3.5.52 \(\int \frac {e^{2 \coth ^{-1}(a x)}}{(c-\frac {c}{a x})^{3/2}} \, dx\) [452]

Optimal. Leaf size=95 \[ -\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{a c \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6302, 6268, 25, 528, 382, 79, 53, 65, 214} \begin {gather*} \frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{a c \sqrt {c-\frac {c}{a x}}}-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 53

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] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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*} \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx\\ &=-\int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{3/2} (1-a x)} \, dx\\ &=\frac {c \int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{5/2} x} \, dx}{a}\\ &=\frac {c \int \frac {a+\frac {1}{x}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx}{a}\\ &=-\frac {c \text {Subst}\left (\int \frac {a+x}{x^2 \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {(7 c) \text {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7 \text {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{a c \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7 \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c}\\ &=-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{a c \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7 \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^2}\\ &=-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{a c \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 55, normalized size = 0.58 \begin {gather*} \frac {x \left (3 a x-7 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1-\frac {1}{a x}\right )\right )}{3 c \sqrt {c-\frac {c}{a x}} (-1+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(3*a*x - 7*Hypergeometric2F1[-3/2, 1, -1/2, 1 - 1/(a*x)]))/(3*c*Sqrt[c - c/(a*x)]*(-1 + a*x))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(259\) vs. \(2(81)=162\).
time = 0.16, size = 260, normalized size = 2.74

method	result	size
risch	\(\frac {a x -1}{a c \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {7 \ln \left (\frac {-\frac {1}{2} a c +c \,a^{2} x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 a^{2} \sqrt {a^{2} c}}-\frac {22 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{3 a^{4} c \left (x -\frac {1}{a}\right )}-\frac {4 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{3 a^{5} c \left (x -\frac {1}{a}\right )^{2}}\right ) a \sqrt {c \left (a x -1\right ) a x}}{c x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\)	\(201\)
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-42 \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}} x^{3}+36 \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} a^{\frac {5}{2}} x -21 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{3}+126 a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, x^{2}-28 a^{\frac {3}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}+63 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2} x^{2}-126 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, x -63 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a x +42 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+21 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{6 \sqrt {\left (a x -1\right ) x}\, c^{2} \left (a x -1\right )^{3} \sqrt {a}}\)	\(260\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(c*(a*x-1)/a/x)^(1/2)*x*(-42*((a*x-1)*x)^(1/2)*a^(7/2)*x^3+36*((a*x-1)*x)^(3/2)*a^(5/2)*x-21*ln(1/2*(2*((
a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*a^3*x^3+126*a^(5/2)*((a*x-1)*x)^(1/2)*x^2-28*a^(3/2)*((a*x-1)*x)^(3/
2)+63*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*a^2*x^2-126*a^(3/2)*((a*x-1)*x)^(1/2)*x-63*ln(1/2*
(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*a*x+42*((a*x-1)*x)^(1/2)*a^(1/2)+21*ln(1/2*(2*((a*x-1)*x)^(1/2)
*a^(1/2)+2*a*x-1)/a^(1/2)))/((a*x-1)*x)^(1/2)/c^2/(a*x-1)^3/a^(1/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 238, normalized size = 2.51 \begin {gather*} \left [\frac {21 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, -\frac {21 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(21*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) + 2*(3*a^3*x^
3 - 28*a^2*x^2 + 21*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), -1/3*(21*(a^2*x^2 - 2*a
*x + 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - (3*a^3*x^3 - 28*a^2*x^2 + 21*a*x)*sqrt((a*c*x -
c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x + 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (a x - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (81) = 162\).
time = 0.52, size = 243, normalized size = 2.56 \begin {gather*} \frac {7 \, \log \left (c^{2} {\left | a \right |} \sqrt {{\left | c \right |}}\right ) \mathrm {sgn}\left (x\right )}{10 \, a c^{\frac {3}{2}}} - \frac {7 \, \log \left ({\left | -2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} {\left | a \right |} + 9 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a \sqrt {c} - 16 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} c {\left | a \right |} + 14 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a c^{\frac {3}{2}} - 6 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} c^{2} {\left | a \right |} + a c^{\frac {5}{2}} \right |}\right ) \mathrm {sgn}\left (x\right )}{10 \, a c^{\frac {3}{2}}} + \frac {\sqrt {a^{2} c x^{2} - a c x} {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{2} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/10*log(c^2*abs(a)*sqrt(abs(c)))*sgn(x)/(a*c^(3/2)) - 7/10*log(abs(-2*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x
))^5*abs(a) + 9*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^4*a*sqrt(c) - 16*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a
*c*x))^3*c*abs(a) + 14*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^2*a*c^(3/2) - 6*(sqrt(a^2*c)*x - sqrt(a^2*c*x
^2 - a*c*x))*c^2*abs(a) + a*c^(5/2)))*sgn(x)/(a*c^(3/2)) + sqrt(a^2*c*x^2 - a*c*x)*abs(a)*sgn(x)/(a^2*c^2)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,x+1}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\left (a\,x-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________