3.8.96 \(\int e^{n \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx\) [796]

Optimal. Leaf size=272 \[ -\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)}}{(1-n) \sqrt {1-a^2 x^2}}+\frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \, _2F_1\left (1,\frac {1}{2} (-1+n);\frac {1+n}{2};\frac {1+a x}{1-a x}\right )}{(1-n) \sqrt {1-a^2 x^2}}+\frac {2^{\frac {1+n}{2}} n \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {1-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{\left (3-4 n+n^2\right ) \sqrt {1-a^2 x^2}} \]

[Out]

-x*(-a*x+1)^(3/2-1/2*n)*(a*x+1)^(-1/2+1/2*n)*(c-c/a^2/x^2)^(1/2)/(1-n)/(-a^2*x^2+1)^(1/2)+2*x*(-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))*(c-c/a^2/x^2)^(1/2)/(1-n)/(-
a^2*x^2+1)^(1/2)+2^(1/2+1/2*n)*n*x*(-a*x+1)^(3/2-1/2*n)*hypergeom([1/2-1/2*n, 3/2-1/2*n],[5/2-1/2*n],-1/2*a*x+
1/2)*(c-c/a^2/x^2)^(1/2)/(n^2-4*n+3)/(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6295, 6285, 131, 80, 71, 133} \begin {gather*} \frac {2^{\frac {n+1}{2}} n x \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {1-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{\left (n^2-4 n+3\right ) \sqrt {1-a^2 x^2}}+\frac {2 x \sqrt {c-\frac {c}{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 )}{(1-n) \sqrt {1-a^2 x^2}}-\frac {x \sqrt {c-\frac {c}{a^2 x^2}} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {3-n}{2}}}{(1-n) \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((-1 + n)/2))/((1 - n)*Sqrt[1 - a^2*x^2])) + (2*Sqr
t[c - c/(a^2*x^2)]*x*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Hypergeometric2F1[1, (-1 + n)/2, (1 + n)/2,
(1 + a*x)/(1 - a*x)])/((1 - n)*Sqrt[1 - a^2*x^2]) + (2^((1 + n)/2)*n*Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x)^((3 - n
)/2)*Hypergeometric2F1[(1 - n)/2, (3 - n)/2, (5 - n)/2, (1 - a*x)/2])/((3 - 4*n + n^2)*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 80

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

Rule 131

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

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 6295

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[x^(2*p)*((c + d/x^2)^p/(
1 + c*(x^2/d))^p), Int[(u/x^(2*p))*(1 + c*(x^2/d))^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] &
& EqQ[c + a^2*d, 0] &&  !IntegerQ[p] &&  !IntegerQ[n/2]

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 208, normalized size = 0.76 \begin {gather*} \frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^{\frac {1}{2} (-1-n)} \left ((-1+n) (1+a x)^{\frac {1+n}{2}} \, _2F_1\left (1,-\frac {1}{2}-\frac {n}{2};\frac {1}{2}-\frac {n}{2};\frac {1-a x}{1+a x}\right )+2^{\frac {1+n}{2}} \left (-\left ((-1+n) \, _2F_1\left (-\frac {1}{2}-\frac {n}{2},-\frac {1}{2}-\frac {n}{2};\frac {1}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )\right )+(1+n) (-1+a x) \, _2F_1\left (-\frac {1}{2}-\frac {n}{2},\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )\right )\right )}{\left (-1+n^2\right ) \sqrt {1-a^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))*(c-c/a^2/x^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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

[Out]

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

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