Integrand size = 27, antiderivative size = 129 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {4 \sqrt {c-a^2 c x^2}}{3 a^3}+\frac {7 x \sqrt {c-a^2 c x^2}}{8 a^2}+\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}+\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {7 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3} \] Output:
4/3*(-a^2*c*x^2+c)^(1/2)/a^3+7/8*x*(-a^2*c*x^2+c)^(1/2)/a^2+2/3*x^2*(-a^2* c*x^2+c)^(1/2)/a+1/4*x^3*(-a^2*c*x^2+c)^(1/2)-7/8*c^(1/2)*arctan(a*c^(1/2) *x/(-a^2*c*x^2+c)^(1/2))/a^3
Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.68 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (32+21 a x+16 a^2 x^2+6 a^3 x^3\right )+21 \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{24 a^3} \] Input:
Integrate[E^(2*ArcCoth[a*x])*x^2*Sqrt[c - a^2*c*x^2],x]
Output:
(Sqrt[c - a^2*c*x^2]*(32 + 21*a*x + 16*a^2*x^2 + 6*a^3*x^3) + 21*Sqrt[c]*A rcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(24*a^3)
Time = 0.97 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.25, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6717, 6701, 541, 25, 27, 533, 27, 533, 27, 455, 224, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 \sqrt {c-a^2 c x^2} e^{2 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int e^{2 \text {arctanh}(a x)} x^2 \sqrt {c-a^2 c x^2}dx\) |
\(\Big \downarrow \) 6701 |
\(\displaystyle -c \int \frac {x^2 (a x+1)^2}{\sqrt {c-a^2 c x^2}}dx\) |
\(\Big \downarrow \) 541 |
\(\displaystyle -c \left (-\frac {\int -\frac {a^2 c x^2 (8 a x+7)}{\sqrt {c-a^2 c x^2}}dx}{4 a^2 c}-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -c \left (\frac {\int \frac {a^2 c x^2 (8 a x+7)}{\sqrt {c-a^2 c x^2}}dx}{4 a^2 c}-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c \left (\frac {1}{4} \int \frac {x^2 (8 a x+7)}{\sqrt {c-a^2 c x^2}}dx-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
\(\Big \downarrow \) 533 |
\(\displaystyle -c \left (\frac {1}{4} \left (\frac {\int \frac {a c x (21 a x+16)}{\sqrt {c-a^2 c x^2}}dx}{3 a^2 c}-\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c \left (\frac {1}{4} \left (\frac {\int \frac {x (21 a x+16)}{\sqrt {c-a^2 c x^2}}dx}{3 a}-\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
\(\Big \downarrow \) 533 |
\(\displaystyle -c \left (\frac {1}{4} \left (\frac {\frac {\int \frac {a c (32 a x+21)}{\sqrt {c-a^2 c x^2}}dx}{2 a^2 c}-\frac {21 x \sqrt {c-a^2 c x^2}}{2 a c}}{3 a}-\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c \left (\frac {1}{4} \left (\frac {\frac {\int \frac {32 a x+21}{\sqrt {c-a^2 c x^2}}dx}{2 a}-\frac {21 x \sqrt {c-a^2 c x^2}}{2 a c}}{3 a}-\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -c \left (\frac {1}{4} \left (\frac {\frac {21 \int \frac {1}{\sqrt {c-a^2 c x^2}}dx-\frac {32 \sqrt {c-a^2 c x^2}}{a c}}{2 a}-\frac {21 x \sqrt {c-a^2 c x^2}}{2 a c}}{3 a}-\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -c \left (\frac {1}{4} \left (\frac {\frac {21 \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}-\frac {32 \sqrt {c-a^2 c x^2}}{a c}}{2 a}-\frac {21 x \sqrt {c-a^2 c x^2}}{2 a c}}{3 a}-\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -c \left (\frac {1}{4} \left (\frac {\frac {\frac {21 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}-\frac {32 \sqrt {c-a^2 c x^2}}{a c}}{2 a}-\frac {21 x \sqrt {c-a^2 c x^2}}{2 a c}}{3 a}-\frac {8 x^2 \sqrt {c-a^2 c x^2}}{3 a c}\right )-\frac {x^3 \sqrt {c-a^2 c x^2}}{4 c}\right )\) |
Input:
Int[E^(2*ArcCoth[a*x])*x^2*Sqrt[c - a^2*c*x^2],x]
Output:
-(c*(-1/4*(x^3*Sqrt[c - a^2*c*x^2])/c + ((-8*x^2*Sqrt[c - a^2*c*x^2])/(3*a *c) + ((-21*x*Sqrt[c - a^2*c*x^2])/(2*a*c) + ((-32*Sqrt[c - a^2*c*x^2])/(a *c) + (21*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(a*Sqrt[c]))/(2*a))/( 3*a))/4))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x ] + Simp[1/(b*(m + n + 2*p + 1)) Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) *x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^(n/2) Int[x^m*(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ [c, 0]) && IGtQ[n/2, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {\left (6 a^{3} x^{3}+16 a^{2} x^{2}+21 a x +32\right ) \left (a^{2} x^{2}-1\right ) c}{24 a^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {7 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c}{8 a^{2} \sqrt {a^{2} c}}\) | \(89\) |
default | \(-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 a^{2} c}+\frac {\frac {9 x \sqrt {-a^{2} c \,x^{2}+c}}{8}+\frac {9 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{8 \sqrt {a^{2} c}}}{a^{2}}-\frac {2 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 a^{3} c}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}-\frac {2 a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}}{a^{3}}\) | \(185\) |
Input:
int((-a^2*c*x^2+c)^(1/2)*(a*x+1)*x^2/(a*x-1),x,method=_RETURNVERBOSE)
Output:
-1/24*(6*a^3*x^3+16*a^2*x^2+21*a*x+32)*(a^2*x^2-1)/a^3/(-c*(a^2*x^2-1))^(1 /2)*c-7/8/a^2/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))*c
Time = 0.12 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.30 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\left [\frac {2 \, {\left (6 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 21 \, a x + 32\right )} \sqrt {-a^{2} c x^{2} + c} + 21 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{48 \, a^{3}}, \frac {{\left (6 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 21 \, a x + 32\right )} \sqrt {-a^{2} c x^{2} + c} + 21 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{24 \, a^{3}}\right ] \] Input:
integrate(1/(a*x-1)*(a*x+1)*x^2*(-a^2*c*x^2+c)^(1/2),x, algorithm="fricas" )
Output:
[1/48*(2*(6*a^3*x^3 + 16*a^2*x^2 + 21*a*x + 32)*sqrt(-a^2*c*x^2 + c) + 21* sqrt(-c)*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a^3, 1/24*((6*a^3*x^3 + 16*a^2*x^2 + 21*a*x + 32)*sqrt(-a^2*c*x^2 + c) + 21*sqr t(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)))/a^3]
\[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^{2} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{a x - 1}\, dx \] Input:
integrate(1/(a*x-1)*(a*x+1)*x**2*(-a**2*c*x**2+c)**(1/2),x)
Output:
Integral(x**2*sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)/(a*x - 1), x)
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {9 \, \sqrt {-a^{2} c x^{2} + c} x}{8 \, a^{2}} - \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{4 \, a^{2} c} - \frac {7 \, \sqrt {c} \arcsin \left (a x\right )}{8 \, a^{3}} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{3}} - \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a^{3} c} \] Input:
integrate(1/(a*x-1)*(a*x+1)*x^2*(-a^2*c*x^2+c)^(1/2),x, algorithm="maxima" )
Output:
9/8*sqrt(-a^2*c*x^2 + c)*x/a^2 - 1/4*(-a^2*c*x^2 + c)^(3/2)*x/(a^2*c) - 7/ 8*sqrt(c)*arcsin(a*x)/a^3 + 2*sqrt(-a^2*c*x^2 + c)/a^3 - 2/3*(-a^2*c*x^2 + c)^(3/2)/(a^3*c)
Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.65 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {1}{24} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left (3 \, x + \frac {8}{a}\right )} x + \frac {21}{a^{2}}\right )} x + \frac {32}{a^{3}}\right )} + \frac {7 \, c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, a^{2} \sqrt {-c} {\left | a \right |}} \] Input:
integrate(1/(a*x-1)*(a*x+1)*x^2*(-a^2*c*x^2+c)^(1/2),x, algorithm="giac")
Output:
1/24*sqrt(-a^2*c*x^2 + c)*((2*(3*x + 8/a)*x + 21/a^2)*x + 32/a^3) + 7/8*c* log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(a^2*sqrt(-c)*abs(a))
Timed out. \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^2\,\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \] Input:
int((x^2*(c - a^2*c*x^2)^(1/2)*(a*x + 1))/(a*x - 1),x)
Output:
int((x^2*(c - a^2*c*x^2)^(1/2)*(a*x + 1))/(a*x - 1), x)
Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.63 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c}\, \left (-21 \mathit {asin} \left (a x \right )+6 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+16 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+21 \sqrt {-a^{2} x^{2}+1}\, a x +32 \sqrt {-a^{2} x^{2}+1}-32\right )}{24 a^{3}} \] Input:
int(1/(a*x-1)*(a*x+1)*x^2*(-a^2*c*x^2+c)^(1/2),x)
Output:
(sqrt(c)*( - 21*asin(a*x) + 6*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 16*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 21*sqrt( - a**2*x**2 + 1)*a*x + 32*sqrt( - a* *2*x**2 + 1) - 32))/(24*a**3)