3.645 \(\int e^{3 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^4 \, dx\)

Optimal. Leaf size=191 \[ -\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {15 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {c^4 (5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}+\frac {c^4 (5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {3 c^4 \sin ^{-1}(a x)}{a} \]

[Out]

1/16*c^4*(5*a*x+16)*(-a^2*x^2+1)^(3/2)/a^4/x^3-1/40*c^4*(5*a*x+24)*(-a^2*x^2+1)^(5/2)/a^6/x^5-1/7*c^4*(-a^2*x^
2+1)^(7/2)/a^8/x^7-1/2*c^4*(-a^2*x^2+1)^(7/2)/a^7/x^6-3*c^4*arcsin(a*x)/a-15/16*c^4*arctanh((-a^2*x^2+1)^(1/2)
)/a-3/16*c^4*(-5*a*x+16)*(-a^2*x^2+1)^(1/2)/a^2/x

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Rubi [A]  time = 0.37, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {6157, 6148, 1807, 811, 813, 844, 216, 266, 63, 208} \[ -\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 (5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}+\frac {c^4 (5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {15 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}-\frac {3 c^4 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^4,x]

[Out]

(-3*c^4*(16 - 5*a*x)*Sqrt[1 - a^2*x^2])/(16*a^2*x) + (c^4*(16 + 5*a*x)*(1 - a^2*x^2)^(3/2))/(16*a^4*x^3) - (c^
4*(24 + 5*a*x)*(1 - a^2*x^2)^(5/2))/(40*a^6*x^5) - (c^4*(1 - a^2*x^2)^(7/2))/(7*a^8*x^7) - (c^4*(1 - a^2*x^2)^
(7/2))/(2*a^7*x^6) - (3*c^4*ArcSin[a*x])/a - (15*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 1807

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

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^{3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=\frac {c^4 \int \frac {e^{3 \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)^3 \left (1-a^2 x^2\right )^{5/2}}{x^8} \, dx}{a^8}\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \int \frac {\left (1-a^2 x^2\right )^{5/2} \left (-21 a-21 a^2 x-7 a^3 x^2\right )}{x^7} \, dx}{7 a^8}\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}+\frac {c^4 \int \frac {\left (126 a^2+21 a^3 x\right ) \left (1-a^2 x^2\right )^{5/2}}{x^6} \, dx}{42 a^8}\\ &=-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {c^4 \int \frac {\left (1008 a^4+210 a^5 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{336 a^8}\\ &=\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}+\frac {c^4 \int \frac {\left (4032 a^6+1260 a^7 x\right ) \sqrt {1-a^2 x^2}}{x^2} \, dx}{1344 a^8}\\ &=-\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {c^4 \int \frac {-2520 a^7+8064 a^8 x}{x \sqrt {1-a^2 x^2}} \, dx}{2688 a^8}\\ &=-\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\left (3 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx+\frac {\left (15 c^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{16 a}\\ &=-\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {3 c^4 \sin ^{-1}(a x)}{a}+\frac {\left (15 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{32 a}\\ &=-\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {3 c^4 \sin ^{-1}(a x)}{a}-\frac {\left (15 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 {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {3 c^4 \sin ^{-1}(a x)}{a}-\frac {15 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}\\ \end {align*}

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Mathematica [C]  time = 0.21, size = 191, normalized size = 1.00 \[ \frac {c^4 \left (-336 a^2 x^2 \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};a^2 x^2\right )-\frac {5 \left (16 a^8 x^8-231 a^7 x^7-64 a^6 x^6+413 a^5 x^5+96 a^4 x^4-238 a^3 x^3-64 a^2 x^2+16 a^7 x^7 \left (a^2 x^2-1\right )^4 \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};1-a^2 x^2\right )-105 a^7 x^7 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+56 a x+16\right )}{\sqrt {1-a^2 x^2}}\right )}{560 a^8 x^7} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^4,x]

[Out]

(c^4*(-336*a^2*x^2*Hypergeometric2F1[-5/2, -5/2, -3/2, a^2*x^2] - (5*(16 + 56*a*x - 64*a^2*x^2 - 238*a^3*x^3 +
96*a^4*x^4 + 413*a^5*x^5 - 64*a^6*x^6 - 231*a^7*x^7 + 16*a^8*x^8 - 105*a^7*x^7*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt
[1 - a^2*x^2]] + 16*a^7*x^7*(-1 + a^2*x^2)^4*Hypergeometric2F1[3, 7/2, 9/2, 1 - a^2*x^2]))/Sqrt[1 - a^2*x^2]))
/(560*a^8*x^7)

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fricas [A]  time = 1.31, size = 175, normalized size = 0.92 \[ \frac {3360 \, a^{7} c^{4} x^{7} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 525 \, a^{7} c^{4} x^{7} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 560 \, a^{7} c^{4} x^{7} + {\left (560 \, a^{7} c^{4} x^{7} - 2496 \, a^{6} c^{4} x^{6} - 525 \, a^{5} c^{4} x^{5} + 992 \, a^{4} c^{4} x^{4} + 770 \, a^{3} c^{4} x^{3} - 96 \, a^{2} c^{4} x^{2} - 280 \, a c^{4} x - 80 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{560 \, a^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="fricas")

[Out]

1/560*(3360*a^7*c^4*x^7*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 525*a^7*c^4*x^7*log((sqrt(-a^2*x^2 + 1) - 1)/
x) + 560*a^7*c^4*x^7 + (560*a^7*c^4*x^7 - 2496*a^6*c^4*x^6 - 525*a^5*c^4*x^5 + 992*a^4*c^4*x^4 + 770*a^3*c^4*x
^3 - 96*a^2*c^4*x^2 - 280*a*c^4*x - 80*c^4)*sqrt(-a^2*x^2 + 1))/(a^8*x^7)

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giac [B]  time = 0.24, size = 505, normalized size = 2.64 \[ \frac {{\left (5 \, c^{4} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} + \frac {49 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}} - \frac {245 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{6} x^{3}} - \frac {875 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{8} x^{4}} + \frac {455 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{10} x^{5}} + \frac {9065 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{12} x^{6}}\right )} a^{14} x^{7}}{4480 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} {\left | a \right |}} - \frac {3 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {15 \, c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{16 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} - \frac {\frac {9065 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{4}}{x} + \frac {455 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac {875 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} - \frac {245 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{2} x^{4}} + \frac {49 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{4} x^{5}} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{6} x^{6}} + \frac {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} c^{4}}{a^{8} x^{7}}}{4480 \, a^{6} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="giac")

[Out]

1/4480*(5*c^4 + 35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/(a^2*x) + 49*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^4/(a^4
*x^2) - 245*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^4/(a^6*x^3) - 875*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a^8*x
^4) + 455*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^4/(a^10*x^5) + 9065*(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)) - 3*c^4*arcsin(a*x)*sgn(a)/abs(a) - 15/16*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 - 1/4480*(9065*(sqrt
(-a^2*x^2 + 1)*abs(a) + a)*a^4*c^4/x + 455*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^4/x^2 - 875*(sqrt(-a^2*x^2
+ 1)*abs(a) + a)^3*c^4/x^3 - 245*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a^2*x^4) + 49*(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) + 5*(sqrt(-a^2*x^2 + 1)*abs(a) +
a)^7*c^4/(a^8*x^7))/(a^6*abs(a))

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maple [A]  time = 0.09, size = 273, normalized size = 1.43 \[ -\frac {218 c^{4}}{35 a^{2} x \sqrt {-a^{2} x^{2}+1}}+\frac {15 c^{4}}{8 a^{5} x^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {c^{4}}{7 a^{8} x^{7} \sqrt {-a^{2} x^{2}+1}}-\frac {c^{4}}{35 a^{6} x^{5} \sqrt {-a^{2} x^{2}+1}}+\frac {68 c^{4}}{35 a^{4} x^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {37 c^{4}}{16 a^{3} x^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {c^{4}}{2 a^{7} x^{6} \sqrt {-a^{2} x^{2}+1}}+\frac {156 c^{4} x}{35 \sqrt {-a^{2} x^{2}+1}}-\frac {c^{4} a \,x^{2}}{\sqrt {-a^{2} x^{2}+1}}-\frac {3 c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {31 c^{4}}{16 a \sqrt {-a^{2} x^{2}+1}}-\frac {15 c^{4} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^4,x)

[Out]

-218/35*c^4/a^2/x/(-a^2*x^2+1)^(1/2)+15/8*c^4/a^5/x^4/(-a^2*x^2+1)^(1/2)-1/7*c^4/a^8/x^7/(-a^2*x^2+1)^(1/2)-1/
35*c^4/a^6/x^5/(-a^2*x^2+1)^(1/2)+68/35*c^4/a^4/x^3/(-a^2*x^2+1)^(1/2)-37/16*c^4/a^3/x^2/(-a^2*x^2+1)^(1/2)-1/
2*c^4/a^7/x^6/(-a^2*x^2+1)^(1/2)+156/35*c^4*x/(-a^2*x^2+1)^(1/2)-c^4*a*x^2/(-a^2*x^2+1)^(1/2)-3*c^4/(a^2)^(1/2
)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+31/16*c^4/a/(-a^2*x^2+1)^(1/2)-15/16*c^4/a*arctanh(1/(-a^2*x^2+1)^(
1/2))

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maxima [B]  time = 0.42, size = 745, normalized size = 3.90 \[ -a^{3} c^{4} {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} + 3 \, a^{2} c^{4} {\left (\frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} - \frac {11 \, c^{4} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {6 \, c^{4} {\left (\frac {1}{\sqrt {-a^{2} x^{2} + 1}} - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )\right )}}{a} + \frac {14 \, {\left (\frac {2 \, a^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x}\right )} c^{4}}{a^{2}} - \frac {c^{4}}{\sqrt {-a^{2} x^{2} + 1} a} - \frac {7 \, {\left (3 \, a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {3 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\right )} c^{4}}{a^{3}} - \frac {2 \, {\left (\frac {8 \, a^{4} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {4 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\right )} c^{4}}{a^{4}} + \frac {11 \, {\left (15 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {15 \, a^{4}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {5 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} + \frac {2}{\sqrt {-a^{2} x^{2} + 1} x^{4}}\right )} c^{4}}{8 \, a^{5}} - \frac {{\left (\frac {16 \, a^{6} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {8 \, a^{4}}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {2 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{5}}\right )} c^{4}}{5 \, a^{6}} - \frac {{\left (105 \, a^{6} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {105 \, a^{6}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {35 \, a^{4}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} + \frac {14 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{4}} + \frac {8}{\sqrt {-a^{2} x^{2} + 1} x^{6}}\right )} c^{4}}{16 \, a^{7}} + \frac {{\left (\frac {128 \, a^{8} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {64 \, a^{6}}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {16 \, a^{4}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} - \frac {8 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} 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)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="maxima")

[Out]

-a^3*c^4*(x^2/(sqrt(-a^2*x^2 + 1)*a^2) - 2/(sqrt(-a^2*x^2 + 1)*a^4)) + 3*a^2*c^4*(x/(sqrt(-a^2*x^2 + 1)*a^2) -
arcsin(a*x)/a^3) - 11*c^4*x/sqrt(-a^2*x^2 + 1) - 6*c^4*(1/sqrt(-a^2*x^2 + 1) - log(2*sqrt(-a^2*x^2 + 1)/abs(x
) + 2/abs(x)))/a + 14*(2*a^2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*x))*c^4/a^2 - c^4/(sqrt(-a^2*x^2 + 1
)*a) - 7*(3*a^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 3*a^2/sqrt(-a^2*x^2 + 1) + 1/(sqrt(-a^2*x^2 + 1)
*x^2))*c^4/a^3 - 2*(8*a^4*x/sqrt(-a^2*x^2 + 1) - 4*a^2/(sqrt(-a^2*x^2 + 1)*x) - 1/(sqrt(-a^2*x^2 + 1)*x^3))*c^
4/a^4 + 11/8*(15*a^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 15*a^4/sqrt(-a^2*x^2 + 1) + 5*a^2/(sqrt(-a^
2*x^2 + 1)*x^2) + 2/(sqrt(-a^2*x^2 + 1)*x^4))*c^4/a^5 - 1/5*(16*a^6*x/sqrt(-a^2*x^2 + 1) - 8*a^4/(sqrt(-a^2*x^
2 + 1)*x) - 2*a^2/(sqrt(-a^2*x^2 + 1)*x^3) - 1/(sqrt(-a^2*x^2 + 1)*x^5))*c^4/a^6 - 1/16*(105*a^6*log(2*sqrt(-a
^2*x^2 + 1)/abs(x) + 2/abs(x)) - 105*a^6/sqrt(-a^2*x^2 + 1) + 35*a^4/(sqrt(-a^2*x^2 + 1)*x^2) + 14*a^2/(sqrt(-
a^2*x^2 + 1)*x^4) + 8/(sqrt(-a^2*x^2 + 1)*x^6))*c^4/a^7 + 1/35*(128*a^8*x/sqrt(-a^2*x^2 + 1) - 64*a^6/(sqrt(-a
^2*x^2 + 1)*x) - 16*a^4/(sqrt(-a^2*x^2 + 1)*x^3) - 8*a^2/(sqrt(-a^2*x^2 + 1)*x^5) - 5/(sqrt(-a^2*x^2 + 1)*x^7)
)*c^4/a^8

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mupad [B]  time = 0.89, size = 228, normalized size = 1.19 \[ \frac {c^4\,\sqrt {1-a^2\,x^2}}{a}-\frac {3\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {156\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^2\,x}-\frac {15\,c^4\,\sqrt {1-a^2\,x^2}}{16\,a^3\,x^2}+\frac {62\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^4\,x^3}+\frac {11\,c^4\,\sqrt {1-a^2\,x^2}}{8\,a^5\,x^4}-\frac {6\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^6\,x^5}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{2\,a^7\,x^6}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{7\,a^8\,x^7}+\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,15{}\mathrm {i}}{16\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - c/(a^2*x^2))^4*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)

[Out]

(c^4*atan((1 - a^2*x^2)^(1/2)*1i)*15i)/(16*a) - (3*c^4*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) + (c^4*(1 - a^2*x^2
)^(1/2))/a - (156*c^4*(1 - a^2*x^2)^(1/2))/(35*a^2*x) - (15*c^4*(1 - a^2*x^2)^(1/2))/(16*a^3*x^2) + (62*c^4*(1
- a^2*x^2)^(1/2))/(35*a^4*x^3) + (11*c^4*(1 - a^2*x^2)^(1/2))/(8*a^5*x^4) - (6*c^4*(1 - a^2*x^2)^(1/2))/(35*a
^6*x^5) - (c^4*(1 - a^2*x^2)^(1/2))/(2*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 = 30.60, size = 935, normalized size = 4.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(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)) - 3*c**4*Piecewise((sqrt(a**(-2))
*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 8*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) - sq
rt(-a**2*x**2 + 1)/(3*x**3), True))/a**4 - 8*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 + 3*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/(1
6*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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