\(\int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx\) [732]

Optimal result

Integrand size = 25, antiderivative size = 82 \[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^m \sqrt {c-a^2 c x^2}}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-\frac {1}{a^2 x^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6327, 6328, 45} \[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^m \sqrt {c-a^2 c x^2}}{a (m+1) \sqrt {1-\frac {1}{a^2 x^2}}} \]

[In]

Int[E^ArcCoth[a*x]*x^m*Sqrt[c - a^2*c*x^2],x]

[Out]

(x^m*Sqrt[c - a^2*c*x^2])/(a*(1 + m)*Sqrt[1 - 1/(a^2*x^2)]) + (x^(1 + m)*Sqrt[c - a^2*c*x^2])/((2 + m)*Sqrt[1
- 1/(a^2*x^2)])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6327

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

Rule 6328

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.68 \[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^m (2+m+a x+a m x) \sqrt {c-a^2 c x^2}}{a (1+m) (2+m) \sqrt {1-\frac {1}{a^2 x^2}}} \]

[In]

Integrate[E^ArcCoth[a*x]*x^m*Sqrt[c - a^2*c*x^2],x]

[Out]

(x^m*(2 + m + a*x + a*m*x)*Sqrt[c - a^2*c*x^2])/(a*(1 + m)*(2 + m)*Sqrt[1 - 1/(a^2*x^2)])

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.76

method	result	size
gosper	\(\frac {x^{1+m} \sqrt {-a^{2} c \,x^{2}+c}\, \left (a m x +a x +m +2\right )}{\left (1+m \right ) \left (2+m \right ) \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(62\)
risch	\(-\frac {\sqrt {-\frac {c \left (a^{2} x^{2}-1\right )}{\left (a x -1\right ) \left (a x +1\right )}}\, \left (a x -1\right ) c \left (a m x +a x +m +2\right ) x \,x^{m}}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-c}\, \left (2+m \right ) \left (1+m \right )}\)	\(95\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90 \[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=-\frac {\sqrt {-a^{2} c x^{2} + c} {\left ({\left (a m + a\right )} x^{2} + {\left (m + 2\right )} x\right )} x^{m} \sqrt {\frac {a x - 1}{a x + 1}}}{m^{2} - {\left (a m^{2} + 3 \, a m + 2 \, a\right )} x + 3 \, m + 2} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x**m*sqrt(-c*(a*x - 1)*(a*x + 1))/sqrt((a*x - 1)/(a*x + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.66 \[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {{\left (a \sqrt {-c} {\left (m + 1\right )} x^{2} + \sqrt {-c} {\left (m + 2\right )} x\right )} {\left (a x + 1\right )} x^{m}}{{\left (m^{2} + 3 \, m + 2\right )} a x + m^{2} + 3 \, m + 2} \]

[In]

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

[Out]

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

Giac [F]

\[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.24 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.13 \[ \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (\frac {x^m\,x^2\,\sqrt {c-a^2\,c\,x^2}\,\left (m+1\right )}{m^2+3\,m+2}+\frac {x\,x^m\,\sqrt {c-a^2\,c\,x^2}\,\left (m+2\right )}{a\,\left (m^2+3\,m+2\right )}\right )}{x-\frac {1}{a}} \]

[In]

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

[Out]

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