3.361 \(\int \frac {e^{\tanh ^{-1}(a x)}}{x^2 (c-a c x)^4} \, dx\)

Optimal. Leaf size=155 \[ \frac {16 a (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {\sqrt {1-a^2 x^2}}{c^4 x}+\frac {a (719 a x+525)}{105 c^4 \sqrt {1-a^2 x^2}}+\frac {a (307 a x+175)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {4 a (17 a x+7)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {5 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^4} \]

[Out]

16/7*a*(a*x+1)/c^4/(-a^2*x^2+1)^(7/2)+4/35*a*(17*a*x+7)/c^4/(-a^2*x^2+1)^(5/2)+1/105*a*(307*a*x+175)/c^4/(-a^2
*x^2+1)^(3/2)-5*a*arctanh((-a^2*x^2+1)^(1/2))/c^4+1/105*a*(719*a*x+525)/c^4/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1
/2)/c^4/x

________________________________________________________________________________________

Rubi [A]  time = 0.45, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6128, 852, 1805, 807, 266, 63, 208} \[ \frac {16 a (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {\sqrt {1-a^2 x^2}}{c^4 x}+\frac {a (719 a x+525)}{105 c^4 \sqrt {1-a^2 x^2}}+\frac {a (307 a x+175)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {4 a (17 a x+7)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {5 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^4} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(x^2*(c - a*c*x)^4),x]

[Out]

(16*a*(1 + a*x))/(7*c^4*(1 - a^2*x^2)^(7/2)) + (4*a*(7 + 17*a*x))/(35*c^4*(1 - a^2*x^2)^(5/2)) + (a*(175 + 307
*a*x))/(105*c^4*(1 - a^2*x^2)^(3/2)) + (a*(525 + 719*a*x))/(105*c^4*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(c^
4*x) - (5*a*ArcTanh[Sqrt[1 - a^2*x^2]])/c^4

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 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 852

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[((f + g*x)^n*(a + c*x^2)^(m + p))/(d - e*x)^m, x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 6128

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[c^n,
Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c +
d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1, 0]) && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^2 (c-a c x)^4} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x^2 (c-a c x)^5} \, dx\\ &=\frac {\int \frac {(c+a c x)^5}{x^2 \left (1-a^2 x^2\right )^{9/2}} \, dx}{c^9}\\ &=\frac {16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {\int \frac {-7 c^5-35 a c^5 x-61 a^2 c^5 x^2+7 a^3 c^5 x^3}{x^2 \left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^9}\\ &=\frac {16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {\int \frac {35 c^5+175 a c^5 x+272 a^2 c^5 x^2}{x^2 \left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^9}\\ &=\frac {16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {-105 c^5-525 a c^5 x-614 a^2 c^5 x^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{105 c^9}\\ &=\frac {16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (525+719 a x)}{105 c^4 \sqrt {1-a^2 x^2}}+\frac {\int \frac {105 c^5+525 a c^5 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{105 c^9}\\ &=\frac {16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (525+719 a x)}{105 c^4 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^4 x}+\frac {(5 a) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c^4}\\ &=\frac {16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (525+719 a x)}{105 c^4 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^4 x}+\frac {(5 a) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 c^4}\\ &=\frac {16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (525+719 a x)}{105 c^4 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^4 x}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a c^4}\\ &=\frac {16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (525+719 a x)}{105 c^4 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^4 x}-\frac {5 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 109, normalized size = 0.70 \[ \frac {824 a^5 x^5-1947 a^4 x^4+485 a^3 x^3+1812 a^2 x^2-525 a x (a x-1)^3 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-1339 a x+105}{105 c^4 x (a x-1)^3 \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]/(x^2*(c - a*c*x)^4),x]

[Out]

(105 - 1339*a*x + 1812*a^2*x^2 + 485*a^3*x^3 - 1947*a^4*x^4 + 824*a^5*x^5 - 525*a*x*(-1 + a*x)^3*Sqrt[1 - a^2*
x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(105*c^4*x*(-1 + a*x)^3*Sqrt[1 - a^2*x^2])

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 188, normalized size = 1.21 \[ \frac {1024 \, a^{5} x^{5} - 4096 \, a^{4} x^{4} + 6144 \, a^{3} x^{3} - 4096 \, a^{2} x^{2} + 1024 \, a x + 525 \, {\left (a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (824 \, a^{4} x^{4} - 2771 \, a^{3} x^{3} + 3256 \, a^{2} x^{2} - 1444 \, a x + 105\right )} \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{4} c^{4} x^{5} - 4 \, a^{3} c^{4} x^{4} + 6 \, a^{2} c^{4} x^{3} - 4 \, a c^{4} x^{2} + c^{4} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

1/105*(1024*a^5*x^5 - 4096*a^4*x^4 + 6144*a^3*x^3 - 4096*a^2*x^2 + 1024*a*x + 525*(a^5*x^5 - 4*a^4*x^4 + 6*a^3
*x^3 - 4*a^2*x^2 + a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (824*a^4*x^4 - 2771*a^3*x^3 + 3256*a^2*x^2 - 1444*a*
x + 105)*sqrt(-a^2*x^2 + 1))/(a^4*c^4*x^5 - 4*a^3*c^4*x^4 + 6*a^2*c^4*x^3 - 4*a*c^4*x^2 + c^4*x)

________________________________________________________________________________________

giac [B]  time = 0.23, size = 323, normalized size = 2.08 \[ -\frac {5 \, a^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{c^{4} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{2 \, c^{4} x {\left | a \right |}} - \frac {{\left (105 \, a^{2} - \frac {4831 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{x} + \frac {24997 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{2} x^{2}} - \frac {61131 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{4} x^{3}} + \frac {82915 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{6} x^{4}} - \frac {66325 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{8} x^{5}} + \frac {29295 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{10} x^{6}} - \frac {5985 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7}}{a^{12} x^{7}}\right )} a^{2} x}{210 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^4,x, algorithm="giac")

[Out]

-5*a^2*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/(c^4*abs(a)) - 1/2*(sqrt(-a^2*x^2 + 1)*ab
s(a) + a)/(c^4*x*abs(a)) - 1/210*(105*a^2 - 4831*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/x + 24997*(sqrt(-a^2*x^2 + 1)
*abs(a) + a)^2/(a^2*x^2) - 61131*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^4*x^3) + 82915*(sqrt(-a^2*x^2 + 1)*abs(a
) + a)^4/(a^6*x^4) - 66325*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^8*x^5) + 29295*(sqrt(-a^2*x^2 + 1)*abs(a) + a)
^6/(a^10*x^6) - 5985*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^7/(a^12*x^7))*a^2*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4*
((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))

________________________________________________________________________________________

maple [B]  time = 0.05, size = 423, normalized size = 2.73 \[ \frac {-5 a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\sqrt {-a^{2} x^{2}+1}}{x}+\frac {4 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {19 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}+\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{2}}-\frac {3 \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{a}}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^4,x)

[Out]

1/c^4*(-5*a*arctanh(1/(-a^2*x^2+1)^(1/2))-(-a^2*x^2+1)^(1/2)/x+4/3/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1
/2)-19/3/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+2/a^2*(1/7/a/(x-1/a)^4*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-
3/7*a*(1/5/a/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-2/5*a*(1/3/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^
(1/2)-1/3/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2))))-3/a*(1/5/a/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2
)-2/5*a*(1/3/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2))))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}^{4} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a*c*x+c)^4,x, algorithm="maxima")

[Out]

integrate((a*x + 1)/(sqrt(-a^2*x^2 + 1)*(a*c*x - c)^4*x^2), x)

________________________________________________________________________________________

mupad [B]  time = 0.83, size = 352, normalized size = 2.27 \[ \frac {26\,a^3\,\sqrt {1-a^2\,x^2}}{15\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {4\,a^5\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{c^4\,x}+\frac {2\,a^3\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {719\,a^2\,\sqrt {1-a^2\,x^2}}{105\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {27\,a^2\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}+\frac {a\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + 1)/(x^2*(1 - a^2*x^2)^(1/2)*(c - a*c*x)^4),x)

[Out]

(26*a^3*(1 - a^2*x^2)^(1/2))/(15*(a^2*c^4 - 2*a^3*c^4*x + a^4*c^4*x^2)) + (4*a^5*(1 - a^2*x^2)^(1/2))/(35*(a^4
*c^4 - 2*a^5*c^4*x + a^6*c^4*x^2)) - (1 - a^2*x^2)^(1/2)/(c^4*x) + (a*atan((1 - a^2*x^2)^(1/2)*1i)*5i)/c^4 + (
2*a^3*(1 - a^2*x^2)^(1/2))/(7*(a^2*c^4 - 4*a^3*c^4*x + 6*a^4*c^4*x^2 - 4*a^5*c^4*x^3 + a^6*c^4*x^4)) + (719*a^
2*(1 - a^2*x^2)^(1/2))/(105*(-a^2)^(1/2)*(c^4*x*(-a^2)^(1/2) - (c^4*(-a^2)^(1/2))/a)) + (27*a^2*(1 - a^2*x^2)^
(1/2))/(35*(-a^2)^(1/2)*(3*c^4*x*(-a^2)^(1/2) - (c^4*(-a^2)^(1/2))/a + a^2*c^4*x^3*(-a^2)^(1/2) - 3*a*c^4*x^2*
(-a^2)^(1/2)))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x}{a^{4} x^{6} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{3} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{6} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{3} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a*c*x+c)**4,x)

[Out]

(Integral(a*x/(a**4*x**6*sqrt(-a**2*x**2 + 1) - 4*a**3*x**5*sqrt(-a**2*x**2 + 1) + 6*a**2*x**4*sqrt(-a**2*x**2
+ 1) - 4*a*x**3*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**6*sqrt(-a**2*x**2
+ 1) - 4*a**3*x**5*sqrt(-a**2*x**2 + 1) + 6*a**2*x**4*sqrt(-a**2*x**2 + 1) - 4*a*x**3*sqrt(-a**2*x**2 + 1) +
x**2*sqrt(-a**2*x**2 + 1)), x))/c**4

________________________________________________________________________________________