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

Optimal. Leaf size=270 \[ \frac {(1-a x)^2}{3 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {2 (a x+1)^3 (37 a x+72) (1-a x)^4}{35 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {2 (a x+1)^{7/2} (1-a x)^{7/2} \sin ^{-1}(a x)}{a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {2 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {82 (a x+1) (1-a x)^4}{105 a^5 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {12 (1-a x)^4}{7 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {10 (1-a x)^3}{3 a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]

[Out]

1/3*(-a*x+1)^2/a^2/(c-c/a^2/x^2)^(7/2)/x-10/3*(-a*x+1)^3/a^3/(c-c/a^2/x^2)^(7/2)/x^2-12/7*(-a*x+1)^4/a^4/(c-c/
a^2/x^2)^(7/2)/x^3-82/105*(-a*x+1)^4*(a*x+1)/a^5/(c-c/a^2/x^2)^(7/2)/x^4-2/35*(-a*x+1)^4*(a*x+1)^2/a^6/(c-c/a^
2/x^2)^(7/2)/x^5-2/35*(-a*x+1)^4*(a*x+1)^3*(37*a*x+72)/a^8/(c-c/a^2/x^2)^(7/2)/x^7-2*(-a*x+1)^(7/2)*(a*x+1)^(7
/2)*arcsin(a*x)/a^8/(c-c/a^2/x^2)^(7/2)/x^7

________________________________________________________________________________________

Rubi [A]  time = 0.43, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6159, 6129, 98, 150, 143, 41, 216} \[ -\frac {2 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {82 (a x+1) (1-a x)^4}{105 a^5 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {2 (a x+1)^3 (37 a x+72) (1-a x)^4}{35 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {12 (1-a x)^4}{7 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {10 (1-a x)^3}{3 a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {(1-a x)^2}{3 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {2 (a x+1)^{7/2} (1-a x)^{7/2} \sin ^{-1}(a x)}{a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(7/2)),x]

[Out]

(1 - a*x)^2/(3*a^2*(c - c/(a^2*x^2))^(7/2)*x) - (10*(1 - a*x)^3)/(3*a^3*(c - c/(a^2*x^2))^(7/2)*x^2) - (12*(1
- a*x)^4)/(7*a^4*(c - c/(a^2*x^2))^(7/2)*x^3) - (82*(1 - a*x)^4*(1 + a*x))/(105*a^5*(c - c/(a^2*x^2))^(7/2)*x^
4) - (2*(1 - a*x)^4*(1 + a*x)^2)/(35*a^6*(c - c/(a^2*x^2))^(7/2)*x^5) - (2*(1 - a*x)^4*(1 + a*x)^3*(72 + 37*a*
x))/(35*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) - (2*(1 - a*x)^(7/2)*(1 + a*x)^(7/2)*ArcSin[a*x])/(a^8*(c - c/(a^2*x^
2))^(7/2)*x^7)

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 98

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

Rule 143

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

Rule 150

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] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

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 6129

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 6159

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*(1 - a*x)^p*(1 + a*x)^p*E^(n*ArcTanh[a*x]))/x^(2*p), 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 \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx &=\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {e^{-2 \tanh ^{-1}(a x)} x^7}{(1-a x)^{7/2} (1+a x)^{7/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^7}{(1-a x)^{5/2} (1+a x)^{9/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^5 (6+4 a x)}{(1-a x)^{3/2} (1+a x)^{9/2}} \, dx}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^4 \left (-50 a-14 a^2 x\right )}{\sqrt {1-a x} (1+a x)^{9/2}} \, dx}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^3 \left (-144 a^2-62 a^3 x\right )}{\sqrt {1-a x} (1+a x)^{7/2}} \, dx}{21 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^2 \left (-246 a^3-228 a^4 x\right )}{\sqrt {1-a x} (1+a x)^{5/2}} \, dx}{105 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}-\frac {2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x \left (-36 a^4-666 a^5 x\right )}{\sqrt {1-a x} (1+a x)^{3/2}} \, dx}{315 a^{10} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}-\frac {2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}-\frac {2 (1-a x)^4 (1+a x)^3 (72+37 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}-\frac {\left (2 (1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}-\frac {2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}-\frac {2 (1-a x)^4 (1+a x)^3 (72+37 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}-\frac {\left (2 (1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}-\frac {10 (1-a x)^3}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {12 (1-a x)^4}{7 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}-\frac {82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}-\frac {2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}-\frac {2 (1-a x)^4 (1+a x)^3 (72+37 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}-\frac {2 (1-a x)^{7/2} (1+a x)^{7/2} \sin ^{-1}(a x)}{a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 131, normalized size = 0.49 \[ \frac {-105 a^6 x^6-562 a^5 x^5-74 a^4 x^4+1226 a^3 x^3+636 a^2 x^2+210 (a x-1) (a x+1)^3 \sqrt {a^2 x^2-1} \log \left (\sqrt {a^2 x^2-1}+a x\right )-654 a x-432}{105 a^2 x (a x-1) \sqrt {c-\frac {c}{a^2 x^2}} (a c x+c)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(7/2)),x]

[Out]

(-432 - 654*a*x + 636*a^2*x^2 + 1226*a^3*x^3 - 74*a^4*x^4 - 562*a^5*x^5 - 105*a^6*x^6 + 210*(-1 + a*x)*(1 + a*
x)^3*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(105*a^2*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x)*(c + a*c*x)
^3)

________________________________________________________________________________________

fricas [A]  time = 0.78, size = 496, normalized size = 1.84 \[ \left [\frac {105 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (105 \, a^{7} x^{7} + 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} - 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} + 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}}, -\frac {210 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (105 \, a^{7} x^{7} + 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} - 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} + 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/105*(105*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*sqrt(c)*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) - (105*a^7*x^7 + 562*a^6*x^6 + 74*a^5*x^5 - 1226*a^4*x^4 - 63
6*a^3*x^3 + 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4
- 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4), -1/105*(210*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3
- a^2*x^2 + 2*a*x + 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + (10
5*a^7*x^7 + 562*a^6*x^6 + 74*a^5*x^5 - 1226*a^4*x^4 - 636*a^3*x^3 + 654*a^2*x^2 + 432*a*x)*sqrt((a^2*c*x^2 - c
)/(a^2*x^2)))/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4)]

________________________________________________________________________________________

giac [A]  time = 3.82, size = 251, normalized size = 0.93 \[ -\frac {{\left (a x + 1\right )} \sqrt {c - \frac {2 \, c}{a x + 1}}}{a c^{4} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} - \frac {4 \, \arctan \left (\frac {\sqrt {c - \frac {2 \, c}{a x + 1}}}{\sqrt {-c}}\right )}{a \sqrt {-c} c^{3} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} + \frac {14 \, c - \frac {27 \, c}{a x + 1}}{48 \, a {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} c^{3} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} - \frac {15 \, a^{6} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {7}{2}} c^{42} + 189 \, a^{6} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {5}{2}} c^{43} + 1330 \, a^{6} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} c^{44} + 10710 \, a^{6} \sqrt {c - \frac {2 \, c}{a x + 1}} c^{45}}{3360 \, a^{7} c^{49} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*x + 1)*sqrt(c - 2*c/(a*x + 1))/(a*c^4*sgn(-1/(a*x + 1) + 1)) - 4*arctan(sqrt(c - 2*c/(a*x + 1))/sqrt(-c))/
(a*sqrt(-c)*c^3*sgn(-1/(a*x + 1) + 1)) + 1/48*(14*c - 27*c/(a*x + 1))/(a*(c - 2*c/(a*x + 1))^(3/2)*c^3*sgn(-1/
(a*x + 1) + 1)) - 1/3360*(15*a^6*(c - 2*c/(a*x + 1))^(7/2)*c^42 + 189*a^6*(c - 2*c/(a*x + 1))^(5/2)*c^43 + 133
0*a^6*(c - 2*c/(a*x + 1))^(3/2)*c^44 + 10710*a^6*sqrt(c - 2*c/(a*x + 1))*c^45)/(a^7*c^49*sgn(-1/(a*x + 1) + 1)
)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 572, normalized size = 2.12 \[ \frac {\left (-105 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{7} a^{7}+96 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x^{6} a^{6}-553 x^{6} c^{\frac {7}{2}} a^{6} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}}+96 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x^{5} a^{5}+392 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{5} a^{5}-240 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x^{4} a^{4}+1540 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{4} a^{4}+210 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x \,a^{6} c -240 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x^{3} a^{3}-350 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{3} a^{3}+210 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} a^{5} c +180 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x^{2} a^{2}-1470 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{2} a^{2}+180 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x a +42 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x a -30 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}}+462 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}}\right ) \left (a x -1\right )}{105 \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{7} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} a^{8} c^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(-105*c^(7/2)*((a*x-1)*(a*x+1)*c/a^2)^(5/2)*x^7*a^7+96*(c*(a^2*x^2-1)/a^2)^(5/2)*c^(7/2)*x^6*a^6-553*x^6
*c^(7/2)*a^6*((a*x-1)*(a*x+1)*c/a^2)^(5/2)+96*(c*(a^2*x^2-1)/a^2)^(5/2)*c^(7/2)*x^5*a^5+392*c^(7/2)*((a*x-1)*(
a*x+1)*c/a^2)^(5/2)*x^5*a^5-240*(c*(a^2*x^2-1)/a^2)^(5/2)*c^(7/2)*x^4*a^4+1540*c^(7/2)*((a*x-1)*(a*x+1)*c/a^2)
^(5/2)*x^4*a^4+210*(c*(a^2*x^2-1)/a^2)^(5/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*((a*x-1)*(a*x+1)*c/a^2)^(
5/2)*x*a^6*c-240*(c*(a^2*x^2-1)/a^2)^(5/2)*c^(7/2)*x^3*a^3-350*c^(7/2)*((a*x-1)*(a*x+1)*c/a^2)^(5/2)*x^3*a^3+2
10*(c*(a^2*x^2-1)/a^2)^(5/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*((a*x-1)*(a*x+1)*c/a^2)^(5/2)*a^5*c+180*(
c*(a^2*x^2-1)/a^2)^(5/2)*c^(7/2)*x^2*a^2-1470*c^(7/2)*((a*x-1)*(a*x+1)*c/a^2)^(5/2)*x^2*a^2+180*(c*(a^2*x^2-1)
/a^2)^(5/2)*c^(7/2)*x*a+42*c^(7/2)*((a*x-1)*(a*x+1)*c/a^2)^(5/2)*x*a-30*(c*(a^2*x^2-1)/a^2)^(5/2)*c^(7/2)+462*
c^(7/2)*((a*x-1)*(a*x+1)*c/a^2)^(5/2))*(a*x-1)/((a*x-1)*(a*x+1)*c/a^2)^(5/2)/x^7/(c*(a^2*x^2-1)/a^2/x^2)^(7/2)
/a^8/c^(7/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {a^{2} x^{2} - 1}{{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________