\(\int (b \coth ^4(c+d x))^{4/3} \, dx\) [44]

Optimal result

Integrand size = 14, antiderivative size = 299 \[ \int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {b \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{d \coth ^{\frac {4}{3}}(c+d x)}+\frac {b \text {arctanh}\left (\frac {\sqrt [3]{\coth (c+d x)}}{1+\coth ^{\frac {2}{3}}(c+d x)}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{7 d}-\frac {3 b \coth ^3(c+d x) \sqrt [3]{b \coth ^4(c+d x)}}{13 d}-\frac {3 b \sqrt [3]{b \coth ^4(c+d x)} \tanh (c+d x)}{d} \] Output:

-1/2*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)/d/coth(d*x+c)^(4/3)+1/2*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)/d/coth(d*x+c)^(4/3)+b*arctanh(coth(d* 
x+c)^(1/3))*(b*coth(d*x+c)^4)^(1/3)/d/coth(d*x+c)^(4/3)+1/2*b*arctanh(coth 
(d*x+c)^(1/3)/(1+coth(d*x+c)^(2/3)))*(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-3*b*(b*coth(d*x+c)^4)^(1/3)*tanh(d*x+c)/d
 

Mathematica [A] (verified)

Time = 1.34 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.82 \[ \int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=-\frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (42 \coth ^4(c+d x) \sqrt [6]{\coth ^2(c+d x)}+13 \left (42 \sqrt [6]{\coth ^2(c+d x)}+6 \coth ^2(c+d x)^{7/6}+7 \log \left (1-\sqrt [6]{\coth ^2(c+d x)}\right )-7 \log \left (1+\sqrt [6]{\coth ^2(c+d x)}\right )-7 (-1)^{2/3} \log \left (1-\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )+7 (-1)^{2/3} \log \left (1+\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )-7 \sqrt [3]{-1} \log \left (1-(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )+7 \sqrt [3]{-1} \log \left (1+(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )\right )\right ) \tanh (c+d x)}{182 d \sqrt [6]{\coth ^2(c+d x)}} \] Input:

Integrate[(b*Coth[c + d*x]^4)^(4/3),x]
 

Output:

-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)*(Co 
th[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))
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.73, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.429, Rules used = {3042, 4141, 3042, 3954, 3042, 3954, 3042, 3954, 3042, 3957, 25, 266, 754, 27, 219, 1142, 25, 1083, 217, 1103}

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.

\(\displaystyle \int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (b \tan \left (i c+i d x+\frac {\pi }{2}\right )^4\right )^{4/3}dx\)

\(\Big \downarrow \) 4141

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \int \coth ^{\frac {16}{3}}(c+d x)dx}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{16/3}dx}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\int \coth ^{\frac {10}{3}}(c+d x)dx-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}+\int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{10/3}dx\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\int \coth ^{\frac {4}{3}}(c+d x)dx-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{4/3}dx-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)}dx-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\int \frac {1}{\left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{2/3}}dx-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (-\frac {\int -\frac {1}{\coth ^{\frac {2}{3}}(c+d x) \left (1-\coth ^2(c+d x)\right )}d\coth (c+d x)}{d}-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\frac {\int \frac {1}{\coth ^{\frac {2}{3}}(c+d x) \left (1-\coth ^2(c+d x)\right )}d\coth (c+d x)}{d}-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \int \frac {1}{1-\coth ^2(c+d x)}d\sqrt [3]{\coth (c+d x)}}{d}-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 754

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{3} \int \frac {1}{1-\coth ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \int \frac {2-\sqrt [3]{\coth (c+d x)}}{2 \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \int \frac {\sqrt [3]{\coth (c+d x)}+2}{2 \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}d\sqrt [3]{\coth (c+d x)}\right )}{d}-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{3} \int \frac {1}{1-\coth ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\coth (c+d x)}+\frac {1}{6} \int \frac {2-\sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{6} \int \frac {\sqrt [3]{\coth (c+d x)}+2}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )}{d}-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \int \frac {2-\sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{6} \int \frac {\sqrt [3]{\coth (c+d x)}+2}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {1}{2} \int -\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-3 \int \frac {1}{-\coth ^{\frac {2}{3}}(c+d x)-3}d\left (2 \sqrt [3]{\coth (c+d x)}-1\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-3 \int \frac {1}{-\coth ^{\frac {2}{3}}(c+d x)-3}d\left (2 \sqrt [3]{\coth (c+d x)}+1\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {b \sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )\right )+\frac {1}{6} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {13}{3}}(c+d x)}{13 d}-\frac {3 \coth ^{\frac {7}{3}}(c+d x)}{7 d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

Input:

Int[(b*Coth[c + d*x]^4)^(4/3),x]
 

Output:

(b*(b*Coth[c + d*x]^4)^(1/3)*((-3*Coth[c + d*x]^(1/3))/d - (3*Coth[c + d*x 
]^(7/3))/(7*d) - (3*Coth[c + d*x]^(13/3))/(13*d) + (3*(ArcTanh[Coth[c + d* 
x]^(1/3)]/3 + (Sqrt[3]*ArcTan[(-1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]] - Log[ 
1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)]/2)/6 + (Sqrt[3]*ArcTan[(1 + 
 2*Coth[c + d*x]^(1/3))/Sqrt[3]] + Log[1 + Coth[c + d*x]^(1/3) + Coth[c + 
d*x]^(2/3)]/2)/6))/d))/Coth[c + d*x]^(4/3)
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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 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 754

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*Cos[(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] + 2*(r/(a*n))   Sum[u, {k, 1, (n - 2)/4}], x]] / 
; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

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 [F]

\[\int \left (b \coth \left (d x +c \right )^{4}\right )^{\frac {4}{3}}d x\]

Input:

int((b*coth(d*x+c)^4)^(4/3),x)
 

Output:

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. \(2 (251) = 502\).

Time = 0.16 (sec) , antiderivative size = 2864, normalized size of antiderivative = 9.58 \[ \int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=\text {Too large to display} \] Input:

integrate((b*coth(d*x+c)^4)^(4/3),x, algorithm="fricas")
 

Output:

-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*co 
sh(d*x + c)^3 - 3*sqrt(3)*b*cosh(d*x + c))*sinh(d*x + c)^5 + 6*sqrt(3)*b*c 
osh(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)/s 
inh(d*x + c))^(1/3))/b) - 182*(sqrt(3)*b*cosh(d*x + c)^8 + 8*sqrt(3)*b*cos 
h(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*sq 
rt(3)*b*cosh(d*x + c)^2 + 3*sqrt(3)*b)*sinh(d*x + c)^4 + 8*(7*sqrt(3)*b*co 
sh(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...
 

Sympy [F]

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

integrate((b*coth(d*x+c)**4)**(4/3),x)
 

Output:

Integral((b*coth(c + d*x)**4)**(4/3), x)
 

Maxima [F]

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

integrate((b*coth(d*x+c)^4)^(4/3),x, algorithm="maxima")
 

Output:

integrate((b*coth(d*x + c)^4)^(4/3), x)
 

Giac [F]

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

integrate((b*coth(d*x+c)^4)^(4/3),x, algorithm="giac")
 

Output:

integrate((b*coth(d*x + c)^4)^(4/3), x)
 

Mupad [F(-1)]

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

int((b*coth(c + d*x)^4)^(4/3),x)
 

Output:

int((b*coth(c + d*x)^4)^(4/3), x)
 

Reduce [F]

\[ \int \left (b \coth ^4(c+d x)\right )^{4/3} \, dx=\frac {b^{\frac {4}{3}} \left (-21 \coth \left (d x +c \right )^{\frac {13}{3}}-39 \coth \left (d x +c \right )^{\frac {7}{3}}-273 \coth \left (d x +c \right )^{\frac {1}{3}}+91 \left (\int \frac {1}{\coth \left (d x +c \right )^{\frac {2}{3}}}d x \right ) d \right )}{91 d} \] Input:

int((b*coth(d*x+c)^4)^(4/3),x)
 

Output:

(b**(1/3)*b*( - 21*coth(c + d*x)**(1/3)*coth(c + d*x)**4 - 39*coth(c + d*x 
)**(1/3)*coth(c + d*x)**2 - 273*coth(c + d*x)**(1/3) + 91*int(coth(c + d*x 
)**(1/3)/coth(c + d*x),x)*d))/(91*d)