\(\int e^{2 \coth ^{-1}(a x)} (c-\frac {c}{a x})^{3/2} \, dx\) [449]

Optimal result

Integrand size = 24, antiderivative size = 70 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=-\frac {c \sqrt {c-\frac {c}{a x}}}{a}+\left (c-\frac {c}{a x}\right )^{3/2} x+\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]

[Out]

(c-c/a/x)^(3/2)*x+c^(3/2)*arctanh((c-c/a/x)^(1/2)/c^(1/2))/a-c*(c-c/a/x)^(1/2)/a

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6302, 6268, 25, 528, 382, 79, 52, 65, 214} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {c \sqrt {c-\frac {c}{a x}}}{a}+x \left (c-\frac {c}{a x}\right )^{3/2} \]

[In]

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

[Out]

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

Rule 25

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[u*((
a + b*x^n)^(m + p)/x^(n*p)), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] &&  !(IntegerQ[m] && NegQ[n])

Rule 52

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 65

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 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)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 214

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

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[(a + b/x^n)^p*((c +
d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rule 528

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 6268

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

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx \\ & = -\int \frac {\left (c-\frac {c}{a x}\right )^{3/2} (1+a x)}{1-a x} \, dx \\ & = \frac {c \int \frac {\sqrt {c-\frac {c}{a x}} (1+a x)}{x} \, dx}{a} \\ & = \frac {c \int \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}} \, dx}{a} \\ & = -\frac {c \text {Subst}\left (\int \frac {(a+x) \sqrt {c-\frac {c x}{a}}}{x^2} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \left (c-\frac {c}{a x}\right )^{3/2} x-\frac {c \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = -\frac {c \sqrt {c-\frac {c}{a x}}}{a}+\left (c-\frac {c}{a x}\right )^{3/2} x-\frac {c^2 \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = -\frac {c \sqrt {c-\frac {c}{a x}}}{a}+\left (c-\frac {c}{a x}\right )^{3/2} x+c \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right ) \\ & = -\frac {c \sqrt {c-\frac {c}{a x}}}{a}+\left (c-\frac {c}{a x}\right )^{3/2} x+\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.79 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\frac {c \sqrt {c-\frac {c}{a x}} (-2+a x)+c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.47

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \left (-2 \sqrt {a \,x^{2}-x}\, a^{\frac {3}{2}} x^{2}+4 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}+\ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \,x^{2}\right )}{2 x \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}}}\)	\(103\)
risch	\(\frac {\left (a^{2} x^{2}-3 a x +2\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a \left (a x -1\right )}+\frac {\ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\)	\(122\)

[In]

int(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(c*(a*x-1)/a/x)^(1/2)/x*c*(-2*(a*x^2-x)^(1/2)*a^(3/2)*x^2+4*(a*x^2-x)^(3/2)*a^(1/2)+ln(1/2*(2*(a*x^2-x)^(1
/2)*a^(1/2)+2*a*x-1)/a^(1/2))*a*x^2)/((a*x-1)*x)^(1/2)/a^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.96 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\left [\frac {c^{\frac {3}{2}} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, a}, -\frac {\sqrt {-c} c \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{a}\right ] \]

[In]

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

[Out]

[1/2*(c^(3/2)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) + 2*(a*c*x - 2*c)*sqrt((a*c*x - c)/(a*
x)))/a, -(sqrt(-c)*c*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (a*c*x - 2*c)*sqrt((a*c*x - c)/(a*x)))/a]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 25.43 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.47 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=c \left (\begin {cases} - \frac {\sqrt {c} \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a} + \frac {\sqrt {c} \sqrt {x} \sqrt {a x - 1}}{\sqrt {a}} & \text {for}\: \left |{a x}\right | > 1 \\- \frac {i \sqrt {a} \sqrt {c} x^{\frac {3}{2}}}{\sqrt {- a x + 1}} + \frac {i \sqrt {c} \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a} + \frac {i \sqrt {c} \sqrt {x}}{\sqrt {a} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) - \frac {c \left (\begin {cases} \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {c - \frac {c}{a x}}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 \sqrt {c - \frac {c}{a x}} & \text {for}\: \frac {c}{a} \neq 0 \\- \sqrt {c} \log {\left (x \right )} & \text {otherwise} \end {cases}\right )}{a} \]

[In]

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

[Out]

c*Piecewise((-sqrt(c)*acosh(sqrt(a)*sqrt(x))/a + sqrt(c)*sqrt(x)*sqrt(a*x - 1)/sqrt(a), Abs(a*x) > 1), (-I*sqr
t(a)*sqrt(c)*x**(3/2)/sqrt(-a*x + 1) + I*sqrt(c)*asin(sqrt(a)*sqrt(x))/a + I*sqrt(c)*sqrt(x)/(sqrt(a)*sqrt(-a*
x + 1)), True)) - c*Piecewise((2*c*atan(sqrt(c - c/(a*x))/sqrt(-c))/sqrt(-c) + 2*sqrt(c - c/(a*x)), Ne(c/a, 0)
), (-sqrt(c)*log(x), True))/a

Maxima [F]

\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}{a x - 1} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]

[In]

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

[Out]

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