3.360 \(\int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx\)

Optimal. Leaf size=127 \[ \frac {2 x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} (c-a c x)^{\frac {n+2}{2}}}{n+4}-\frac {2 (n+6) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} (c-a c x)^{\frac {n+2}{2}}}{a (n+2) (n+4)} \]

[Out]

-2*(6+n)*(1-1/a/x)^(-1-1/2*n)*(1+1/a/x)^(1+1/2*n)*(-a*c*x+c)^(1+1/2*n)/a/(n^2+6*n+8)+2*(1-1/a/x)^(-1-1/2*n)*(1
+1/a/x)^(1+1/2*n)*x*(-a*c*x+c)^(1+1/2*n)/(4+n)

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Rubi [A]  time = 0.16, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6176, 6181, 79, 37} \[ \frac {2 x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} (c-a c x)^{\frac {n+2}{2}}}{n+4}-\frac {2 (n+6) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} (c-a c x)^{\frac {n+2}{2}}}{a (n+2) (n+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 37

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

Rule 79

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 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 e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx &=\left (\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} x^{-1-\frac {n}{2}} (c-a c x)^{1+\frac {n}{2}}\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{1+\frac {n}{2}} x^{1+\frac {n}{2}} \, dx\\ &=-\left (\left (\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (\frac {1}{x}\right )^{1+\frac {n}{2}} (c-a c x)^{1+\frac {n}{2}}\right ) \operatorname {Subst}\left (\int x^{-3-\frac {n}{2}} \left (1-\frac {x}{a}\right ) \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {2 \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {2+n}{2}}}{4+n}+\frac {\left ((6+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (\frac {1}{x}\right )^{1+\frac {n}{2}} (c-a c x)^{1+\frac {n}{2}}\right ) \operatorname {Subst}\left (\int x^{-2-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{a (4+n)}\\ &=-\frac {2 (6+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {2+n}{2}}}{a (2+n) (4+n)}+\frac {2 \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {2+n}{2}}}{4+n}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 78, normalized size = 0.61 \[ -\frac {2 c (a x+1) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} (n (a x-1)+2 a x-6) (c-a c x)^{n/2}}{a (n+2) (n+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-a c x + c\right )}^{\frac {1}{2} \, n + 1} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 61, normalized size = 0.48 \[ \frac {2 \left (-a c x +c \right )^{1+\frac {n}{2}} {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (a n x +2 a x -n -6\right ) \left (a x +1\right )}{\left (a x -1\right ) a \left (n^{2}+6 n +8\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(-a*c*x+c)^(1+1/2*n)*exp(n*arccoth(a*x))*(a*n*x+2*a*x-n-6)*(a*x+1)/(a*x-1)/a/(n^2+6*n+8)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.34, size = 140, normalized size = 1.10 \[ -\frac {\left (\frac {\left (2\,n+12\right )\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+1}}{a^2\,\left (n^2+6\,n+8\right )}-\frac {x^2\,\left (2\,n+4\right )\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+1}}{n^2+6\,n+8}+\frac {8\,x\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+1}}{a\,\left (n^2+6\,n+8\right )}\right )\,{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}}{\left (x-\frac {1}{a}\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((((2*n + 12)*(c - a*c*x)^(n/2 + 1))/(a^2*(6*n + n^2 + 8)) - (x^2*(2*n + 4)*(c - a*c*x)^(n/2 + 1))/(6*n + n^2
+ 8) + (8*x*(c - a*c*x)^(n/2 + 1))/(a*(6*n + n^2 + 8)))*((a*x + 1)/(a*x))^(n/2))/((x - 1/a)*((a*x - 1)/(a*x))
^(n/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} c^{\frac {n}{2} + 1} x e^{\frac {i \pi n}{2}} & \text {for}\: a = 0 \\- \frac {\int \frac {1}{a x e^{4 \operatorname {acoth}{\left (a x \right )}} - e^{4 \operatorname {acoth}{\left (a x \right )}}}\, dx}{c} & \text {for}\: n = -4 \\\int e^{- 2 \operatorname {acoth}{\left (a x \right )}}\, dx & \text {for}\: n = -2 \\- \frac {2 a^{2} c n x^{2} \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{2} + 6 a n + 8 a} - \frac {4 a^{2} c x^{2} \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{2} + 6 a n + 8 a} + \frac {8 a c x \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{2} + 6 a n + 8 a} + \frac {2 c n \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{2} + 6 a n + 8 a} + \frac {12 c \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{2} + 6 a n + 8 a} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**(n/2 + 1)*x*exp(I*pi*n/2), Eq(a, 0)), (-Integral(1/(a*x*exp(4*acoth(a*x)) - exp(4*acoth(a*x))),
x)/c, Eq(n, -4)), (Integral(exp(-2*acoth(a*x)), x), Eq(n, -2)), (-2*a**2*c*n*x**2*(-a*c*x + c)**(n/2)*exp(n*ac
oth(a*x))/(a*n**2 + 6*a*n + 8*a) - 4*a**2*c*x**2*(-a*c*x + c)**(n/2)*exp(n*acoth(a*x))/(a*n**2 + 6*a*n + 8*a)
+ 8*a*c*x*(-a*c*x + c)**(n/2)*exp(n*acoth(a*x))/(a*n**2 + 6*a*n + 8*a) + 2*c*n*(-a*c*x + c)**(n/2)*exp(n*acoth
(a*x))/(a*n**2 + 6*a*n + 8*a) + 12*c*(-a*c*x + c)**(n/2)*exp(n*acoth(a*x))/(a*n**2 + 6*a*n + 8*a), True))

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