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

Optimal. Leaf size=169 \[ \frac {c^4 (16-35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}-\frac {c^4 (7 a x+6) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 (35 a x+24) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (35 a x+16) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 \sin ^{-1}(a x)}{a} \]

[Out]

-1/48*c^4*(35*a*x+16)*(-a^2*x^2+1)^(3/2)/a^4/x^3+1/120*c^4*(35*a*x+24)*(-a^2*x^2+1)^(5/2)/a^6/x^5-1/42*c^4*(7*
a*x+6)*(-a^2*x^2+1)^(7/2)/a^8/x^7+c^4*arcsin(a*x)/a+35/16*c^4*arctanh((-a^2*x^2+1)^(1/2))/a+1/16*c^4*(-35*a*x+
16)*(-a^2*x^2+1)^(1/2)/a^2/x

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Rubi [A]  time = 0.22, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6157, 6148, 811, 813, 844, 216, 266, 63, 208} \[ -\frac {c^4 (7 a x+6) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 (35 a x+24) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (35 a x+16) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (16-35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}+\frac {c^4 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^4*(16 - 35*a*x)*Sqrt[1 - a^2*x^2])/(16*a^2*x) - (c^4*(16 + 35*a*x)*(1 - a^2*x^2)^(3/2))/(48*a^4*x^3) + (c^4
*(24 + 35*a*x)*(1 - a^2*x^2)^(5/2))/(120*a^6*x^5) - (c^4*(6 + 7*a*x)*(1 - a^2*x^2)^(7/2))/(42*a^8*x^7) + (c^4*
ArcSin[a*x])/a + (35*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(16*a)

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 216

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]

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 811

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

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 6148

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a^2*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] || Gt
Q[c, 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]

Rule 6157

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

Rubi steps

\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=\frac {c^4 \int \frac {e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=\frac {c^4 \int \frac {(1+a x) \left (1-a^2 x^2\right )^{7/2}}{x^8} \, dx}{a^8}\\ &=-\frac {c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}-\frac {c^4 \int \frac {\left (12 a^2+14 a^3 x\right ) \left (1-a^2 x^2\right )^{5/2}}{x^6} \, dx}{12 a^8}\\ &=\frac {c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \int \frac {\left (96 a^4+140 a^5 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{96 a^8}\\ &=-\frac {c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}-\frac {c^4 \int \frac {\left (384 a^6+840 a^7 x\right ) \sqrt {1-a^2 x^2}}{x^2} \, dx}{384 a^8}\\ &=\frac {c^4 (16-35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \int \frac {-1680 a^7+768 a^8 x}{x \sqrt {1-a^2 x^2}} \, dx}{768 a^8}\\ &=\frac {c^4 (16-35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+c^4 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\frac {\left (35 c^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{16 a}\\ &=\frac {c^4 (16-35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \sin ^{-1}(a x)}{a}-\frac {\left (35 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{32 a}\\ &=\frac {c^4 (16-35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \sin ^{-1}(a x)}{a}+\frac {\left (35 c^4\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 )}{16 a^3}\\ &=\frac {c^4 (16-35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \sin ^{-1}(a x)}{a}+\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 70, normalized size = 0.41 \[ \frac {c^4 \left (-\frac {9 \, _2F_1\left (-\frac {7}{2},-\frac {7}{2};-\frac {5}{2};a^2 x^2\right )}{x^7}-7 a^7 \left (1-a^2 x^2\right )^{9/2} \, _2F_1\left (4,\frac {9}{2};\frac {11}{2};1-a^2 x^2\right )\right )}{63 a^8} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^4*((-9*Hypergeometric2F1[-7/2, -7/2, -5/2, a^2*x^2])/x^7 - 7*a^7*(1 - a^2*x^2)^(9/2)*Hypergeometric2F1[4, 9
/2, 11/2, 1 - a^2*x^2]))/(63*a^8)

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fricas [A]  time = 0.64, size = 175, normalized size = 1.04 \[ -\frac {3360 \, a^{7} c^{4} x^{7} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3675 \, a^{7} c^{4} x^{7} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 1680 \, a^{7} c^{4} x^{7} + {\left (1680 \, a^{7} c^{4} x^{7} - 2816 \, a^{6} c^{4} x^{6} + 3045 \, a^{5} c^{4} x^{5} + 1952 \, a^{4} c^{4} x^{4} - 1330 \, a^{3} c^{4} x^{3} - 1056 \, a^{2} c^{4} x^{2} + 280 \, a c^{4} x + 240 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{1680 \, a^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1680*(3360*a^7*c^4*x^7*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3675*a^7*c^4*x^7*log((sqrt(-a^2*x^2 + 1) -
1)/x) + 1680*a^7*c^4*x^7 + (1680*a^7*c^4*x^7 - 2816*a^6*c^4*x^6 + 3045*a^5*c^4*x^5 + 1952*a^4*c^4*x^4 - 1330*a
^3*c^4*x^3 - 1056*a^2*c^4*x^2 + 280*a*c^4*x + 240*c^4)*sqrt(-a^2*x^2 + 1))/(a^8*x^7)

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giac [B]  time = 0.21, size = 505, normalized size = 2.99 \[ \frac {{\left (15 \, c^{4} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} - \frac {189 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}} - \frac {525 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{6} x^{3}} + \frac {1295 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{8} x^{4}} + \frac {4935 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{10} x^{5}} - \frac {9765 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{12} x^{6}}\right )} a^{14} x^{7}}{13440 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} {\left | a \right |}} + \frac {c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {35 \, c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{16 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} + \frac {\frac {9765 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{4}}{x} - \frac {4935 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac {1295 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} + \frac {525 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{2} x^{4}} + \frac {189 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{4} x^{5}} - \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{6} x^{6}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} c^{4}}{a^{8} x^{7}}}{13440 \, a^{6} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13440*(15*c^4 + 35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/(a^2*x) - 189*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^4/(
a^4*x^2) - 525*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^4/(a^6*x^3) + 1295*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a
^8*x^4) + 4935*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^4/(a^10*x^5) - 9765*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6*c^4/(
a^12*x^6))*a^14*x^7/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^7*abs(a)) + c^4*arcsin(a*x)*sgn(a)/abs(a) + 35/16*c^4*log
(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - sqrt(-a^2*x^2 + 1)*c^4/a + 1/13440*(9765*(
sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4*c^4/x - 4935*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^4/x^2 - 1295*(sqrt(-a^
2*x^2 + 1)*abs(a) + a)^3*c^4/x^3 + 525*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a^2*x^4) + 189*(sqrt(-a^2*x^2 +
1)*abs(a) + a)^5*c^4/(a^4*x^5) - 35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6*c^4/(a^6*x^6) - 15*(sqrt(-a^2*x^2 + 1)*a
bs(a) + a)^7*c^4/(a^8*x^7))/(a^6*abs(a))

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maple [A]  time = 0.06, size = 233, normalized size = 1.38 \[ -\frac {c^{4} \sqrt {-a^{2} x^{2}+1}}{a}+\frac {c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {35 c^{4} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a}+\frac {176 c^{4} \sqrt {-a^{2} x^{2}+1}}{105 a^{2} x}-\frac {122 c^{4} \sqrt {-a^{2} x^{2}+1}}{105 a^{4} x^{3}}-\frac {29 c^{4} \sqrt {-a^{2} x^{2}+1}}{16 x^{2} a^{3}}-\frac {c^{4} \sqrt {-a^{2} x^{2}+1}}{7 a^{8} x^{7}}+\frac {22 c^{4} \sqrt {-a^{2} x^{2}+1}}{35 a^{6} x^{5}}-\frac {c^{4} \sqrt {-a^{2} x^{2}+1}}{6 a^{7} x^{6}}+\frac {19 c^{4} \sqrt {-a^{2} x^{2}+1}}{24 a^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^4*(-a^2*x^2+1)^(1/2)/a+c^4/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+35/16*c^4/a*arctanh(1/(-a^2
*x^2+1)^(1/2))+176/105*c^4*(-a^2*x^2+1)^(1/2)/a^2/x-122/105*c^4*(-a^2*x^2+1)^(1/2)/a^4/x^3-29/16*c^4*(-a^2*x^2
+1)^(1/2)/x^2/a^3-1/7*c^4/a^8/x^7*(-a^2*x^2+1)^(1/2)+22/35*c^4/a^6/x^5*(-a^2*x^2+1)^(1/2)-1/6*c^4/a^7/x^6*(-a^
2*x^2+1)^(1/2)+19/24*c^4/a^5/x^4*(-a^2*x^2+1)^(1/2)

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maxima [B]  time = 0.42, size = 515, normalized size = 3.05 \[ \frac {c^{4} \arcsin \left (a x\right )}{a} + \frac {4 \, c^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} - \frac {3 \, {\left (a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{2}}\right )} c^{4}}{a^{3}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{a^{2} x} - \frac {2 \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x} + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{3}}\right )} c^{4}}{a^{4}} + \frac {{\left (3 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{x^{4}}\right )} c^{4}}{2 \, a^{5}} + \frac {4 \, {\left (\frac {8 \, \sqrt {-a^{2} x^{2} + 1} a^{4}}{x} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x^{3}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{x^{5}}\right )} c^{4}}{15 \, a^{6}} - \frac {{\left (15 \, a^{6} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {15 \, \sqrt {-a^{2} x^{2} + 1} a^{4}}{x^{2}} + \frac {10 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{x^{6}}\right )} c^{4}}{48 \, a^{7}} - \frac {{\left (\frac {16 \, \sqrt {-a^{2} x^{2} + 1} a^{6}}{x} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1} a^{4}}{x^{3}} + \frac {6 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x^{5}} + \frac {5 \, \sqrt {-a^{2} x^{2} + 1}}{x^{7}}\right )} c^{4}}{35 \, a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*arcsin(a*x)/a + 4*c^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x))/a - sqrt(-a^2*x^2 + 1)*c^4/a - 3*(a^2*lo
g(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-a^2*x^2 + 1)/x^2)*c^4/a^3 + 4*sqrt(-a^2*x^2 + 1)*c^4/(a^2*x)
- 2*(2*sqrt(-a^2*x^2 + 1)*a^2/x + sqrt(-a^2*x^2 + 1)/x^3)*c^4/a^4 + 1/2*(3*a^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x
) + 2/abs(x)) + 3*sqrt(-a^2*x^2 + 1)*a^2/x^2 + 2*sqrt(-a^2*x^2 + 1)/x^4)*c^4/a^5 + 4/15*(8*sqrt(-a^2*x^2 + 1)*
a^4/x + 4*sqrt(-a^2*x^2 + 1)*a^2/x^3 + 3*sqrt(-a^2*x^2 + 1)/x^5)*c^4/a^6 - 1/48*(15*a^6*log(2*sqrt(-a^2*x^2 +
1)/abs(x) + 2/abs(x)) + 15*sqrt(-a^2*x^2 + 1)*a^4/x^2 + 10*sqrt(-a^2*x^2 + 1)*a^2/x^4 + 8*sqrt(-a^2*x^2 + 1)/x
^6)*c^4/a^7 - 1/35*(16*sqrt(-a^2*x^2 + 1)*a^6/x + 8*sqrt(-a^2*x^2 + 1)*a^4/x^3 + 6*sqrt(-a^2*x^2 + 1)*a^2/x^5
+ 5*sqrt(-a^2*x^2 + 1)/x^7)*c^4/a^8

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mupad [B]  time = 0.88, size = 228, normalized size = 1.35 \[ \frac {c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{a}+\frac {176\,c^4\,\sqrt {1-a^2\,x^2}}{105\,a^2\,x}-\frac {29\,c^4\,\sqrt {1-a^2\,x^2}}{16\,a^3\,x^2}-\frac {122\,c^4\,\sqrt {1-a^2\,x^2}}{105\,a^4\,x^3}+\frac {19\,c^4\,\sqrt {1-a^2\,x^2}}{24\,a^5\,x^4}+\frac {22\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^6\,x^5}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{6\,a^7\,x^6}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{7\,a^8\,x^7}-\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,35{}\mathrm {i}}{16\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^4*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) - (c^4*atan((1 - a^2*x^2)^(1/2)*1i)*35i)/(16*a) - (c^4*(1 - a^2*x^2)^
(1/2))/a + (176*c^4*(1 - a^2*x^2)^(1/2))/(105*a^2*x) - (29*c^4*(1 - a^2*x^2)^(1/2))/(16*a^3*x^2) - (122*c^4*(1
- a^2*x^2)^(1/2))/(105*a^4*x^3) + (19*c^4*(1 - a^2*x^2)^(1/2))/(24*a^5*x^4) + (22*c^4*(1 - a^2*x^2)^(1/2))/(3
5*a^6*x^5) - (c^4*(1 - a^2*x^2)^(1/2))/(6*a^7*x^6) - (c^4*(1 - a^2*x^2)^(1/2))/(7*a^8*x^7)

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sympy [A]  time = 26.21, size = 1119, normalized size = 6.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c**4*Piecewise((sqrt(a**(-2))*as
in(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - 4*c**4*Piecewise((-acosh(1/(a*x
)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a - 4*c**4*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x
**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 + 6*c**4*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a
**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*
x**3*sqrt(1 - 1/(a**2*x**2))), True))/a**3 + 6*c**4*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**4 - 4*c**4*Piecewise((-3*a**4*acosh(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*sqrt(-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**5 - 4*c**4*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))/a**6 + c
**4*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*sqrt(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))/a**7 + c**
4*Piecewise((-16*a**7*sqrt(-1 + 1/(a**2*x**2))/35 - 8*a**5*sqrt(-1 + 1/(a**2*x**2))/(35*x**2) - 6*a**3*sqrt(-1
+ 1/(a**2*x**2))/(35*x**4) - a*sqrt(-1 + 1/(a**2*x**2))/(7*x**6), 1/Abs(a**2*x**2) > 1), (-16*I*a**7*sqrt(1 -
1/(a**2*x**2))/35 - 8*I*a**5*sqrt(1 - 1/(a**2*x**2))/(35*x**2) - 6*I*a**3*sqrt(1 - 1/(a**2*x**2))/(35*x**4) -
I*a*sqrt(1 - 1/(a**2*x**2))/(7*x**6), True))/a**8

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