3.14.32 \(\int \frac {e^{n \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx\) [1332]

Optimal. Leaf size=268 \[ -\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}}-\frac {2 a n (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {c-a^2 c x^2} \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\frac {1-a x}{1+a x}\right )}{(1-n) \sqrt {1-a^2 x^2}}+\frac {2^{\frac {1+n}{2}} a (1-a x)^{\frac {1-n}{2}} \sqrt {c-a^2 c x^2} \, _2F_1\left (\frac {1-n}{2},\frac {1-n}{2};\frac {3-n}{2};\frac {1}{2} (1-a x)\right )}{(1-n) \sqrt {1-a^2 x^2}} \]

[Out]

-(-a*x+1)^(1/2-1/2*n)*(a*x+1)^(1/2+1/2*n)*(-a^2*c*x^2+c)^(1/2)/x/(-a^2*x^2+1)^(1/2)-2*a*n*(-a*x+1)^(1/2-1/2*n)
*(a*x+1)^(-1/2+1/2*n)*hypergeom([1, 1/2-1/2*n],[3/2-1/2*n],(-a*x+1)/(a*x+1))*(-a^2*c*x^2+c)^(1/2)/(1-n)/(-a^2*
x^2+1)^(1/2)+2^(1/2+1/2*n)*a*(-a*x+1)^(1/2-1/2*n)*hypergeom([1/2-1/2*n, 1/2-1/2*n],[3/2-1/2*n],-1/2*a*x+1/2)*(
-a^2*c*x^2+c)^(1/2)/(1-n)/(-a^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6288, 6285, 130, 71, 98, 133} \begin {gather*} -\frac {2 a n \sqrt {c-a^2 c x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\frac {1-a x}{a x+1}\right )}{(1-n) \sqrt {1-a^2 x^2}}+\frac {a 2^{\frac {n+1}{2}} \sqrt {c-a^2 c x^2} (1-a x)^{\frac {1-n}{2}} \, _2F_1\left (\frac {1-n}{2},\frac {1-n}{2};\frac {3-n}{2};\frac {1}{2} (1-a x)\right )}{(1-n) \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2} (a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1-n}{2}}}{x \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[c - a^2*c*x^2])/(x*Sqrt[1 - a^2*x^2])) - (2*a*n*(1 - a*x)^
((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, (1 - a*x)/(1
+ a*x)])/((1 - n)*Sqrt[1 - a^2*x^2]) + (2^((1 + n)/2)*a*(1 - a*x)^((1 - n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeomet
ric2F1[(1 - n)/2, (1 - n)/2, (3 - n)/2, (1 - a*x)/2])/((1 - n)*Sqrt[1 - a^2*x^2])

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 130

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_))^2, x_Symbol] :> Dist[b*(d/f^2),
Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1), x], x] + Dist[(b*e - a*f)*((d*e - c*f)/f^2), Int[(a + b*x)^(m - 1)*(
(c + d*x)^(n - 1)/(e + f*x)^2), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[m + n, 0] && EqQ[2*b*d*e
- f*(b*c + a*d), 0]

Rule 133

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*c - a
*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rule 6285

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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 6288

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[c^IntPart[p]*((c +
d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[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]) &&  !IntegerQ[n/2]

Rubi steps

\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \frac {e^{n \tanh ^{-1}(a x)} \sqrt {1-a^2 x^2}}{x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {(1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{\frac {1}{2}+\frac {n}{2}}}{x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {2^{\frac {3}{2}-\frac {n}{2}} a (1+a x)^{\frac {3+n}{2}} \sqrt {c-a^2 c x^2} F_1\left (\frac {3+n}{2};\frac {1}{2} (-1+n),2;\frac {5+n}{2};\frac {1}{2} (1+a x),1+a x\right )}{(3+n) \sqrt {1-a^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.34, size = 138, normalized size = 0.51 \begin {gather*} -\frac {c e^{n \tanh ^{-1}(a x)} \sqrt {1-a^2 x^2} \left ((1+n) \sqrt {1-a^2 x^2}+2 a e^{\tanh ^{-1}(a x)} x \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-e^{2 \tanh ^{-1}(a x)}\right )+2 a e^{\tanh ^{-1}(a x)} n x \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};e^{2 \tanh ^{-1}(a x)}\right )\right )}{(1+n) x \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*E^(n*ArcTanh[a*x])*Sqrt[1 - a^2*x^2]*((1 + n)*Sqrt[1 - a^2*x^2] + 2*a*E^ArcTanh[a*x]*x*Hypergeometric2F1[
1, (1 + n)/2, (3 + n)/2, -E^(2*ArcTanh[a*x])] + 2*a*E^ArcTanh[a*x]*n*x*Hypergeometric2F1[1, (1 + n)/2, (3 + n)
/2, E^(2*ArcTanh[a*x])]))/((1 + n)*x*Sqrt[c - a^2*c*x^2]))

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} \sqrt {-a^{2} c \,x^{2}+c}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2)/x^2,x)

[Out]

int(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2)/x^2,x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2)/x^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2)/x^2,x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*c*x^2 + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/x^2, x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))*(-a**2*c*x**2+c)**(1/2)/x**2,x)

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*atanh(a*x))/x**2, x)

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2)/x^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\sqrt {c-a^2\,c\,x^2}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(n*atanh(a*x))*(c - a^2*c*x^2)^(1/2))/x^2,x)

[Out]

int((exp(n*atanh(a*x))*(c - a^2*c*x^2)^(1/2))/x^2, x)

________________________________________________________________________________________