3.305 \(\int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^2}{x^7} \, dx\)

Optimal. Leaf size=156 \[ -\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac {a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac {1}{16} a^6 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {a^4 c^2 \sqrt {1-a^2 x^2}}{16 x^2}+\frac {2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3} \]

[Out]

-1/6*c^2*(-a^2*x^2+1)^(3/2)/x^6+1/5*a*c^2*(-a^2*x^2+1)^(3/2)/x^5-1/8*a^2*c^2*(-a^2*x^2+1)^(3/2)/x^4+2/15*a^3*c
^2*(-a^2*x^2+1)^(3/2)/x^3+1/16*a^6*c^2*arctanh((-a^2*x^2+1)^(1/2))-1/16*a^4*c^2*(-a^2*x^2+1)^(1/2)/x^2

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Rubi [A]  time = 0.17, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6128, 835, 807, 266, 47, 63, 208} \[ -\frac {a^4 c^2 \sqrt {1-a^2 x^2}}{16 x^2}+\frac {2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac {1}{16} a^6 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^7,x]

[Out]

-(a^4*c^2*Sqrt[1 - a^2*x^2])/(16*x^2) - (c^2*(1 - a^2*x^2)^(3/2))/(6*x^6) + (a*c^2*(1 - a^2*x^2)^(3/2))/(5*x^5
) - (a^2*c^2*(1 - a^2*x^2)^(3/2))/(8*x^4) + (2*a^3*c^2*(1 - a^2*x^2)^(3/2))/(15*x^3) + (a^6*c^2*ArcTanh[Sqrt[1
- a^2*x^2]])/16

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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 835

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))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

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)} (c-a c x)^2}{x^7} \, dx &=c \int \frac {(c-a c x) \sqrt {1-a^2 x^2}}{x^7} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}-\frac {1}{6} c \int \frac {\left (6 a c-3 a^2 c x\right ) \sqrt {1-a^2 x^2}}{x^6} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac {1}{30} c \int \frac {\left (15 a^2 c-12 a^3 c x\right ) \sqrt {1-a^2 x^2}}{x^5} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac {a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}-\frac {1}{120} c \int \frac {\left (48 a^3 c-15 a^4 c x\right ) \sqrt {1-a^2 x^2}}{x^4} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac {a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac {2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac {1}{8} \left (a^4 c^2\right ) \int \frac {\sqrt {1-a^2 x^2}}{x^3} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac {a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac {2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac {1}{16} \left (a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {a^4 c^2 \sqrt {1-a^2 x^2}}{16 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac {a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac {2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac {1}{32} \left (a^6 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {a^4 c^2 \sqrt {1-a^2 x^2}}{16 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac {a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac {2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac {1}{16} \left (a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {a^4 c^2 \sqrt {1-a^2 x^2}}{16 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac {a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac {2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac {1}{16} a^6 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 115, normalized size = 0.74 \[ \frac {c^2 \left (32 a^7 x^7-15 a^6 x^6-16 a^5 x^5+5 a^4 x^4-64 a^3 x^3+50 a^2 x^2+15 a^6 x^6 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+48 a x-40\right )}{240 x^6 \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^7,x]

[Out]

(c^2*(-40 + 48*a*x + 50*a^2*x^2 - 64*a^3*x^3 + 5*a^4*x^4 - 16*a^5*x^5 - 15*a^6*x^6 + 32*a^7*x^7 + 15*a^6*x^6*S
qrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(240*x^6*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 0.47, size = 106, normalized size = 0.68 \[ -\frac {15 \, a^{6} c^{2} x^{6} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (32 \, a^{5} c^{2} x^{5} - 15 \, a^{4} c^{2} x^{4} + 16 \, a^{3} c^{2} x^{3} - 10 \, a^{2} c^{2} x^{2} - 48 \, a c^{2} x + 40 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{240 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^7,x, algorithm="fricas")

[Out]

-1/240*(15*a^6*c^2*x^6*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (32*a^5*c^2*x^5 - 15*a^4*c^2*x^4 + 16*a^3*c^2*x^3 - 1
0*a^2*c^2*x^2 - 48*a*c^2*x + 40*c^2)*sqrt(-a^2*x^2 + 1))/x^6

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giac [B]  time = 0.65, size = 424, normalized size = 2.72 \[ \frac {{\left (5 \, a^{7} c^{2} - \frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} c^{2}}{x} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} c^{2}}{x^{2}} - \frac {20 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a c^{2}}{x^{3}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{2}}{a x^{4}} + \frac {120 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{2}}{a^{3} x^{5}}\right )} a^{12} x^{6}}{1920 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} {\left | a \right |}} + \frac {a^{7} c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{16 \, {\left | a \right |}} - \frac {\frac {120 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{9} c^{2} {\left | a \right |}}{x} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{7} c^{2} {\left | a \right |}}{x^{2}} - \frac {20 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a^{5} c^{2} {\left | a \right |}}{x^{3}} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} a^{3} c^{2} {\left | a \right |}}{x^{4}} - \frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} a c^{2} {\left | a \right |}}{x^{5}} + \frac {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{2} {\left | a \right |}}{a x^{6}}}{1920 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^7,x, algorithm="giac")

[Out]

1/1920*(5*a^7*c^2 - 12*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^5*c^2/x + 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^3*c^
2/x^2 - 20*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*a*c^2/x^3 - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^2/(a*x^4) + 12
0*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^2/(a^3*x^5))*a^12*x^6/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^6*abs(a)) + 1/16*
a^7*c^2*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/1920*(120*(sqrt(-a^2*x^2 + 1)
*abs(a) + a)*a^9*c^2*abs(a)/x - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^7*c^2*abs(a)/x^2 - 20*(sqrt(-a^2*x^2 +
1)*abs(a) + a)^3*a^5*c^2*abs(a)/x^3 + 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*a^3*c^2*abs(a)/x^4 - 12*(sqrt(-a^2*
x^2 + 1)*abs(a) + a)^5*a*c^2*abs(a)/x^5 + 5*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6*c^2*abs(a)/(a*x^6))/a^6

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maple [A]  time = 0.05, size = 193, normalized size = 1.24 \[ c^{2} \left (a^{3} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )-\frac {a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}+\frac {3 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4}\right )}{6}-\frac {\sqrt {-a^{2} x^{2}+1}}{6 x^{6}}-a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{5 x^{5}}+\frac {4 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )}{5}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^7,x)

[Out]

c^2*(a^3*(-1/3*(-a^2*x^2+1)^(1/2)/x^3-2/3*a^2*(-a^2*x^2+1)^(1/2)/x)-1/6*a^2*(-1/4*(-a^2*x^2+1)^(1/2)/x^4+3/4*a
^2*(-1/2*(-a^2*x^2+1)^(1/2)/x^2-1/2*a^2*arctanh(1/(-a^2*x^2+1)^(1/2))))-1/6/x^6*(-a^2*x^2+1)^(1/2)-a*(-1/5/x^5
*(-a^2*x^2+1)^(1/2)+4/5*a^2*(-1/3*(-a^2*x^2+1)^(1/2)/x^3-2/3*a^2*(-a^2*x^2+1)^(1/2)/x)))

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maxima [A]  time = 0.40, size = 168, normalized size = 1.08 \[ \frac {1}{16} \, a^{6} c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{5} c^{2}}{15 \, x} + \frac {\sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}}{16 \, x^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} a^{3} c^{2}}{15 \, x^{3}} + \frac {\sqrt {-a^{2} x^{2} + 1} a^{2} c^{2}}{24 \, x^{4}} + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{2}}{5 \, x^{5}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^7,x, algorithm="maxima")

[Out]

1/16*a^6*c^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 2/15*sqrt(-a^2*x^2 + 1)*a^5*c^2/x + 1/16*sqrt(-a^2*
x^2 + 1)*a^4*c^2/x^2 - 1/15*sqrt(-a^2*x^2 + 1)*a^3*c^2/x^3 + 1/24*sqrt(-a^2*x^2 + 1)*a^2*c^2/x^4 + 1/5*sqrt(-a
^2*x^2 + 1)*a*c^2/x^5 - 1/6*sqrt(-a^2*x^2 + 1)*c^2/x^6

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mupad [B]  time = 0.05, size = 159, normalized size = 1.02 \[ \frac {a\,c^2\,\sqrt {1-a^2\,x^2}}{5\,x^5}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{6\,x^6}+\frac {a^2\,c^2\,\sqrt {1-a^2\,x^2}}{24\,x^4}-\frac {a^3\,c^2\,\sqrt {1-a^2\,x^2}}{15\,x^3}+\frac {a^4\,c^2\,\sqrt {1-a^2\,x^2}}{16\,x^2}-\frac {2\,a^5\,c^2\,\sqrt {1-a^2\,x^2}}{15\,x}-\frac {a^6\,c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a*c*x)^2*(a*x + 1))/(x^7*(1 - a^2*x^2)^(1/2)),x)

[Out]

(a*c^2*(1 - a^2*x^2)^(1/2))/(5*x^5) - (c^2*(1 - a^2*x^2)^(1/2))/(6*x^6) - (a^6*c^2*atan((1 - a^2*x^2)^(1/2)*1i
)*1i)/16 + (a^2*c^2*(1 - a^2*x^2)^(1/2))/(24*x^4) - (a^3*c^2*(1 - a^2*x^2)^(1/2))/(15*x^3) + (a^4*c^2*(1 - a^2
*x^2)^(1/2))/(16*x^2) - (2*a^5*c^2*(1 - a^2*x^2)^(1/2))/(15*x)

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sympy [C]  time = 11.61, size = 644, normalized size = 4.13 \[ a^{3} c^{2} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \frac {8 a^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {8 i a^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} - \frac {5 a^{6} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{16} + \frac {5 a^{5}}{16 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {5 a^{3}}{48 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{24 x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{6 a x^{7} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {5 i a^{6} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{16} - \frac {5 i a^{5}}{16 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {5 i a^{3}}{48 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{24 x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{6 a x^{7} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2/x**7,x)

[Out]

a**3*c**2*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1)
, (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) - a**2*c**2*Piecewise((-3*a**4*a
cosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sq
rt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**
2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) - a*c**2*Piecewise(
(-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/
(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*
x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True)) + c**2*Piecewise((-5*a**6*acosh(1/(a*x))/16 + 5*a**5/(16*
x*sqrt(-1 + 1/(a**2*x**2))) - 5*a**3/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) - a/(24*x**5*sqrt(-1 + 1/(a**2*x**2)))
- 1/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (5*I*a**6*asin(1/(a*x))/16 - 5*I*a**5/(16*x*s
qrt(1 - 1/(a**2*x**2))) + 5*I*a**3/(48*x**3*sqrt(1 - 1/(a**2*x**2))) + I*a/(24*x**5*sqrt(1 - 1/(a**2*x**2))) +
I/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True))

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