\(\int \frac {a+b \text {arctanh}(c \sqrt {x})}{d+e x} \, dx\) [47]

Optimal result

Integrand size = 20, antiderivative size = 318 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d+e x} \, dx=-\frac {2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{e}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c \sqrt {x}}\right )}{e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e} \] Output:

-2*(a+b*arctanh(c*x^(1/2)))*ln(2/(1+c*x^(1/2)))/e+(a+b*arctanh(c*x^(1/2))) 
*ln(2*c*((-d)^(1/2)-e^(1/2)*x^(1/2))/(c*(-d)^(1/2)-e^(1/2))/(1+c*x^(1/2))) 
/e+(a+b*arctanh(c*x^(1/2)))*ln(2*c*((-d)^(1/2)+e^(1/2)*x^(1/2))/(c*(-d)^(1 
/2)+e^(1/2))/(1+c*x^(1/2)))/e+b*polylog(2,1-2/(1+c*x^(1/2)))/e-1/2*b*polyl 
og(2,1-2*c*((-d)^(1/2)-e^(1/2)*x^(1/2))/(c*(-d)^(1/2)-e^(1/2))/(1+c*x^(1/2 
)))/e-1/2*b*polylog(2,1-2*c*((-d)^(1/2)+e^(1/2)*x^(1/2))/(c*(-d)^(1/2)+e^( 
1/2))/(1+c*x^(1/2)))/e
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.16 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.36 \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d+e x} \, dx=\frac {a \log (d+e x)}{e}-\frac {b \left (-2 \text {arctanh}\left (c \sqrt {x}\right )^2+4 i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right ) \text {arctanh}\left (\frac {c e \sqrt {x}}{\sqrt {-c^2 d e}}\right )+2 \text {arctanh}\left (c \sqrt {x}\right ) \left (\text {arctanh}\left (c \sqrt {x}\right )+2 \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )-2 \left (-i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (-2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )-2 \left (i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )-2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e-2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )+\operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e+2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )\right )}{2 e} \] Input:

Integrate[(a + b*ArcTanh[c*Sqrt[x]])/(d + e*x),x]
 

Output:

(a*Log[d + e*x])/e - (b*(-2*ArcTanh[c*Sqrt[x]]^2 + (4*I)*ArcSin[Sqrt[(c^2* 
d)/(c^2*d + e)]]*ArcTanh[(c*e*Sqrt[x])/Sqrt[-(c^2*d*e)]] + 2*ArcTanh[c*Sqr 
t[x]]*(ArcTanh[c*Sqrt[x]] + 2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) - 2*((-I 
)*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]] + ArcTanh[c*Sqrt[x]])*Log[(-2*Sqrt[-(c 
^2*d*e)] + e*(-1 + E^(2*ArcTanh[c*Sqrt[x]])) + c^2*d*(1 + E^(2*ArcTanh[c*S 
qrt[x]])))/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] - 2*(I*ArcSin[Sqrt[(c^2 
*d)/(c^2*d + e)]] + ArcTanh[c*Sqrt[x]])*Log[(2*Sqrt[-(c^2*d*e)] + e*(-1 + 
E^(2*ArcTanh[c*Sqrt[x]])) + c^2*d*(1 + E^(2*ArcTanh[c*Sqrt[x]])))/((c^2*d 
+ e)*E^(2*ArcTanh[c*Sqrt[x]]))] - 2*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])] 
 + PolyLog[2, (-(c^2*d) + e - 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTan 
h[c*Sqrt[x]]))] + PolyLog[2, (-(c^2*d) + e + 2*Sqrt[-(c^2*d*e)])/((c^2*d + 
 e)*E^(2*ArcTanh[c*Sqrt[x]]))]))/(2*e)
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6484, 6606, 2009}

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.

\(\displaystyle \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d+e x} \, dx\)

\(\Big \downarrow \) 6484

\(\displaystyle 2 \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}\)

\(\Big \downarrow \) 6606

\(\displaystyle 2 \int \left (\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{2 e}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{2 e}-\frac {\log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (\sqrt {-d} c+\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt {x} c+1}\right )}{2 e}\right )\)

Input:

Int[(a + b*ArcTanh[c*Sqrt[x]])/(d + e*x),x]
 

Output:

2*(-(((a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 + c*Sqrt[x])])/e) + ((a + b*ArcT 
anh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt 
[e])*(1 + c*Sqrt[x]))])/(2*e) + ((a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt 
[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(2*e) 
+ (b*PolyLog[2, 1 - 2/(1 + c*Sqrt[x])])/(2*e) - (b*PolyLog[2, 1 - (2*c*(Sq 
rt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(4*e 
) - (b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sq 
rt[e])*(1 + c*Sqrt[x]))])/(4*e))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6484

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] 
 :> With[{k = Denominator[n]}, Simp[k   Subst[Int[x^(k - 1)*((a + b*ArcTanh 
[c*x^(k*n)])/(d + e*x^k)), x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e}, x 
] && FractionQ[n]
 

rule 6606

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*ArcTanh[c* 
x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, 
 d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])
 

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.36

method	result	size
parts	\(\frac {a \ln \left (e x +d \right )}{e}+\frac {2 b \left (\frac {c^{2} \ln \left (c^{2} e x +c^{2} d \right ) \operatorname {arctanh}\left (c \sqrt {x}\right )}{2 e}-\frac {c^{2} \left (-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )+\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )\right )}{2 e}\right )}{c^{2}}\)	\(433\)
derivativedivides	\(\frac {\frac {a \,c^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e}+2 b \,c^{2} \left (\frac {\ln \left (c^{2} e x +c^{2} d \right ) \operatorname {arctanh}\left (c \sqrt {x}\right )}{2 e}-\frac {-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )+\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )}{2 e}\right )}{c^{2}}\)	\(442\)
default	\(\frac {\frac {a \,c^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e}+2 b \,c^{2} \left (\frac {\ln \left (c^{2} e x +c^{2} d \right ) \operatorname {arctanh}\left (c \sqrt {x}\right )}{2 e}-\frac {-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )+\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )}{2 e}\right )}{c^{2}}\)	\(442\)

Input:

int((a+b*arctanh(c*x^(1/2)))/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

a*ln(e*x+d)/e+2*b/c^2*(1/2*c^2/e*ln(c^2*e*x+c^2*d)*arctanh(c*x^(1/2))-1/2* 
c^2/e*(-1/2*ln(c*x^(1/2)-1)*ln(c^2*e*x+c^2*d)+e*(1/2*ln(c*x^(1/2)-1)*(ln(( 
c*(-d*e)^(1/2)-e*(c*x^(1/2)-1)-e)/(c*(-d*e)^(1/2)-e))+ln((c*(-d*e)^(1/2)+e 
*(c*x^(1/2)-1)+e)/(c*(-d*e)^(1/2)+e)))/e+1/2*(dilog((c*(-d*e)^(1/2)-e*(c*x 
^(1/2)-1)-e)/(c*(-d*e)^(1/2)-e))+dilog((c*(-d*e)^(1/2)+e*(c*x^(1/2)-1)+e)/ 
(c*(-d*e)^(1/2)+e)))/e)+1/2*ln(1+c*x^(1/2))*ln(c^2*e*x+c^2*d)-e*(1/2*ln(1+ 
c*x^(1/2))*(ln((c*(-d*e)^(1/2)-e*(1+c*x^(1/2))+e)/(c*(-d*e)^(1/2)+e))+ln(( 
c*(-d*e)^(1/2)+e*(1+c*x^(1/2))-e)/(c*(-d*e)^(1/2)-e)))/e+1/2*(dilog((c*(-d 
*e)^(1/2)-e*(1+c*x^(1/2))+e)/(c*(-d*e)^(1/2)+e))+dilog((c*(-d*e)^(1/2)+e*( 
1+c*x^(1/2))-e)/(c*(-d*e)^(1/2)-e)))/e)))
 

Fricas [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arctanh(c*x^(1/2)))/(e*x+d),x, algorithm="fricas")
 

Output:

integral((b*arctanh(c*sqrt(x)) + a)/(e*x + d), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d+e x} \, dx=\int \frac {a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}}{d + e x}\, dx \] Input:

integrate((a+b*atanh(c*x**(1/2)))/(e*x+d),x)
 

Output:

Integral((a + b*atanh(c*sqrt(x)))/(d + e*x), x)
 

Maxima [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arctanh(c*x^(1/2)))/(e*x+d),x, algorithm="maxima")
 

Output:

b*integrate(1/2*log(c*sqrt(x) + 1)/(e*x + d), x) - b*integrate(1/2*log(-c* 
sqrt(x) + 1)/(e*x + d), x) + a*log(e*x + d)/e
 

Giac [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arctanh(c*x^(1/2)))/(e*x+d),x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*sqrt(x)) + a)/(e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d+e x} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{d+e\,x} \,d x \] Input:

int((a + b*atanh(c*x^(1/2)))/(d + e*x),x)
 

Output:

int((a + b*atanh(c*x^(1/2)))/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{d+e x} \, dx=\frac {\left (\int \frac {\mathit {atanh} \left (\sqrt {x}\, c \right )}{e x +d}d x \right ) b e +\mathrm {log}\left (e x +d \right ) a}{e} \] Input:

int((a+b*atanh(c*x^(1/2)))/(e*x+d),x)
                                                                                    
                                                                                    
 

Output:

(int(atanh(sqrt(x)*c)/(d + e*x),x)*b*e + log(d + e*x)*a)/e