Integrand size = 23, antiderivative size = 200 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=-\frac {c \sqrt {1-a^2 x^2}}{3 x^3 \sqrt {c-a c x}}-\frac {13 a c \sqrt {1-a^2 x^2}}{12 x^2 \sqrt {c-a c x}}-\frac {19 a^2 c \sqrt {1-a^2 x^2}}{8 x \sqrt {c-a c x}}-\frac {45}{8} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )+4 \sqrt {2} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right ) \] Output:
-1/3*c*(-a^2*x^2+1)^(1/2)/x^3/(-a*c*x+c)^(1/2)-13/12*a*c*(-a^2*x^2+1)^(1/2 )/x^2/(-a*c*x+c)^(1/2)-19/8*a^2*c*(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^(1/2)-45 /8*a^3*c^(1/2)*arctanh(c^(1/2)*(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(1/2))+4*2^(1 /2)*a^3*c^(1/2)*arctanh(1/2*c^(1/2)*(-a^2*x^2+1)^(1/2)*2^(1/2)/(-a*c*x+c)^ (1/2))
Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.50 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=-\frac {\sqrt {c-a c x} \left (\sqrt {1+a x} \left (8+26 a x+57 a^2 x^2\right )+135 a^3 x^3 \text {arctanh}\left (\sqrt {1+a x}\right )-96 \sqrt {2} a^3 x^3 \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )\right )}{24 x^3 \sqrt {1-a x}} \] Input:
Integrate[(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^4,x]
Output:
-1/24*(Sqrt[c - a*c*x]*(Sqrt[1 + a*x]*(8 + 26*a*x + 57*a^2*x^2) + 135*a^3* x^3*ArcTanh[Sqrt[1 + a*x]] - 96*Sqrt[2]*a^3*x^3*ArcTanh[Sqrt[1 + a*x]/Sqrt [2]]))/(x^3*Sqrt[1 - a*x])
Time = 0.63 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.61, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6680, 37, 109, 27, 168, 27, 168, 27, 174, 73, 219, 221}
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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {(a x+1)^{3/2} \sqrt {c-a c x}}{x^4 (1-a x)^{3/2}}dx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle \frac {\sqrt {c-a c x} \int \frac {(a x+1)^{3/2}}{x^4 (1-a x)}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{3} \int -\frac {a (11 a x+13)}{2 x^3 (1-a x) \sqrt {a x+1}}dx-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \int \frac {11 a x+13}{x^3 (1-a x) \sqrt {a x+1}}dx-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (-\frac {1}{2} \int -\frac {3 a (13 a x+19)}{2 x^2 (1-a x) \sqrt {a x+1}}dx-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \int \frac {13 a x+19}{x^2 (1-a x) \sqrt {a x+1}}dx-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \left (-\int -\frac {a (19 a x+45)}{2 x (1-a x) \sqrt {a x+1}}dx-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {19 a x+45}{x (1-a x) \sqrt {a x+1}}dx-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \left (45 \int \frac {1}{x \sqrt {a x+1}}dx+64 a \int \frac {1}{(1-a x) \sqrt {a x+1}}dx\right )-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \left (128 \int \frac {1}{1-a x}d\sqrt {a x+1}+\frac {90 \int \frac {1}{\frac {a x+1}{a}-\frac {1}{a}}d\sqrt {a x+1}}{a}\right )-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \left (\frac {90 \int \frac {1}{\frac {a x+1}{a}-\frac {1}{a}}d\sqrt {a x+1}}{a}+64 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )\right )-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\left (\frac {1}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \left (64 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )-90 \text {arctanh}\left (\sqrt {a x+1}\right )\right )-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right ) \sqrt {c-a c x}}{\sqrt {1-a x}}\) |
Input:
Int[(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^4,x]
Output:
(Sqrt[c - a*c*x]*(-1/3*Sqrt[1 + a*x]/x^3 + (a*((-13*Sqrt[1 + a*x])/(2*x^2) + (3*a*((-19*Sqrt[1 + a*x])/x + (a*(-90*ArcTanh[Sqrt[1 + a*x]] + 64*Sqrt[ 2]*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]]))/2))/4))/6))/Sqrt[1 - a*x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S imp[(a + b*x)^m/(c + d*x)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && !SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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]
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, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.13 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.76
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (96 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{3} c \,x^{3}-135 c \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{3} x^{3}-57 a^{2} x^{2} \sqrt {c}\, \sqrt {c \left (a x +1\right )}-26 a x \sqrt {c \left (a x +1\right )}\, \sqrt {c}-8 \sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{24 \sqrt {c}\, \left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, x^{3}}\) | \(151\) |
risch | \(\frac {\left (57 a^{3} x^{3}+83 a^{2} x^{2}+34 a x +8\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{24 x^{3} \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {a^{3} \left (\frac {64 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {90 \,\operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{16 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(187\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^4,x,method=_RETURNVERB OSE)
Output:
-1/24*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(96*2^(1/2)*arctanh(1/2*(c*(a* x+1))^(1/2)*2^(1/2)/c^(1/2))*a^3*c*x^3-135*c*arctanh((c*(a*x+1))^(1/2)/c^( 1/2))*a^3*x^3-57*a^2*x^2*c^(1/2)*(c*(a*x+1))^(1/2)-26*a*x*(c*(a*x+1))^(1/2 )*c^(1/2)-8*(c*(a*x+1))^(1/2)*c^(1/2))/c^(1/2)/(a*x-1)/(c*(a*x+1))^(1/2)/x ^3
Time = 0.16 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.00 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\left [\frac {96 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 135 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, {\left (57 \, a^{2} x^{2} + 26 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{48 \, {\left (a x^{4} - x^{3}\right )}}, \frac {96 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{2 \, {\left (a c x - c\right )}}\right ) - 135 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + {\left (57 \, a^{2} x^{2} + 26 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{24 \, {\left (a x^{4} - x^{3}\right )}}\right ] \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^4,x, algorithm=" fricas")
Output:
[1/48*(96*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2* a*x + 1)) + 135*(a^4*x^4 - a^3*x^3)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sq rt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*(57*a^2* x^2 + 26*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^4 - x^3), 1/24 *(96*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) - 135*(a^4*x^4 - a^3*x^3)*sqr t(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) + ( 57*a^2*x^2 + 26*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^4 - x^3 )]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )^{3}}{x^{4} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(1/2)/x**4,x)
Output:
Integral(sqrt(-c*(a*x - 1))*(a*x + 1)**3/(x**4*(-(a*x - 1)*(a*x + 1))**(3/ 2)), x)
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int { \frac {\sqrt {-a c x + c} {\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^4,x, algorithm=" maxima")
Output:
integrate(sqrt(-a*c*x + c)*(a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*x^4), x)
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^4,x, algorithm=" giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^3}{x^4\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(((c - a*c*x)^(1/2)*(a*x + 1)^3)/(x^4*(1 - a^2*x^2)^(3/2)),x)
Output:
int(((c - a*c*x)^(1/2)*(a*x + 1)^3)/(x^4*(1 - a^2*x^2)^(3/2)), x)
Time = 0.17 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.98 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {\sqrt {c}\, \left (-114 \sqrt {a x +1}\, a^{2} x^{2}-52 \sqrt {a x +1}\, a x -16 \sqrt {a x +1}-192 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right ) a^{3} x^{3}+182 \sqrt {2}\, a^{3} x^{3}-135 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{3} x^{3}+135 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{3} x^{3}+135 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{3} x^{3}-135 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{3} x^{3}\right )}{48 x^{3}} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^4,x)
Output:
(sqrt(c)*( - 114*sqrt(a*x + 1)*a**2*x**2 - 52*sqrt(a*x + 1)*a*x - 16*sqrt( a*x + 1) - 192*sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2))/2))*a**3*x** 3 + 182*sqrt(2)*a**3*x**3 - 135*log( - sqrt(2) + tan(asin(sqrt( - a*x + 1) /sqrt(2))/2) - 1)*a**3*x**3 + 135*log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1)*a**3*x**3 + 135*log(sqrt(2) + tan(asin(sqrt( - a*x + 1 )/sqrt(2))/2) - 1)*a**3*x**3 - 135*log(sqrt(2) + tan(asin(sqrt( - a*x + 1) /sqrt(2))/2) + 1)*a**3*x**3))/(48*x**3)