x) + 1)^(5/2),x)

[Out]

8^(1/2)*log(- 2*8^(1/2)*(coth(x) + 1)^(1/2) - 8) - (2*(coth(x) + 1)^(3/2))/3 - 2*2^(1/2)*log(4*2^(1/2)*(coth(x
) + 1)^(1/2) - 8) - 4*(coth(x) + 1)^(1/2)