3.7.96 \(\int e^{2 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^{7/2} \, dx\) [696]

Optimal. Leaf size=372 \[ -\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}+\frac {57 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}-\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}-\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}+\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {2 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \text {ArcSin}(a x)}{(1-a x)^{7/2} (1+a x)^{7/2}}-\frac {25 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}} \]

[Out]

-11/30*a^3*(c-c/a^2/x^2)^(7/2)*x^4/(-a*x+1)^3+57/16*a^6*(c-c/a^2/x^2)^(7/2)*x^7/(-a*x+1)^3/(a*x+1)^3-41/24*a^5
*(c-c/a^2/x^2)^(7/2)*x^6/(-a*x+1)^3/(a*x+1)^2-57/80*a^4*(c-c/a^2/x^2)^(7/2)*x^5/(-a*x+1)^3/(a*x+1)+13/40*a^2*(
c-c/a^2/x^2)^(7/2)*x^3*(a*x+1)/(-a*x+1)^3-1/15*a*(c-c/a^2/x^2)^(7/2)*x^2*(a*x+1)/(-a*x+1)^2-1/6*(c-c/a^2/x^2)^
(7/2)*x*(a*x+1)/(-a*x+1)-2*a^6*(c-c/a^2/x^2)^(7/2)*x^7*arcsin(a*x)/(-a*x+1)^(7/2)/(a*x+1)^(7/2)-25/16*a^6*(c-c
/a^2/x^2)^(7/2)*x^7*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))/(-a*x+1)^(7/2)/(a*x+1)^(7/2)

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Rubi [A]
time = 0.31, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6294, 6264, 99, 154, 159, 163, 41, 222, 94, 214} \begin {gather*} -\frac {a x^2 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{15 (1-a x)^2}-\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{6 (1-a x)}+\frac {13 a^2 x^3 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{40 (1-a x)^3}-\frac {2 a^6 x^7 \text {ArcSin}(a x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{(1-a x)^{7/2} (a x+1)^{7/2}}+\frac {57 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)^3}-\frac {25 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{16 (1-a x)^{7/2} (a x+1)^{7/2}}-\frac {41 a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{24 (1-a x)^3 (a x+1)^2}-\frac {57 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{80 (1-a x)^3 (a x+1)}-\frac {11 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{30 (1-a x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(30*(1 - a*x)^3) + (57*a^6*(c - c/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^3*
(1 + a*x)^3) - (41*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(24*(1 - a*x)^3*(1 + a*x)^2) - (57*a^4*(c - c/(a^2*x^2))^(
7/2)*x^5)/(80*(1 - a*x)^3*(1 + a*x)) + (13*a^2*(c - c/(a^2*x^2))^(7/2)*x^3*(1 + a*x))/(40*(1 - a*x)^3) - (a*(c
- c/(a^2*x^2))^(7/2)*x^2*(1 + a*x))/(15*(1 - a*x)^2) - ((c - c/(a^2*x^2))^(7/2)*x*(1 + a*x))/(6*(1 - a*x)) -
(2*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*ArcSin[a*x])/((1 - a*x)^(7/2)*(1 + a*x)^(7/2)) - (25*a^6*(c - c/(a^2*x^2))^
(7/2)*x^7*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(16*(1 - a*x)^(7/2)*(1 + a*x)^(7/2))

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 99

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

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 214

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]

Rule 222

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 6264

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

Rule 6294

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

Rubi steps

\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^{7/2} (1+a x)^{7/2}}{x^7} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{5/2} (1+a x)^{9/2}}{x^7} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{3/2} (1+a x)^{7/2} \left (2 a-7 a^2 x\right )}{x^6} \, dx}{6 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {\sqrt {1-a x} (1+a x)^{7/2} \left (-39 a^2+33 a^3 x\right )}{x^5} \, dx}{30 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1+a x)^{7/2} \left (132 a^3-93 a^4 x\right )}{x^4 \sqrt {1-a x}} \, dx}{120 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}+\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1+a x)^{5/2} \left (513 a^4-411 a^5 x\right )}{x^3 \sqrt {1-a x}} \, dx}{360 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}+\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1+a x)^{3/2} \left (1230 a^5-1335 a^6 x\right )}{x^2 \sqrt {1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}-\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}+\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {\sqrt {1+a x} \left (1125 a^6-2565 a^7 x\right )}{x \sqrt {1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}+\frac {57 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}-\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}-\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}+\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {-1125 a^7+1440 a^8 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{720 a (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}+\frac {57 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}-\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}-\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}+\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}+\frac {\left (25 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{16 (1-a x)^{7/2} (1+a x)^{7/2}}-\frac {\left (2 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}+\frac {57 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}-\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}-\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}+\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {\left (25 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \text {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}}-\frac {\left (2 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}+\frac {57 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}-\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}-\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}+\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {2 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \sin ^{-1}(a x)}{(1-a x)^{7/2} (1+a x)^{7/2}}-\frac {25 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 150, normalized size = 0.40 \begin {gather*} -\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (-40-96 a x+70 a^2 x^2+352 a^3 x^3+105 a^4 x^4-736 a^5 x^5+240 a^6 x^6\right )+375 a^6 x^6 \text {ArcTan}\left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )+480 a^6 x^6 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{240 a^6 x^5 \sqrt {-1+a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/240*(c^3*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-40 - 96*a*x + 70*a^2*x^2 + 352*a^3*x^3 + 105*a^4*x^4 -
736*a^5*x^5 + 240*a^6*x^6) + 375*a^6*x^6*ArcTan[1/Sqrt[-1 + a^2*x^2]] + 480*a^6*x^6*Log[a*x + Sqrt[-1 + a^2*x
^2]]))/(a^6*x^5*Sqrt[-1 + a^2*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(794\) vs. \(2(326)=652\).
time = 0.94, size = 795, normalized size = 2.14

method	result	size
risch	\(\frac {\left (736 a^{7} x^{7}-105 a^{6} x^{6}-1088 a^{5} x^{5}+35 a^{4} x^{4}+448 a^{3} x^{3}+110 a^{2} x^{2}-96 a x -40\right ) c^{3} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{240 x^{5} a^{6} \left (a^{2} x^{2}-1\right )}-\frac {\left (\frac {a^{6} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c}+\frac {2 a^{7} \ln \left (\frac {x \,a^{2} c}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}+\frac {25 a^{6} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{16 \sqrt {-c}}\right ) c^{3} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {c \left (a^{2} x^{2}-1\right )}}{a^{6} \left (a^{2} x^{2}-1\right )}\)	\(251\)
default	\(-\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} x \left (2016 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} a^{9} c \,x^{7}-2016 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} a^{9} x^{5}-375 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} a^{8} c \,x^{6}+480 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} a^{8} c \,x^{6}-105 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} a^{8} x^{4}-2352 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{7} c^{2} x^{7}+560 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} a^{7} c^{2} x^{7}-224 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} a^{7} x^{3}+525 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{6} c^{2} x^{6}+2940 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{5} c^{3} x^{7}-700 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} a^{5} c^{3} x^{7}-630 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} a^{6} x^{2}-875 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{4} c^{3} x^{6}-672 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} a^{5} x -4410 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c^{4} x^{7}+1050 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, a^{3} c^{4} x^{7}+4410 c^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{6}-1050 c^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) a \,x^{6}-280 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} a^{4} \sqrt {-\frac {c}{a^{2}}}+2625 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c^{4} x^{6}+2625 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{5} x^{6}\right )}{1680 a^{2} \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} c}\)	\(795\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1680*(c*(a^2*x^2-1)/a^2/x^2)^(7/2)*x/a^2*(2016*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(7/2)*a^9*c*x^7-2016*(-c/
a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(9/2)*a^9*x^5-375*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(7/2)*a^8*c*x^6+480*(-c/a^
2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(7/2)*a^8*c*x^6-105*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(9/2)*a^8*x^4-2352*(-c
/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(5/2)*a^7*c^2*x^7+560*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(5/2)*a^7*c^2*x^7
-224*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(9/2)*a^7*x^3+525*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(5/2)*a^6*c^2*x^6
+2940*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*a^5*c^3*x^7-700*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*a^
5*c^3*x^7-630*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(9/2)*a^6*x^2-875*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*a^
4*c^3*x^6-672*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(9/2)*a^5*x-4410*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a^3
*c^4*x^7+1050*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*a^3*c^4*x^7+4410*c^(9/2)*(-c/a^2)^(1/2)*ln(x*c^(1/2
)+(c*(a^2*x^2-1)/a^2)^(1/2))*a*x^6-1050*c^(9/2)*(-c/a^2)^(1/2)*ln((c^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)+c*x)/
c^(1/2))*a*x^6-280*(c*(a^2*x^2-1)/a^2)^(9/2)*a^4*(-c/a^2)^(1/2)+2625*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*
a^2*c^4*x^6+2625*ln(2*((-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a^2-c)/a^2/x)*c^5*x^6)/(-c/a^2)^(1/2)/(c*(a^2*
x^2-1)/a^2)^(7/2)/c

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.41, size = 438, normalized size = 1.18 \begin {gather*} \left [\frac {960 \, a^{5} \sqrt {-c} c^{3} x^{5} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 375 \, a^{5} \sqrt {-c} c^{3} x^{5} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (240 \, a^{6} c^{3} x^{6} - 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} + 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} - 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{480 \, a^{6} x^{5}}, -\frac {375 \, a^{5} c^{\frac {7}{2}} x^{5} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 240 \, a^{5} c^{\frac {7}{2}} x^{5} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (240 \, a^{6} c^{3} x^{6} - 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} + 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} - 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{240 \, a^{6} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/480*(960*a^5*sqrt(-c)*c^3*x^5*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 37
5*a^5*sqrt(-c)*c^3*x^5*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(240*a
^6*c^3*x^6 - 736*a^5*c^3*x^5 + 105*a^4*c^3*x^4 + 352*a^3*c^3*x^3 + 70*a^2*c^3*x^2 - 96*a*c^3*x - 40*c^3)*sqrt(
(a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5), -1/240*(375*a^5*c^(7/2)*x^5*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^
2*x^2))/(a^2*c*x^2 - c)) - 240*a^5*c^(7/2)*x^5*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x
^2)) - c) + (240*a^6*c^3*x^6 - 736*a^5*c^3*x^5 + 105*a^4*c^3*x^4 + 352*a^3*c^3*x^3 + 70*a^2*c^3*x^2 - 96*a*c^3
*x - 40*c^3)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5)]

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

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Giac [A]
time = 24.72, size = 561, normalized size = 1.51 \begin {gather*} \frac {1}{120} \, {\left (\frac {375 \, c^{\frac {7}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {240 \, c^{\frac {7}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a {\left | a \right |}} - \frac {120 \, \sqrt {a^{2} c x^{2} - c} c^{3} \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{11} c^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 1440 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{10} a c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} c^{5} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 4320 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{8} a c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 7360 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {13}{2}} \mathrm {sgn}\left (x\right ) + 150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{7} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 6720 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {15}{2}} \mathrm {sgn}\left (x\right ) - 595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{8} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 2976 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {17}{2}} \mathrm {sgn}\left (x\right ) - 105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{9} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 736 \, a c^{\frac {19}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{6} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(375*c^(7/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x)/a^2 + 240*c^(7/2)*log(abs(-sq
rt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/(a*abs(a)) - 120*sqrt(a^2*c*x^2 - c)*c^3*sgn(x)/a^2 + (105*(sqrt(a^
2*c)*x - sqrt(a^2*c*x^2 - c))^11*c^4*abs(a)*sgn(x) + 1440*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^10*a*c^(9/2)*s
gn(x) + 595*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^9*c^5*abs(a)*sgn(x) + 4320*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 -
c))^8*a*c^(11/2)*sgn(x) - 150*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^7*c^6*abs(a)*sgn(x) + 7360*(sqrt(a^2*c)*x
- sqrt(a^2*c*x^2 - c))^6*a*c^(13/2)*sgn(x) + 150*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^5*c^7*abs(a)*sgn(x) +
6720*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a*c^(15/2)*sgn(x) - 595*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*c
^8*abs(a)*sgn(x) + 2976*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*a*c^(17/2)*sgn(x) - 105*(sqrt(a^2*c)*x - sqrt(
a^2*c*x^2 - c))*c^9*abs(a)*sgn(x) + 736*a*c^(19/2)*sgn(x))/(((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^6*a^
2*abs(a)))*abs(a)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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