\(\int \sqrt {b \coth ^3(c+d x)} \, dx\) [30]

Optimal result

Integrand size = 14, antiderivative size = 104 \[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\frac {\arctan \left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}+\frac {\text {arctanh}\left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2 \sqrt {b \coth ^3(c+d x)} \tanh (c+d x)}{d} \] Output:

arctan(coth(d*x+c)^(1/2))*(b*coth(d*x+c)^3)^(1/2)/d/coth(d*x+c)^(3/2)+arct 
anh(coth(d*x+c)^(1/2))*(b*coth(d*x+c)^3)^(1/2)/d/coth(d*x+c)^(3/2)-2*(b*co 
th(d*x+c)^3)^(1/2)*tanh(d*x+c)/d
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.61 \[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\frac {\left (\arctan \left (\sqrt {\coth (c+d x)}\right )+\text {arctanh}\left (\sqrt {\coth (c+d x)}\right )-2 \sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)} \] Input:

Integrate[Sqrt[b*Coth[c + d*x]^3],x]
 

Output:

((ArcTan[Sqrt[Coth[c + d*x]]] + ArcTanh[Sqrt[Coth[c + d*x]]] - 2*Sqrt[Coth 
[c + d*x]])*Sqrt[b*Coth[c + d*x]^3])/(d*Coth[c + d*x]^(3/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.74, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {3042, 4141, 3042, 3954, 3042, 3957, 25, 266, 756, 216, 219}

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.

\(\displaystyle \int \sqrt {b \coth ^3(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {i b \tan \left (i c+i d x+\frac {\pi }{2}\right )^3}dx\)

\(\Big \downarrow \) 4141

\(\displaystyle \frac {\sqrt {b \coth ^3(c+d x)} \int \coth ^{\frac {3}{2}}(c+d x)dx}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {b \coth ^3(c+d x)} \int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}dx}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {\sqrt {b \coth ^3(c+d x)} \left (\int \frac {1}{\sqrt {\coth (c+d x)}}dx-\frac {2 \sqrt {\coth (c+d x)}}{d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {b \coth ^3(c+d x)} \left (-\frac {2 \sqrt {\coth (c+d x)}}{d}+\int \frac {1}{\sqrt {-i \tan \left (i c+i d x+\frac {\pi }{2}\right )}}dx\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {\sqrt {b \coth ^3(c+d x)} \left (-\frac {\int -\frac {1}{\sqrt {\coth (c+d x)} \left (1-\coth ^2(c+d x)\right )}d\coth (c+d x)}{d}-\frac {2 \sqrt {\coth (c+d x)}}{d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {b \coth ^3(c+d x)} \left (\frac {\int \frac {1}{\sqrt {\coth (c+d x)} \left (1-\coth ^2(c+d x)\right )}d\coth (c+d x)}{d}-\frac {2 \sqrt {\coth (c+d x)}}{d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\sqrt {b \coth ^3(c+d x)} \left (\frac {2 \int \frac {1}{1-\coth ^2(c+d x)}d\sqrt {\coth (c+d x)}}{d}-\frac {2 \sqrt {\coth (c+d x)}}{d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {\sqrt {b \coth ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (c+d x)}d\sqrt {\coth (c+d x)}+\frac {1}{2} \int \frac {1}{\coth (c+d x)+1}d\sqrt {\coth (c+d x)}\right )}{d}-\frac {2 \sqrt {\coth (c+d x)}}{d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {b \coth ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (c+d x)}d\sqrt {\coth (c+d x)}+\frac {1}{2} \arctan \left (\sqrt {\coth (c+d x)}\right )\right )}{d}-\frac {2 \sqrt {\coth (c+d x)}}{d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {b \coth ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \arctan \left (\sqrt {\coth (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\coth (c+d x)}\right )\right )}{d}-\frac {2 \sqrt {\coth (c+d x)}}{d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

Input:

Int[Sqrt[b*Coth[c + d*x]^3],x]
 

Output:

(((2*(ArcTan[Sqrt[Coth[c + d*x]]]/2 + ArcTanh[Sqrt[Coth[c + d*x]]]/2))/d - 
 (2*Sqrt[Coth[c + d*x]])/d)*Sqrt[b*Coth[c + d*x]^3])/Coth[c + d*x]^(3/2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 216

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

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3954

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

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[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 4141

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff 
= FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ 
n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p]))   Int[ActivateTrig[u]*(Ta 
n[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]])
 

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\sqrt {b \coth \left (d x +c \right )^{3}}\, \left (-2 \sqrt {b \coth \left (d x +c \right )}+\sqrt {b}\, \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )+\sqrt {b}\, \operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )\right )}{d \coth \left (d x +c \right ) \sqrt {b \coth \left (d x +c \right )}}\)	\(86\)
default	\(\frac {\sqrt {b \coth \left (d x +c \right )^{3}}\, \left (-2 \sqrt {b \coth \left (d x +c \right )}+\sqrt {b}\, \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )+\sqrt {b}\, \operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )\right )}{d \coth \left (d x +c \right ) \sqrt {b \coth \left (d x +c \right )}}\)	\(86\)

Input:

int((b*coth(d*x+c)^3)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/d*(b*coth(d*x+c)^3)^(1/2)/coth(d*x+c)/(b*coth(d*x+c))^(1/2)*(-2*(b*coth( 
d*x+c))^(1/2)+b^(1/2)*arctan((b*coth(d*x+c))^(1/2)/b^(1/2))+b^(1/2)*arctan 
h((b*coth(d*x+c))^(1/2)/b^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (90) = 180\).

Time = 0.11 (sec) , antiderivative size = 627, normalized size of antiderivative = 6.03 \[ \int \sqrt {b \coth ^3(c+d x)} \, dx =\text {Too large to display} \] Input:

integrate((b*coth(d*x+c)^3)^(1/2),x, algorithm="fricas")
 

Output:

[-1/4*(2*sqrt(-b)*arctan((cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) 
+ sinh(d*x + c)^2 - 1)*sqrt(-b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c))/(b*cos 
h(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2)) - sqr 
t(-b)*log(-(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)^3*sinh(d*x + c) + 6*b*co 
sh(d*x + c)^2*sinh(d*x + c)^2 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh 
(d*x + c)^4 - 2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d* 
x + c)^2 - 1)*sqrt(-b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c)) - 2*b)/(cosh(d* 
x + c)^4 + 4*cosh(d*x + c)^3*sinh(d*x + c) + 6*cosh(d*x + c)^2*sinh(d*x + 
c)^2 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4)) + 8*sqrt(b*cosh 
(d*x + c)/sinh(d*x + c)))/d, -1/4*(2*sqrt(b)*arctan((cosh(d*x + c)^2 + 2*c 
osh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(b*cosh(d*x + c)/sin 
h(d*x + c))/sqrt(b)) - sqrt(b)*log(2*b*cosh(d*x + c)^4 + 8*b*cosh(d*x + c) 
^3*sinh(d*x + c) + 12*b*cosh(d*x + c)^2*sinh(d*x + c)^2 + 8*b*cosh(d*x + c 
)*sinh(d*x + c)^3 + 2*b*sinh(d*x + c)^4 + 2*(cosh(d*x + c)^4 + 4*cosh(d*x 
+ c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + (6*cosh(d*x + c)^2 - 1)*sinh(d*x 
+ c)^2 - cosh(d*x + c)^2 + 2*(2*cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x 
+ c))*sqrt(b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c)) - b) + 8*sqrt(b*cosh(d*x 
 + c)/sinh(d*x + c)))/d]
 

Sympy [F]

\[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\int \sqrt {b \coth ^{3}{\left (c + d x \right )}}\, dx \] Input:

integrate((b*coth(d*x+c)**3)**(1/2),x)
 

Output:

Integral(sqrt(b*coth(c + d*x)**3), x)
 

Maxima [F]

\[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\int { \sqrt {b \coth \left (d x + c\right )^{3}} \,d x } \] Input:

integrate((b*coth(d*x+c)^3)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*coth(d*x + c)^3), x)
 

Giac [F(-2)]

Exception generated. \[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*coth(d*x+c)^3)^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\int \sqrt {b\,{\mathrm {coth}\left (c+d\,x\right )}^3} \,d x \] Input:

int((b*coth(c + d*x)^3)^(1/2),x)
 

Output:

int((b*coth(c + d*x)^3)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {b \coth ^3(c+d x)} \, dx=\frac {\sqrt {b}\, \left (-2 \sqrt {\coth \left (d x +c \right )}+\left (\int \frac {\sqrt {\coth \left (d x +c \right )}}{\coth \left (d x +c \right )}d x \right ) d \right )}{d} \] Input:

int((b*coth(d*x+c)^3)^(1/2),x)
 

Output:

(sqrt(b)*( - 2*sqrt(coth(c + d*x)) + int(sqrt(coth(c + d*x))/coth(c + d*x) 
,x)*d))/d