### 3.439 $$\int e^{\coth ^{-1}(a x)} (c-\frac{c}{a x})^{7/2} \, dx$$

Optimal. Leaf size=196 $\frac{49 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{15 a \sqrt{c-\frac{c}{a x}}}+\frac{31 c^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}}{15 a}+\frac{7 c^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 a}-\frac{5 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a}+c x \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{5/2}$

[Out]

(49*c^4*Sqrt[1 - 1/(a^2*x^2)])/(15*a*Sqrt[c - c/(a*x)]) + (31*c^3*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)])/(15
*a) + (7*c^2*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(3/2))/(5*a) + c*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(5/2)*x
- (5*c^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a

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Rubi [A]  time = 0.148126, antiderivative size = 221, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 7, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.318, Rules used = {6182, 6179, 97, 153, 147, 63, 208} $\frac{x \left (a-\frac{1}{x}\right )^3 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{7 \left (a-\frac{1}{x}\right )^2 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (80 a-\frac{31}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{5 \left (c-\frac{c}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{7/2}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((80*a - 31/x)*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2))/(15*a^2*(1 - 1/(a*x))^(7/2)) + (7*(a - x^(-1))^2*Sqrt[1
+ 1/(a*x)]*(c - c/(a*x))^(7/2))/(5*a^3*(1 - 1/(a*x))^(7/2)) + ((a - x^(-1))^3*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^
(7/2)*x)/(a^3*(1 - 1/(a*x))^(7/2)) - (5*(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 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

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^{\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^{\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 )^3 \sqrt{1+\frac{x}{a}}}{x^2} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{7/2}}\\ &=\frac{\left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2} x}{a^3 \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{5}{2 a}-\frac{7 x}{2 a^2}\right ) \left (1-\frac{x}{a}\right )^2}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{7/2}}\\ &=\frac{7 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{\left (2 a \left (c-\frac{c}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{25}{4 a^2}-\frac{31 x}{4 a^3}\right ) \left (1-\frac{x}{a}\right )}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{5 \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=\frac{\left (80 a-\frac{31}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{7 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (5 \left (c-\frac{c}{a x}\right )^{7/2}\right ) \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{\left (80 a-\frac{31}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{7 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (5 \left (c-\frac{c}{a x}\right )^{7/2}\right ) \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{\left (80 a-\frac{31}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{7 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{5 \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 = 0.0818589, size = 101, normalized size = 0.52 $\frac{c^3 \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} \left (15 a^3 x^3+56 a^2 x^2-28 a x+6\right )-75 a^2 x^2 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{15 a^3 x^2 \sqrt{1-\frac{1}{a x}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(6 - 28*a*x + 56*a^2*x^2 + 15*a^3*x^3) - 75*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 = 0.18, size = 149, normalized size = 0.8 \begin{align*} -{\frac{{c}^{3}}{30\,{x}^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -30\,{a}^{7/2}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}+75\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{3}{a}^{3}-112\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+56\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-12\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{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))^(1/2)*(c-c/a/x)^(7/2),x)

[Out]

-1/30/((a*x-1)/(a*x+1))^(1/2)*(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)+75*ln(1
/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*x^3*a^3-112*a^(5/2)*x^2*((a*x+1)*x)^(1/2)+56*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}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.2213, size = 861, normalized size = 4.39 \begin{align*} \left [\frac{75 \,{\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} + 71 \, a^{3} c^{3} x^{3} + 28 \, a^{2} c^{3} x^{2} - 22 \, 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{75 \,{\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} + 71 \, a^{3} c^{3} x^{3} + 28 \, a^{2} c^{3} x^{2} - 22 \, 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))^(1/2)*(c-c/a/x)^(7/2),x, algorithm="fricas")

[Out]

[1/60*(75*(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 + 71*a^3*c^3*x^3 +
28*a^2*c^3*x^2 - 22*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*(75*(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 + 71*a^3*c^3*x^3 + 28*a^2*c^3*x^2 - 22*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))**(1/2)*(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}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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