3.378 \(\int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\)

Optimal. Leaf size=245 \[ -\frac {3 a^2 x^3 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n+3}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}} \, _2F_1\left (\frac {1}{2},\frac {n+3}{2};\frac {3}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 \left (n^2+8 n+15\right ) (c-a c x)^{7/2}}+\frac {3 a^2 x^3 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}}}{2 \left (n^2+8 n+15\right ) (c-a c x)^{7/2}}-\frac {a x^2 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}}}{(n+5) (c-a c x)^{7/2}} \]

[Out]

-a*(1-1/a/x)^(1-1/2*n)*(1+1/a/x)^(1+1/2*n)*x^2/(5+n)/(-a*c*x+c)^(7/2)+3/2*a^2*(1-1/a/x)^(2-1/2*n)*(1+1/a/x)^(1
+1/2*n)*x^3/(n^2+8*n+15)/(-a*c*x+c)^(7/2)-3/2*a^2*((a-1/x)/(a+1/x))^(3/2+1/2*n)*(1-1/a/x)^(2-1/2*n)*(1+1/a/x)^
(1+1/2*n)*x^3*hypergeom([1/2, 3/2+1/2*n],[3/2],2/(a+1/x)/x)/(n^2+8*n+15)/(-a*c*x+c)^(7/2)

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Rubi [A]  time = 0.27, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6176, 6181, 94, 132} \[ -\frac {3 a^2 x^3 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n+3}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}} \, _2F_1\left (\frac {1}{2},\frac {n+3}{2};\frac {3}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 \left (n^2+8 n+15\right ) (c-a c x)^{7/2}}+\frac {3 a^2 x^3 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}}}{2 \left (n^2+8 n+15\right ) (c-a c x)^{7/2}}-\frac {a x^2 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}}}{(n+5) (c-a c x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcCoth[a*x])/(c - a*c*x)^(7/2),x]

[Out]

-((a*(1 - 1/(a*x))^((2 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2)/((5 + n)*(c - a*c*x)^(7/2))) + (3*a^2*(1 - 1/(a*
x))^((4 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^3)/(2*(15 + 8*n + n^2)*(c - a*c*x)^(7/2)) - (3*a^2*((a - x^(-1))/(
a + x^(-1)))^((3 + n)/2)*(1 - 1/(a*x))^((4 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^3*Hypergeometric2F1[1/2, (3 + n
)/2, 3/2, 2/((a + x^(-1))*x)])/(2*(15 + 8*n + n^2)*(c - a*c*x)^(7/2))

Rule 94

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

Rule 132

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

Rule 6176

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

Rule 6181

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

Rubi steps

\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{7/2} x^{7/2}} \, dx}{(c-a c x)^{7/2}}\\ &=-\frac {\left (1-\frac {1}{a x}\right )^{7/2} \operatorname {Subst}\left (\int x^{3/2} \left (1-\frac {x}{a}\right )^{-\frac {7}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac {a \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{(5+n) (c-a c x)^{7/2}}+\frac {\left (3 a \left (1-\frac {1}{a x}\right )^{7/2}\right ) \operatorname {Subst}\left (\int \sqrt {x} \left (1-\frac {x}{a}\right )^{-\frac {5}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{2 (5+n) \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac {a \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{(5+n) (c-a c x)^{7/2}}+\frac {3 a^2 \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3}{2 \left (15+8 n+n^2\right ) (c-a c x)^{7/2}}-\frac {\left (3 a^2 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{\sqrt {x}} \, dx,x,\frac {1}{x}\right )}{4 (3+n) (5+n) \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac {a \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{(5+n) (c-a c x)^{7/2}}+\frac {3 a^2 \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3}{2 \left (15+8 n+n^2\right ) (c-a c x)^{7/2}}-\frac {3 a^2 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {3+n}{2}} \left (1-\frac {1}{a x}\right )^{\frac {4-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^3 \, _2F_1\left (\frac {1}{2},\frac {3+n}{2};\frac {3}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 \left (15+8 n+n^2\right ) (c-a c x)^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 138, normalized size = 0.56 \[ \frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \left (3 (a x-1)^2 \left (\frac {a x-1}{a x+1}\right )^{\frac {n+1}{2}} \, _2F_1\left (\frac {1}{2},\frac {n+3}{2};\frac {3}{2};\frac {2}{a x+1}\right )+(a x+1) (-3 a x+2 n+9)\right )}{2 a c^3 (n+3) (n+5) (a x-1)^2 \sqrt {c-a c x}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*ArcCoth[a*x])/(c - a*c*x)^(7/2),x]

[Out]

((1 + 1/(a*x))^(n/2)*((9 + 2*n - 3*a*x)*(1 + a*x) + 3*(-1 + a*x)^2*((-1 + a*x)/(1 + a*x))^((1 + n)/2)*Hypergeo
metric2F1[1/2, (3 + n)/2, 3/2, 2/(1 + a*x)]))/(2*a*c^3*(3 + n)*(5 + n)*(1 - 1/(a*x))^(n/2)*(-1 + a*x)^2*Sqrt[c
- a*c*x])

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fricas [F]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{4} x^{4} - 4 \, a^{3} c^{4} x^{3} + 6 \, a^{2} c^{4} x^{2} - 4 \, a c^{4} x + c^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))/(-a*c*x+c)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a*c*x + c)*((a*x - 1)/(a*x + 1))^(1/2*n)/(a^4*c^4*x^4 - 4*a^3*c^4*x^3 + 6*a^2*c^4*x^2 - 4*a*c^4
*x + c^4), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))/(-a*c*x+c)^(7/2),x, algorithm="giac")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(1/2*n)/(-a*c*x + c)^(7/2), x)

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maple [F]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{\left (-a c x +c \right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccoth(a*x))/(-a*c*x+c)^(7/2),x)

[Out]

int(exp(n*arccoth(a*x))/(-a*c*x+c)^(7/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))/(-a*c*x+c)^(7/2),x, algorithm="maxima")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(1/2*n)/(-a*c*x + c)^(7/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a\,c\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*acoth(a*x))/(c - a*c*x)^(7/2),x)

[Out]

int(exp(n*acoth(a*x))/(c - a*c*x)^(7/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*acoth(a*x))/(-a*c*x+c)**(7/2),x)

[Out]

Timed out

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