3.14.46 \(\int \frac {e^{n \tanh ^{-1}(a x)}}{x^2 (c-a^2 c x^2)^{3/2}} \, dx\) [1346]

Optimal. Leaf size=321 \[ \frac {a (2+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1+n) \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c x \sqrt {c-a^2 c x^2}}-\frac {a \left (2+2 n+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {2 a n (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2} \, _2F_1\left (1,\frac {1}{2} (-1+n);\frac {1+n}{2};\frac {1+a x}{1-a x}\right )}{c (1-n) \sqrt {c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.22, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6288, 6285, 105, 160, 12, 133} \begin {gather*} \frac {2 a n \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {n-1}{2};\frac {n+1}{2};\frac {a x+1}{1-a x}\right )}{c (1-n) \sqrt {c-a^2 c x^2}}-\frac {a \left (n^2+2 n+2\right ) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}}}{c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {a (n+2) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{c (n+1) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{c x \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)^(3/2)),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 105

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[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 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 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

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)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \tanh ^{-1}(a x)}}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}}}{x^2} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=-\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c x \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}} \left (-a n-2 a^2 x\right )}{x} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {a (2+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1+n) \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c x \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{-\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}} \left (a^2 n (1+n)+a^3 (2+n) x\right )}{x} \, dx}{a c (1+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {a (2+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1+n) \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c x \sqrt {c-a^2 c x^2}}-\frac {a \left (2+2 n+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1-n) (1+n) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {a^3 (1-n) n (1+n) (1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}}}{x} \, dx}{a^2 c (1-n) (1+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {a (2+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1+n) \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c x \sqrt {c-a^2 c x^2}}-\frac {a \left (2+2 n+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1-n) (1+n) \sqrt {c-a^2 c x^2}}+\frac {\left (a n \sqrt {1-a^2 x^2}\right ) \int \frac {(1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}}}{x} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {a (2+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1+n) \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c x \sqrt {c-a^2 c x^2}}-\frac {a \left (2+2 n+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1-n) (1+n) \sqrt {c-a^2 c x^2}}-\frac {2 a n (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2} \, _2F_1\left (1,\frac {3-n}{2};\frac {5-n}{2};\frac {1-a x}{1+a x}\right )}{c (3-n) \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 173, normalized size = 0.54 \begin {gather*} \frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2} \left (-\left ((-3+n) (1+a x) \left (-1+2 a^2 x^2+n^2 (-1+a x)^2+a n x (-3+2 a x)\right )\right )+2 a n \left (-1+n^2\right ) x (-1+a x)^2 \, _2F_1\left (1,\frac {3}{2}-\frac {n}{2};\frac {5}{2}-\frac {n}{2};\frac {1-a x}{1+a x}\right )\right )}{c (-3+n) (-1+n) (1+n) x \sqrt {c-a^2 c x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)^(3/2)),x]

[Out]

((1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2]*(-((-3 + n)*(1 + a*x)*(-1 + 2*a^2*x^2 + n^2*(
-1 + a*x)^2 + a*n*x*(-3 + 2*a*x))) + 2*a*n*(-1 + n^2)*x*(-1 + a*x)^2*Hypergeometric2F1[1, 3/2 - n/2, 5/2 - n/2
, (1 - a*x)/(1 + a*x)]))/(c*(-3 + n)*(-1 + n)*(1 + n)*x*Sqrt[c - a^2*c*x^2])

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c)^(3/2),x)

[Out]

int(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c)^(3/2),x)

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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))/x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/((-a^2*c*x^2 + c)^(3/2)*x^2), x)

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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))/x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*c*x^2 + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^6 - 2*a^2*c^2*x^4 + c^2*x^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{2} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))/x**2/(-a**2*c*x**2+c)**(3/2),x)

[Out]

Integral(exp(n*atanh(a*x))/(x**2*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)

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Giac [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))/x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/((-a^2*c*x^2 + c)^(3/2)*x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x^2\,{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*atanh(a*x))/(x^2*(c - a^2*c*x^2)^(3/2)),x)

[Out]

int(exp(n*atanh(a*x))/(x^2*(c - a^2*c*x^2)^(3/2)), x)

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