\(\int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 (c-a^2 c x^2)^{5/4}} \, dx\) [1339]

Optimal result

Integrand size = 29, antiderivative size = 196 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/4}} \, dx=\frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} c \sqrt [4]{c-a^2 c x^2}} \] Output:

2*a*(-a^2*x^2+1)^(1/4)/c/(-a*x+1)^(1/2)/(-a^2*c*x^2+c)^(1/4)-(-a^2*x^2+1)^ 
(1/4)/c/x/(-a*x+1)^(1/2)/(-a^2*c*x^2+c)^(1/4)-a*(-a^2*x^2+1)^(1/4)*arctanh 
((-a*x+1)^(1/2))/c/(-a^2*c*x^2+c)^(1/4)-1/2*a*(-a^2*x^2+1)^(1/4)*arctanh(1 
/2*(-a*x+1)^(1/2)*2^(1/2))*2^(1/2)/c/(-a^2*c*x^2+c)^(1/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.44 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/4}} \, dx=\frac {\sqrt [4]{1-a^2 x^2} \left (-1+a x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {1}{2} (1-a x)\right )+a x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1-a x\right )\right )}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}} \] Input:

Integrate[E^(ArcTanh[a*x]/2)/(x^2*(c - a^2*c*x^2)^(5/4)),x]
 

Output:

((1 - a^2*x^2)^(1/4)*(-1 + a*x*Hypergeometric2F1[-1/2, 1, 1/2, (1 - a*x)/2 
] + a*x*Hypergeometric2F1[-1/2, 1, 1/2, 1 - a*x]))/(c*x*Sqrt[1 - a*x]*(c - 
 a^2*c*x^2)^(1/4))
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.53, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {6703, 6700, 114, 27, 169, 25, 27, 174, 73, 219, 221}

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.

\(\displaystyle \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/4}} \, dx\)

\(\Big \downarrow \) 6703

\(\displaystyle \frac {\sqrt [4]{1-a^2 x^2} \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (1-a^2 x^2\right )^{5/4}}dx}{c \sqrt [4]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 6700

\(\displaystyle \frac {\sqrt [4]{1-a^2 x^2} \int \frac {1}{x^2 (1-a x)^{3/2} (a x+1)}dx}{c \sqrt [4]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {\sqrt [4]{1-a^2 x^2} \left (-\int -\frac {a (3 a x+1)}{2 x (1-a x)^{3/2} (a x+1)}dx-\frac {1}{x \sqrt {1-a x}}\right )}{c \sqrt [4]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt [4]{1-a^2 x^2} \left (\frac {1}{2} a \int \frac {3 a x+1}{x (1-a x)^{3/2} (a x+1)}dx-\frac {1}{x \sqrt {1-a x}}\right )}{c \sqrt [4]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\sqrt [4]{1-a^2 x^2} \left (\frac {1}{2} a \left (\frac {4}{\sqrt {1-a x}}-\frac {\int -\frac {a (2 a x+1)}{x \sqrt {1-a x} (a x+1)}dx}{a}\right )-\frac {1}{x \sqrt {1-a x}}\right )}{c \sqrt [4]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt [4]{1-a^2 x^2} \left (\frac {1}{2} a \left (\frac {\int \frac {a (2 a x+1)}{x \sqrt {1-a x} (a x+1)}dx}{a}+\frac {4}{\sqrt {1-a x}}\right )-\frac {1}{x \sqrt {1-a x}}\right )}{c \sqrt [4]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt [4]{1-a^2 x^2} \left (\frac {1}{2} a \left (\int \frac {2 a x+1}{x \sqrt {1-a x} (a x+1)}dx+\frac {4}{\sqrt {1-a x}}\right )-\frac {1}{x \sqrt {1-a x}}\right )}{c \sqrt [4]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\sqrt [4]{1-a^2 x^2} \left (\frac {1}{2} a \left (\int \frac {1}{x \sqrt {1-a x}}dx+a \int \frac {1}{\sqrt {1-a x} (a x+1)}dx+\frac {4}{\sqrt {1-a x}}\right )-\frac {1}{x \sqrt {1-a x}}\right )}{c \sqrt [4]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\sqrt [4]{1-a^2 x^2} \left (\frac {1}{2} a \left (-2 \int \frac {1}{a x+1}d\sqrt {1-a x}-\frac {2 \int \frac {1}{\frac {1}{a}-\frac {1-a x}{a}}d\sqrt {1-a x}}{a}+\frac {4}{\sqrt {1-a x}}\right )-\frac {1}{x \sqrt {1-a x}}\right )}{c \sqrt [4]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt [4]{1-a^2 x^2} \left (\frac {1}{2} a \left (-\frac {2 \int \frac {1}{\frac {1}{a}-\frac {1-a x}{a}}d\sqrt {1-a x}}{a}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+\frac {4}{\sqrt {1-a x}}\right )-\frac {1}{x \sqrt {1-a x}}\right )}{c \sqrt [4]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt [4]{1-a^2 x^2} \left (\frac {1}{2} a \left (-2 \text {arctanh}\left (\sqrt {1-a x}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+\frac {4}{\sqrt {1-a x}}\right )-\frac {1}{x \sqrt {1-a x}}\right )}{c \sqrt [4]{c-a^2 c x^2}}\)

Input:

Int[E^(ArcTanh[a*x]/2)/(x^2*(c - a^2*c*x^2)^(5/4)),x]
 

Output:

((1 - a^2*x^2)^(1/4)*(-(1/(x*Sqrt[1 - a*x])) + (a*(4/Sqrt[1 - a*x] - 2*Arc 
Tanh[Sqrt[1 - a*x]] - Sqrt[2]*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]]))/2))/(c*(c - 
 a^2*c*x^2)^(1/4))
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 6700

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[c^p   Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], 
 x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || 
 GtQ[c, 0])
 

rule 6703

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar 
t[p])   Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, 
d, m, n, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) &&  !I 
ntegerQ[n/2]
 

Maple [F]

\[\int \frac {\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{4}}}d x\]

Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(5/4),x)
 

Output:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(5/4),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/4}} \, dx=\text {Timed out} \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(5/4),x, a 
lgorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/4}} \, dx=\text {Timed out} \] Input:

integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**2/(-a**2*c*x**2+c)**(5 
/4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/4}} \, dx=\int { \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{4}} x^{2}} \,d x } \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(5/4),x, a 
lgorithm="maxima")
 

Output:

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/((-a^2*c*x^2 + c)^(5/4)*x^2), 
 x)
 

Giac [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/4}} \, dx=\int { \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{4}} x^{2}} \,d x } \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(5/4),x, a 
lgorithm="giac")
 

Output:

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/((-a^2*c*x^2 + c)^(5/4)*x^2), 
 x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/4}} \, dx=\int \frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{x^2\,{\left (c-a^2\,c\,x^2\right )}^{5/4}} \,d x \] Input:

int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/(x^2*(c - a^2*c*x^2)^(5/4)),x)
                                                                                    
                                                                                    
 

Output:

int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/(x^2*(c - a^2*c*x^2)^(5/4)), x)
 

Reduce [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/4}} \, dx=-\frac {\int \frac {\sqrt {a x +1}}{\left (a x +1\right )^{\frac {1}{4}} \left (-a x +1\right )^{\frac {1}{4}} \left (-a^{2} x^{2}+1\right )^{\frac {1}{4}} a^{2} x^{4}-\left (a x +1\right )^{\frac {1}{4}} \left (-a x +1\right )^{\frac {1}{4}} \left (-a^{2} x^{2}+1\right )^{\frac {1}{4}} x^{2}}d x}{c^{\frac {5}{4}}} \] Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(5/4),x)
 

Output:

( - int(sqrt(a*x + 1)/((a*x + 1)**(1/4)*( - a*x + 1)**(1/4)*( - a**2*x**2 
+ 1)**(1/4)*a**2*x**4 - (a*x + 1)**(1/4)*( - a*x + 1)**(1/4)*( - a**2*x**2 
 + 1)**(1/4)*x**2),x))/(c**(1/4)*c)