3.449 \(\int e^{2 \coth ^{-1}(a x)} (c-\frac {c}{a x})^{3/2} \, dx\)
Optimal. Leaf size=70 \[ \frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {c \sqrt {c-\frac {c}{a x}}}{a}+x \left (c-\frac {c}{a x}\right )^{3/2} \]
[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
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Rubi [A] time = 0.18, antiderivative size = 70, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used =
{6167, 6133, 25, 514, 375, 78, 50, 63, 208} \[ \frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {c \sqrt {c-\frac {c}{a x}}}{a}+x \left (c-\frac {c}{a x}\right )^{3/2} \]
Antiderivative was successfully verified.
[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 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 78
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] || LtQ
[p, n]))))
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]
Rule 375
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 514
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 6133
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 6167
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 e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx &=-\int e^{2 \tanh ^{-1}(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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 55, normalized size = 0.79 \[ \frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )+c (a x-2) \sqrt {c-\frac {c}{a x}}}{a} \]
Antiderivative was successfully verified.
[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
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fricas [A] time = 0.68, size = 137, normalized size = 1.96 \[ \left [\frac {c^{\frac {3}{2}} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, a}, -\frac {\sqrt {-c} c \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{a}\right ] \]
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]
[1/2*(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]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Warning, choosing root of [1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%
}+%%%{-2,[3,2,3]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [86,-97,-82]
Warning, choosing root of [1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%
{-2,[3,2,3]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [7,-27,26]Warning
, choosing root of [1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,
2,3]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-89,63,-49]Warning, cho
osing root of [1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%
%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-86,-64,-30]Warning, choosin
g root of [1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+
%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [70,22,42]Warning, choosing root
of [1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{-1,
[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [56,-9,-13]Sign error (%%%{sqrt(c)*a,0%%
%}+%%%{2*sqrt(-a*c)*abs(a),1/2%%%}+%%%{-2*sqrt(c)*a^2,1%%%}+%%%{-a*sqrt(-a*c)*abs(a),3/2%%%}+%%%{-a^2*sqrt(-a*
c)*abs(a)/4,5/2%%%}+%%%{undef,7/2%%%})Limit: Max order reached or unable to make series expansion Error: Bad A
rgument Value
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maple [A] time = 0.05, size = 103, normalized size = 1.47 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \left (-2 \sqrt {a \,x^{2}-x}\, a^{\frac {3}{2}} x^{2}+4 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}+\ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{2} a \right )}{2 x \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*x+1)/(a*x-1)*(c-c/a/x)^(3/2),x)
[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))*x^2*a)/((a*x-1)*x)^(1/2)/a^(3/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}{a x - 1}\,{d x} \]
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)*(c - c/(a*x))^(3/2)/(a*x - 1), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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]
Exception raised: TypeError
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