Integrand size = 27, antiderivative size = 130 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x}{3 a^3}+\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{8 a^2}+\frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x^3}{3 a}+\frac {1}{4} \sqrt {c-\frac {c}{a^2 x^2}} x^4+\frac {7 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{8 a^4} \] Output:
4/3*(c-c/a^2/x^2)^(1/2)*x/a^3+7/8*(c-c/a^2/x^2)^(1/2)*x^2/a^2+2/3*(c-c/a^2 /x^2)^(1/2)*x^3/a+1/4*(c-c/a^2/x^2)^(1/2)*x^4+7/8*c^(1/2)*arctanh((c-c/a^2 /x^2)^(1/2)/c^(1/2))/a^4
Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (\sqrt {-1+a^2 x^2} \left (32+21 a x+16 a^2 x^2+6 a^3 x^3\right )+21 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{24 a^3 \sqrt {-1+a^2 x^2}} \] Input:
Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^3,x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2]*(32 + 21*a*x + 16*a^2*x^2 + 6 *a^3*x^3) + 21*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(24*a^3*Sqrt[-1 + a^2*x^2])
Time = 1.07 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6717, 6709, 541, 25, 27, 533, 27, 533, 27, 455, 223}
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.
| \(\displaystyle \int x^3 \sqrt {c-\frac {c}{a^2 x^2}} e^{2 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {x^2 (a x+1)^2}{\sqrt {1-a^2 x^2}}dx}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {\int -\frac {a^2 x^2 (8 a x+7)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {\int \frac {a^2 x^2 (8 a x+7)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \int \frac {x^2 (8 a x+7)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {\int \frac {a x (21 a x+16)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {\int \frac {x (21 a x+16)}{\sqrt {1-a^2 x^2}}dx}{3 a}-\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {a (32 a x+21)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {21 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {32 a x+21}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {21 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {\frac {21 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {32 \sqrt {1-a^2 x^2}}{a}}{2 a}-\frac {21 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {x \left (\frac {1}{4} \left (\frac {\frac {\frac {21 \arcsin (a x)}{a}-\frac {32 \sqrt {1-a^2 x^2}}{a}}{2 a}-\frac {21 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^3,x]
Output:
-((Sqrt[c - c/(a^2*x^2)]*x*(-1/4*(x^3*Sqrt[1 - a^2*x^2]) + ((-8*x^2*Sqrt[1 - a^2*x^2])/(3*a) + ((-21*x*Sqrt[1 - a^2*x^2])/(2*a) + ((-32*Sqrt[1 - a^2 *x^2])/a + (21*ArcSin[a*x])/a)/(2*a))/(3*a))/4))/Sqrt[1 - a^2*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
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_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p) Int[u*((1 + a*x)^n/(x^ (2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && IntegerQ[n/2] && !GtQ[c, 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 = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(\frac {\left (6 a^{3} x^{3}+16 a^{2} x^{2}+21 a x +32\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}{24 a^{3}}+\frac {7 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {c \left (a^{2} x^{2}-1\right )}\, x}{8 a^{2} \sqrt {a^{2} c}\, \left (a^{2} x^{2}-1\right )}\) | \(134\) |
| default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (-6 x {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{4}-16 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3}-27 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c x +27 c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )-48 c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+x c}{\sqrt {c}}\right )-48 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a c \right )}{24 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{4} c}\) | \(196\) |
Input:
int(1/(a*x-1)*(a*x+1)*(c-c/a^2/x^2)^(1/2)*x^3,x,method=_RETURNVERBOSE)
Output:
1/24*(6*a^3*x^3+16*a^2*x^2+21*a*x+32)/a^3*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)*x+ 7/8/a^2*ln(a^2*c*x/(a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)*(c*(a^ 2*x^2-1)/a^2/x^2)^(1/2)*(c*(a^2*x^2-1))^(1/2)/(a^2*x^2-1)*x
Time = 0.11 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.71 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\left [\frac {2 \, {\left (6 \, a^{4} x^{4} + 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 32 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + 21 \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{48 \, a^{4}}, \frac {{\left (6 \, a^{4} x^{4} + 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 32 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 21 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{24 \, a^{4}}\right ] \] Input:
integrate(1/(a*x-1)*(a*x+1)*(c-c/a^2/x^2)^(1/2)*x^3,x, algorithm="fricas")
Output:
[1/48*(2*(6*a^4*x^4 + 16*a^3*x^3 + 21*a^2*x^2 + 32*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) + 21*sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c *x^2 - c)/(a^2*x^2)) - c))/a^4, 1/24*((6*a^4*x^4 + 16*a^3*x^3 + 21*a^2*x^2 + 32*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 21*sqrt(-c)*arctan(a^2*sqrt(- c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)))/a^4]
\[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\int \frac {x^{3} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{a x - 1}\, dx \] Input:
integrate(1/(a*x-1)*(a*x+1)*(c-c/a**2/x**2)**(1/2)*x**3,x)
Output:
Integral(x**3*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)/(a*x - 1), x )
\[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{3}}{a x - 1} \,d x } \] Input:
integrate(1/(a*x-1)*(a*x+1)*(c-c/a^2/x^2)^(1/2)*x^3,x, algorithm="maxima")
Output:
integrate((a*x + 1)*sqrt(c - c/(a^2*x^2))*x^3/(a*x - 1), x)
Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\frac {1}{48} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left ({\left (2 \, x {\left (\frac {3 \, x \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {8 \, \mathrm {sgn}\left (x\right )}{a^{3}}\right )} + \frac {21 \, \mathrm {sgn}\left (x\right )}{a^{4}}\right )} x + \frac {32 \, \mathrm {sgn}\left (x\right )}{a^{5}}\right )} - \frac {42 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a^{4} {\left | a \right |}} + \frac {{\left (21 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) - 64 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\left (x\right )}{a^{5} {\left | a \right |}}\right )} {\left | a \right |} \] Input:
integrate(1/(a*x-1)*(a*x+1)*(c-c/a^2/x^2)^(1/2)*x^3,x, algorithm="giac")
Output:
1/48*(2*sqrt(a^2*c*x^2 - c)*((2*x*(3*x*sgn(x)/a^2 + 8*sgn(x)/a^3) + 21*sgn (x)/a^4)*x + 32*sgn(x)/a^5) - 42*sqrt(c)*log(abs(-sqrt(a^2*c)*x + sqrt(a^2 *c*x^2 - c)))*sgn(x)/(a^4*abs(a)) + (21*a*sqrt(c)*log(abs(c)) - 64*sqrt(-c )*abs(a))*sgn(x)/(a^5*abs(a)))*abs(a)
Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\int \frac {x^3\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )}{a\,x-1} \,d x \] Input:
int((x^3*(c - c/(a^2*x^2))^(1/2)*(a*x + 1))/(a*x - 1),x)
Output:
int((x^3*(c - c/(a^2*x^2))^(1/2)*(a*x + 1))/(a*x - 1), x)
Time = 0.14 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.67 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\frac {\sqrt {c}\, \left (6 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}+16 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+21 \sqrt {a^{2} x^{2}-1}\, a x +32 \sqrt {a^{2} x^{2}-1}+21 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )\right )}{24 a^{4}} \] Input:
int(1/(a*x-1)*(a*x+1)*(c-c/a^2/x^2)^(1/2)*x^3,x)
Output:
(sqrt(c)*(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) + 21*log(sqrt(a** 2*x**2 - 1) + a*x)))/(24*a**4)