\(\int (b \coth ^3(c+d x))^{3/2} \, dx\) [29]

Optimal result

Integrand size = 14, antiderivative size = 134 \[ \int \left (b \coth ^3(c+d x)\right )^{3/2} \, dx=-\frac {2 b \sqrt {b \coth ^3(c+d x)}}{3 d}-\frac {b \arctan \left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}+\frac {b \text {arctanh}\left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2 b \coth ^2(c+d x) \sqrt {b \coth ^3(c+d x)}}{7 d} \] Output:

-2/3*b*(b*coth(d*x+c)^3)^(1/2)/d-b*arctan(coth(d*x+c)^(1/2))*(b*coth(d*x+c 
)^3)^(1/2)/d/coth(d*x+c)^(3/2)+b*arctanh(coth(d*x+c)^(1/2))*(b*coth(d*x+c) 
^3)^(1/2)/d/coth(d*x+c)^(3/2)-2/7*b*coth(d*x+c)^2*(b*coth(d*x+c)^3)^(1/2)/ 
d
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.61 \[ \int \left (b \coth ^3(c+d x)\right )^{3/2} \, dx=-\frac {\left (b \coth ^3(c+d x)\right )^{3/2} \left (\arctan \left (\sqrt {\coth (c+d x)}\right )-\text {arctanh}\left (\sqrt {\coth (c+d x)}\right )+\frac {2}{3} \coth ^{\frac {3}{2}}(c+d x)+\frac {2}{7} \coth ^{\frac {7}{2}}(c+d x)\right )}{d \coth ^{\frac {9}{2}}(c+d x)} \] Input:

Integrate[(b*Coth[c + d*x]^3)^(3/2),x]
 

Output:

-(((b*Coth[c + d*x]^3)^(3/2)*(ArcTan[Sqrt[Coth[c + d*x]]] - ArcTanh[Sqrt[C 
oth[c + d*x]]] + (2*Coth[c + d*x]^(3/2))/3 + (2*Coth[c + d*x]^(7/2))/7))/( 
d*Coth[c + d*x]^(9/2)))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.72, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {3042, 4141, 3042, 3954, 3042, 3954, 3042, 3957, 25, 266, 827, 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.

\(\displaystyle \int \left (b \coth ^3(c+d x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4141

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \int \coth ^{\frac {9}{2}}(c+d x)dx}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{9/2}dx}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \left (\int \coth ^{\frac {5}{2}}(c+d x)dx-\frac {2 \coth ^{\frac {7}{2}}(c+d x)}{7 d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \left (-\frac {2 \coth ^{\frac {7}{2}}(c+d x)}{7 d}+\int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{5/2}dx\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \left (\int \sqrt {\coth (c+d x)}dx-\frac {2 \coth ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {2 \coth ^{\frac {3}{2}}(c+d x)}{3 d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \left (\int \sqrt {-i \tan \left (i c+i d x+\frac {\pi }{2}\right )}dx-\frac {2 \coth ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {2 \coth ^{\frac {3}{2}}(c+d x)}{3 d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \left (-\frac {\int -\frac {\sqrt {\coth (c+d x)}}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}-\frac {2 \coth ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {2 \coth ^{\frac {3}{2}}(c+d x)}{3 d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \left (\frac {\int \frac {\sqrt {\coth (c+d x)}}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}-\frac {2 \coth ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {2 \coth ^{\frac {3}{2}}(c+d x)}{3 d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \left (\frac {2 \int \frac {\coth (c+d x)}{1-\coth ^2(c+d x)}d\sqrt {\coth (c+d x)}}{d}-\frac {2 \coth ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {2 \coth ^{\frac {3}{2}}(c+d x)}{3 d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (c+d x)}d\sqrt {\coth (c+d x)}-\frac {1}{2} \int \frac {1}{\coth (c+d x)+1}d\sqrt {\coth (c+d x)}\right )}{d}-\frac {2 \coth ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {2 \coth ^{\frac {3}{2}}(c+d x)}{3 d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (c+d x)}d\sqrt {\coth (c+d x)}-\frac {1}{2} \arctan \left (\sqrt {\coth (c+d x)}\right )\right )}{d}-\frac {2 \coth ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {2 \coth ^{\frac {3}{2}}(c+d x)}{3 d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {b \sqrt {b \coth ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\coth (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\coth (c+d x)}\right )\right )}{d}-\frac {2 \coth ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {2 \coth ^{\frac {3}{2}}(c+d x)}{3 d}\right )}{\coth ^{\frac {3}{2}}(c+d x)}\)

Input:

Int[(b*Coth[c + d*x]^3)^(3/2),x]
 

Output:

(b*Sqrt[b*Coth[c + d*x]^3]*((2*(-1/2*ArcTan[Sqrt[Coth[c + d*x]]] + ArcTanh 
[Sqrt[Coth[c + d*x]]]/2))/d - (2*Coth[c + d*x]^(3/2))/(3*d) - (2*Coth[c + 
d*x]^(7/2))/(7*d)))/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 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   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 = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(-\frac {\left (b \coth \left (d x +c \right )^{3}\right )^{\frac {3}{2}} \left (21 b^{\frac {7}{2}} \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )-21 b^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )+6 \left (b \coth \left (d x +c \right )\right )^{\frac {7}{2}}+14 b^{2} \left (b \coth \left (d x +c \right )\right )^{\frac {3}{2}}\right )}{21 d \coth \left (d x +c \right )^{3} \left (b \coth \left (d x +c \right )\right )^{\frac {3}{2}} b^{2}}\)	\(107\)
default	\(-\frac {\left (b \coth \left (d x +c \right )^{3}\right )^{\frac {3}{2}} \left (21 b^{\frac {7}{2}} \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )-21 b^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )+6 \left (b \coth \left (d x +c \right )\right )^{\frac {7}{2}}+14 b^{2} \left (b \coth \left (d x +c \right )\right )^{\frac {3}{2}}\right )}{21 d \coth \left (d x +c \right )^{3} \left (b \coth \left (d x +c \right )\right )^{\frac {3}{2}} b^{2}}\)	\(107\)

Input:

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

Output:

-1/21/d*(b*coth(d*x+c)^3)^(3/2)*(21*b^(7/2)*arctan((b*coth(d*x+c))^(1/2)/b 
^(1/2))-21*b^(7/2)*arctanh((b*coth(d*x+c))^(1/2)/b^(1/2))+6*(b*coth(d*x+c) 
)^(7/2)+14*b^2*(b*coth(d*x+c))^(3/2))/coth(d*x+c)^3/(b*coth(d*x+c))^(3/2)/ 
b^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1044 vs. \(2 (114) = 228\).

Time = 0.15 (sec) , antiderivative size = 2145, normalized size of antiderivative = 16.01 \[ \int \left (b \coth ^3(c+d x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/84*(42*(b*cosh(d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b*sinh 
(d*x + c)^6 - 3*b*cosh(d*x + c)^4 + 3*(5*b*cosh(d*x + c)^2 - b)*sinh(d*x + 
 c)^4 + 4*(5*b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^3 + 3*b* 
cosh(d*x + c)^2 + 3*(5*b*cosh(d*x + c)^4 - 6*b*cosh(d*x + c)^2 + b)*sinh(d 
*x + c)^2 + 6*(b*cosh(d*x + c)^5 - 2*b*cosh(d*x + c)^3 + b*cosh(d*x + c))* 
sinh(d*x + c) - b)*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*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c) 
^2)) - 21*(b*cosh(d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b*sinh( 
d*x + c)^6 - 3*b*cosh(d*x + c)^4 + 3*(5*b*cosh(d*x + c)^2 - b)*sinh(d*x + 
c)^4 + 4*(5*b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^3 + 3*b*c 
osh(d*x + c)^2 + 3*(5*b*cosh(d*x + c)^4 - 6*b*cosh(d*x + c)^2 + b)*sinh(d* 
x + c)^2 + 6*(b*cosh(d*x + c)^5 - 2*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*s 
inh(d*x + c) - b)*sqrt(-b)*log(-(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)^3*s 
inh(d*x + c) + 6*b*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b*cosh(d*x + c)*sin 
h(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*si 
nh(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)) + 16*(5*b*cosh(d*x + c)^6 + 30*b*cosh(d*x + c)*sinh(d*x + c)^5 + ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \left (b \coth ^3(c+d x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*coth(d*x+c)^3)^(3/2),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. \[ \int \left (b \coth ^3(c+d x)\right )^{3/2} \, dx=\int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^3\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \left (b \coth ^3(c+d x)\right )^{3/2} \, dx=\sqrt {b}\, \left (\int \sqrt {\coth \left (d x +c \right )}\, \coth \left (d x +c \right )^{4}d x \right ) b \] Input:

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

Output:

sqrt(b)*int(sqrt(coth(c + d*x))*coth(c + d*x)**4,x)*b