3.456 \(\int e^{3 \coth ^{-1}(a x)} (c-\frac{c}{a x})^{7/2} \, dx\)

Optimal. Leaf size=237 \[ -\frac{2 \left (\frac{1}{a x}+1\right )^{5/2} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{7/2}}{3 a \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{x \left (\frac{1}{a x}+1\right )^{5/2} \left (c-\frac{c}{a x}\right )^{7/2}}{\left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{a \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{\left (c-\frac{c}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{7/2}} \]

[Out]

(Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2))/(a*(1 - 1/(a*x))^(7/2)) + ((1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(7/2))/(3
*a*(1 - 1/(a*x))^(7/2)) - (2*(1 + 1/(a*x))^(5/2)*(c - c/(a*x))^(7/2))/(5*a*(1 - 1/(a*x))^(7/2)) + ((1 + 1/(a*x
))^(5/2)*(c - c/(a*x))^(7/2)*x)/(1 - 1/(a*x))^(7/2) - ((c - c/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 -
1/(a*x))^(7/2))

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Rubi [A]  time = 0.145512, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6182, 6179, 89, 80, 50, 63, 208} \[ -\frac{2 \left (\frac{1}{a x}+1\right )^{5/2} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{7/2}}{3 a \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{x \left (\frac{1}{a x}+1\right )^{5/2} \left (c-\frac{c}{a x}\right )^{7/2}}{\left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{a \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{\left (c-\frac{c}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2),x]

[Out]

(Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2))/(a*(1 - 1/(a*x))^(7/2)) + ((1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(7/2))/(3
*a*(1 - 1/(a*x))^(7/2)) - (2*(1 + 1/(a*x))^(5/2)*(c - c/(a*x))^(7/2))/(5*a*(1 - 1/(a*x))^(7/2)) + ((1 + 1/(a*x
))^(5/2)*(c - c/(a*x))^(7/2)*x)/(1 - 1/(a*x))^(7/2) - ((c - c/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 -
1/(a*x))^(7/2))

Rule 6182

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

Rule 6179

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

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0885057, size = 101, normalized size = 0.43 \[ \frac{c^3 \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} \left (15 a^3 x^3+44 a^2 x^2+8 a x-6\right )-15 a^2 x^2 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{15 a^3 x^2 \sqrt{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2),x]

[Out]

(c^3*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(-6 + 8*a*x + 44*a^2*x^2 + 15*a^3*x^3) - 15*a^2*x^2*ArcTanh[Sqrt[1 +
1/(a*x)]]))/(15*a^3*Sqrt[1 - 1/(a*x)]*x^2)

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Maple [A]  time = 0.181, size = 161, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-1 \right ){c}^{3}}{ \left ( 30\,ax+30 \right ){x}^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -30\,{a}^{7/2}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{3}{a}^{3}-88\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}-16\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}+12\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(7/2),x)

[Out]

-1/30/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)/x^2*c^3/a^(7/2)*(-30*a^(7/2)*x^3*((a*x+1)*
x)^(1/2)+15*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*x^3*a^3-88*a^(5/2)*x^2*((a*x+1)*x)^(1/2)-16*
a^(3/2)*x*((a*x+1)*x)^(1/2)+12*((a*x+1)*x)^(1/2)*a^(1/2))/((a*x+1)*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(7/2),x, algorithm="maxima")

[Out]

integrate((c - c/(a*x))^(7/2)/((a*x - 1)/(a*x + 1))^(3/2), x)

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Fricas [A]  time = 2.228, size = 859, normalized size = 3.62 \begin{align*} \left [\frac{15 \,{\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (15 \, a^{4} c^{3} x^{4} + 59 \, a^{3} c^{3} x^{3} + 52 \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x - 6 \, c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{60 \,{\left (a^{4} x^{3} - a^{3} x^{2}\right )}}, \frac{15 \,{\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (15 \, a^{4} c^{3} x^{4} + 59 \, a^{3} c^{3} x^{3} + 52 \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x - 6 \, c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{30 \,{\left (a^{4} x^{3} - a^{3} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(7/2),x, algorithm="fricas")

[Out]

[1/60*(15*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sq
rt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(15*a^4*c^3*x^4 + 59*a^3*c^3*x^3 +
52*a^2*c^3*x^2 + 2*a*c^3*x - 6*c^3)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2), 1
/30*(15*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt(
(a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(15*a^4*c^3*x^4 + 59*a^3*c^3*x^3 + 52*a^2*c^3*x^2 + 2*a*c^3*
x - 6*c^3)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a/x)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(7/2),x, algorithm="giac")

[Out]

integrate((c - c/(a*x))^(7/2)/((a*x - 1)/(a*x + 1))^(3/2), x)