\(\int \frac {x^{7/2}}{\text {arctanh}(\tanh (a+b x))} \, dx\) [191]

Optimal result

Integrand size = 15, antiderivative size = 143 \[ \int \frac {x^{7/2}}{\text {arctanh}(\tanh (a+b x))} \, dx=\frac {2 x^{7/2}}{7 b}+\frac {2 x^{5/2} (b x-\text {arctanh}(\tanh (a+b x)))}{5 b^2}+\frac {2 x^{3/2} (b x-\text {arctanh}(\tanh (a+b x)))^2}{3 b^3}+\frac {2 \sqrt {x} (b x-\text {arctanh}(\tanh (a+b x)))^3}{b^4}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right ) (b x-\text {arctanh}(\tanh (a+b x)))^{7/2}}{b^{9/2}} \] Output:

2/7*x^(7/2)/b+2/5*x^(5/2)*(b*x-arctanh(tanh(b*x+a)))/b^2+2/3*x^(3/2)*(b*x- 
arctanh(tanh(b*x+a)))^2/b^3+2*x^(1/2)*(b*x-arctanh(tanh(b*x+a)))^3/b^4-2*a 
rctanh(b^(1/2)*x^(1/2)/(b*x-arctanh(tanh(b*x+a)))^(1/2))*(b*x-arctanh(tanh 
(b*x+a)))^(7/2)/b^(9/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.90 \[ \int \frac {x^{7/2}}{\text {arctanh}(\tanh (a+b x))} \, dx=\frac {2 \left (176 b^{7/2} x^{7/2}-406 b^{5/2} x^{5/2} \text {arctanh}(\tanh (a+b x))+350 b^{3/2} x^{3/2} \text {arctanh}(\tanh (a+b x))^2-105 \sqrt {b} \sqrt {x} \text {arctanh}(\tanh (a+b x))^3+105 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-b x+\text {arctanh}(\tanh (a+b x))}}\right ) (-b x+\text {arctanh}(\tanh (a+b x)))^{7/2}\right )}{105 b^{9/2}} \] Input:

Integrate[x^(7/2)/ArcTanh[Tanh[a + b*x]],x]
 

Output:

(2*(176*b^(7/2)*x^(7/2) - 406*b^(5/2)*x^(5/2)*ArcTanh[Tanh[a + b*x]] + 350 
*b^(3/2)*x^(3/2)*ArcTanh[Tanh[a + b*x]]^2 - 105*Sqrt[b]*Sqrt[x]*ArcTanh[Ta 
nh[a + b*x]]^3 + 105*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[-(b*x) + ArcTanh[Tanh[a 
 + b*x]]]]*(-(b*x) + ArcTanh[Tanh[a + b*x]])^(7/2)))/(105*b^(9/2))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2590, 2590, 2590, 2590, 2593}

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.

\(\displaystyle \int \frac {x^{7/2}}{\text {arctanh}(\tanh (a+b x))} \, dx\)

\(\Big \downarrow \) 2590

\(\displaystyle \frac {(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {x^{5/2}}{\text {arctanh}(\tanh (a+b x))}dx}{b}+\frac {2 x^{7/2}}{7 b}\)

\(\Big \downarrow \) 2590

\(\displaystyle \frac {(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {x^{3/2}}{\text {arctanh}(\tanh (a+b x))}dx}{b}+\frac {2 x^{5/2}}{5 b}\right )}{b}+\frac {2 x^{7/2}}{7 b}\)

\(\Big \downarrow \) 2590

\(\displaystyle \frac {(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {\sqrt {x}}{\text {arctanh}(\tanh (a+b x))}dx}{b}+\frac {2 x^{3/2}}{3 b}\right )}{b}+\frac {2 x^{5/2}}{5 b}\right )}{b}+\frac {2 x^{7/2}}{7 b}\)

\(\Big \downarrow \) 2590

\(\displaystyle \frac {(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))}dx}{b}+\frac {2 \sqrt {x}}{b}\right )}{b}+\frac {2 x^{3/2}}{3 b}\right )}{b}+\frac {2 x^{5/2}}{5 b}\right )}{b}+\frac {2 x^{7/2}}{7 b}\)

\(\Big \downarrow \) 2593

\(\displaystyle \frac {\left (\frac {\left (\frac {\left (\frac {2 \sqrt {x}}{b}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right ) \sqrt {b x-\text {arctanh}(\tanh (a+b x))}}{b^{3/2}}\right ) (b x-\text {arctanh}(\tanh (a+b x)))}{b}+\frac {2 x^{3/2}}{3 b}\right ) (b x-\text {arctanh}(\tanh (a+b x)))}{b}+\frac {2 x^{5/2}}{5 b}\right ) (b x-\text {arctanh}(\tanh (a+b x)))}{b}+\frac {2 x^{7/2}}{7 b}\)

Input:

Int[x^(7/2)/ArcTanh[Tanh[a + b*x]],x]
 

Output:

(2*x^(7/2))/(7*b) + (((2*x^(5/2))/(5*b) + (((2*x^(3/2))/(3*b) + (((2*Sqrt[ 
x])/b - (2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*S 
qrt[b*x - ArcTanh[Tanh[a + b*x]]])/b^(3/2))*(b*x - ArcTanh[Tanh[a + b*x]]) 
)/b)*(b*x - ArcTanh[Tanh[a + b*x]]))/b)*(b*x - ArcTanh[Tanh[a + b*x]]))/b
 

Defintions of rubi rules used

rule 2590

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[ 
D[v, x]]}, Simp[v^n/(a*n), x] - Simp[(b*u - a*v)/a   Int[v^(n - 1)/u, x], x 
] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && GtQ[n, 0] && NeQ[n, 
 1]
 

rule 2593

Int[1/((u_)*Sqrt[v_]), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simpli 
fy[D[v, x]]}, Simp[-2*(ArcTanh[Sqrt[v]/Rt[-(b*u - a*v)/a, 2]]/(a*Rt[-(b*u - 
 a*v)/a, 2])), x] /; NeQ[b*u - a*v, 0] && NegQ[(b*u - a*v)/a]] /; Piecewise 
LinearQ[u, v, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(261\) vs. \(2(119)=238\).

Time = 0.35 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.83

method	result	size
derivativedivides	\(-\frac {2 \left (-\frac {b^{3} x^{\frac {7}{2}}}{7}+\frac {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b^{2} x^{\frac {5}{2}}}{5}-\frac {\left (a^{2}+2 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right ) x^{\frac {3}{2}} b}{3}+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) \left (a^{2}+2 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right ) \sqrt {x}\right )}{b^{4}}+\frac {2 \left (a^{4}+4 a^{3} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )+6 a^{2} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}+4 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{4}\right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right )}{b^{4} \sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\)	\(262\)
default	\(-\frac {2 \left (-\frac {b^{3} x^{\frac {7}{2}}}{7}+\frac {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b^{2} x^{\frac {5}{2}}}{5}-\frac {\left (a^{2}+2 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right ) x^{\frac {3}{2}} b}{3}+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) \left (a^{2}+2 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right ) \sqrt {x}\right )}{b^{4}}+\frac {2 \left (a^{4}+4 a^{3} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )+6 a^{2} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}+4 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{4}\right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right )}{b^{4} \sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\)	\(262\)

Input:

int(x^(7/2)/arctanh(tanh(b*x+a)),x,method=_RETURNVERBOSE)
 

Output:

-2/b^4*(-1/7*b^3*x^(7/2)+1/5*(arctanh(tanh(b*x+a))-b*x)*b^2*x^(5/2)-1/3*(a 
^2+2*a*(arctanh(tanh(b*x+a))-b*x-a)+(arctanh(tanh(b*x+a))-b*x-a)^2)*x^(3/2 
)*b+(arctanh(tanh(b*x+a))-b*x)*(a^2+2*a*(arctanh(tanh(b*x+a))-b*x-a)+(arct 
anh(tanh(b*x+a))-b*x-a)^2)*x^(1/2))+2*(a^4+4*a^3*(arctanh(tanh(b*x+a))-b*x 
-a)+6*a^2*(arctanh(tanh(b*x+a))-b*x-a)^2+4*a*(arctanh(tanh(b*x+a))-b*x-a)^ 
3+(arctanh(tanh(b*x+a))-b*x-a)^4)/b^4/((arctanh(tanh(b*x+a))-b*x)*b)^(1/2) 
*arctan(b*x^(1/2)/((arctanh(tanh(b*x+a))-b*x)*b)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.07 \[ \int \frac {x^{7/2}}{\text {arctanh}(\tanh (a+b x))} \, dx=\left [\frac {105 \, a^{3} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, {\left (15 \, b^{3} x^{3} - 21 \, a b^{2} x^{2} + 35 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt {x}}{105 \, b^{4}}, \frac {2 \, {\left (105 \, a^{3} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (15 \, b^{3} x^{3} - 21 \, a b^{2} x^{2} + 35 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt {x}\right )}}{105 \, b^{4}}\right ] \] Input:

integrate(x^(7/2)/arctanh(tanh(b*x+a)),x, algorithm="fricas")
 

Output:

[1/105*(105*a^3*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a 
)) + 2*(15*b^3*x^3 - 21*a*b^2*x^2 + 35*a^2*b*x - 105*a^3)*sqrt(x))/b^4, 2/ 
105*(105*a^3*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) + (15*b^3*x^3 - 21*a* 
b^2*x^2 + 35*a^2*b*x - 105*a^3)*sqrt(x))/b^4]
 

Sympy [F]

\[ \int \frac {x^{7/2}}{\text {arctanh}(\tanh (a+b x))} \, dx=\int \frac {x^{\frac {7}{2}}}{\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \] Input:

integrate(x**(7/2)/atanh(tanh(b*x+a)),x)
 

Output:

Integral(x**(7/2)/atanh(tanh(a + b*x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.45 \[ \int \frac {x^{7/2}}{\text {arctanh}(\tanh (a+b x))} \, dx=\frac {2 \, a^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {2 \, {\left (15 \, b^{3} x^{\frac {7}{2}} - 21 \, a b^{2} x^{\frac {5}{2}} + 35 \, a^{2} b x^{\frac {3}{2}} - 105 \, a^{3} \sqrt {x}\right )}}{105 \, b^{4}} \] Input:

integrate(x^(7/2)/arctanh(tanh(b*x+a)),x, algorithm="maxima")
 

Output:

2*a^4*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) + 2/105*(15*b^3*x^(7/2) 
- 21*a*b^2*x^(5/2) + 35*a^2*b*x^(3/2) - 105*a^3*sqrt(x))/b^4
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.49 \[ \int \frac {x^{7/2}}{\text {arctanh}(\tanh (a+b x))} \, dx=\frac {2 \, a^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {2 \, {\left (15 \, b^{6} x^{\frac {7}{2}} - 21 \, a b^{5} x^{\frac {5}{2}} + 35 \, a^{2} b^{4} x^{\frac {3}{2}} - 105 \, a^{3} b^{3} \sqrt {x}\right )}}{105 \, b^{7}} \] Input:

integrate(x^(7/2)/arctanh(tanh(b*x+a)),x, algorithm="giac")
 

Output:

2*a^4*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) + 2/105*(15*b^6*x^(7/2) 
- 21*a*b^5*x^(5/2) + 35*a^2*b^4*x^(3/2) - 105*a^3*b^3*sqrt(x))/b^7
 

Mupad [B] (verification not implemented)

Time = 3.72 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.32 \[ \int \frac {x^{7/2}}{\text {arctanh}(\tanh (a+b x))} \, dx=\frac {2\,x^{7/2}}{7\,b}+\frac {x^{5/2}\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{5\,b^2}+\frac {x^{3/2}\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{6\,b^3}+\frac {\sqrt {x}\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}{4\,b^4}+\frac {\sqrt {2}\,\ln \left (\frac {64\,b^{19/2}\,\left (\sqrt {2}\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )-4\,\sqrt {b}\,\sqrt {x}\,\sqrt {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x}+2\,\sqrt {2}\,b\,x\right )}{\left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )\,\sqrt {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x}}\right )\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^{7/2}}{16\,b^{9/2}} \] Input:

int(x^(7/2)/atanh(tanh(a + b*x)),x)
 

Output:

(2*x^(7/2))/(7*b) + (x^(5/2)*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*ex 
p(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x))/(5*b^2) + (x^(3/2) 
*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a) 
*exp(2*b*x) + 1)) + 2*b*x)^2)/(6*b^3) + (x^(1/2)*(log(2/(exp(2*a)*exp(2*b* 
x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x) 
^3)/(4*b^4) + (2^(1/2)*log((64*b^(19/2)*(2^(1/2)*(log(2/(exp(2*a)*exp(2*b* 
x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x) 
 - 4*b^(1/2)*x^(1/2)*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*exp(2*a)*e 
xp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)^(1/2) + 2*2^(1/2)*b*x))/((l 
og((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) - log(2/(exp(2*a)*ex 
p(2*b*x) + 1)))*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*exp(2*a)*exp(2* 
b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)^(1/2)))*(log(2/(exp(2*a)*exp(2*b 
*x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x 
)^(7/2))/(16*b^(9/2))
 

Reduce [F]

\[ \int \frac {x^{7/2}}{\text {arctanh}(\tanh (a+b x))} \, dx=\int \frac {\sqrt {x}\, x^{3}}{\mathit {atanh} \left (\tanh \left (b x +a \right )\right )}d x \] Input:

int(x^(7/2)/atanh(tanh(b*x+a)),x)
 

Output:

int((sqrt(x)*x**3)/atanh(tanh(a + b*x)),x)