3.42 \(\int (c+d x^2)^3 \cosh ^{-1}(a x) \, dx\)

Optimal. Leaf size=267 \[ -\frac{d \left (1-a^2 x^2\right )^2 \left (35 a^4 c^2+42 a^2 c d+15 d^2\right )}{105 a^7 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\left (1-a^2 x^2\right ) \left (35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2+5 d^3\right )}{35 a^7 \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 d^2 \left (1-a^2 x^2\right )^3 \left (7 a^2 c+5 d\right )}{175 a^7 \sqrt{a x-1} \sqrt{a x+1}}-\frac{d^3 \left (1-a^2 x^2\right )^4}{49 a^7 \sqrt{a x-1} \sqrt{a x+1}}+c^2 d x^3 \cosh ^{-1}(a x)+c^3 x \cosh ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cosh ^{-1}(a x) \]

[Out]

((35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*(1 - a^2*x^2))/(35*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (d*
(35*a^4*c^2 + 42*a^2*c*d + 15*d^2)*(1 - a^2*x^2)^2)/(105*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*d^2*(7*a^2*c +
5*d)*(1 - a^2*x^2)^3)/(175*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (d^3*(1 - a^2*x^2)^4)/(49*a^7*Sqrt[-1 + a*x]*S
qrt[1 + a*x]) + c^3*x*ArcCosh[a*x] + c^2*d*x^3*ArcCosh[a*x] + (3*c*d^2*x^5*ArcCosh[a*x])/5 + (d^3*x^7*ArcCosh[
a*x])/7

________________________________________________________________________________________

Rubi [A]  time = 0.350182, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {194, 5705, 12, 1610, 1799, 1850} \[ -\frac{d \left (1-a^2 x^2\right )^2 \left (35 a^4 c^2+42 a^2 c d+15 d^2\right )}{105 a^7 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\left (1-a^2 x^2\right ) \left (35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2+5 d^3\right )}{35 a^7 \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 d^2 \left (1-a^2 x^2\right )^3 \left (7 a^2 c+5 d\right )}{175 a^7 \sqrt{a x-1} \sqrt{a x+1}}-\frac{d^3 \left (1-a^2 x^2\right )^4}{49 a^7 \sqrt{a x-1} \sqrt{a x+1}}+c^2 d x^3 \cosh ^{-1}(a x)+c^3 x \cosh ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3*ArcCosh[a*x],x]

[Out]

((35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*(1 - a^2*x^2))/(35*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (d*
(35*a^4*c^2 + 42*a^2*c*d + 15*d^2)*(1 - a^2*x^2)^2)/(105*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*d^2*(7*a^2*c +
5*d)*(1 - a^2*x^2)^3)/(175*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (d^3*(1 - a^2*x^2)^4)/(49*a^7*Sqrt[-1 + a*x]*S
qrt[1 + a*x]) + c^3*x*ArcCosh[a*x] + c^2*d*x^3*ArcCosh[a*x] + (3*c*d^2*x^5*ArcCosh[a*x])/5 + (d^3*x^7*ArcCosh[
a*x])/7

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 5705

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \left (c+d x^2\right )^3 \cosh ^{-1}(a x) \, dx &=c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cosh ^{-1}(a x)-a \int \frac{x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{35 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cosh ^{-1}(a x)-\frac{1}{35} a \int \frac{x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cosh ^{-1}(a x)-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \int \frac{x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{\sqrt{-1+a^2 x^2}} \, dx}{35 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cosh ^{-1}(a x)-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{35 c^3+35 c^2 d x+21 c d^2 x^2+5 d^3 x^3}{\sqrt{-1+a^2 x}} \, dx,x,x^2\right )}{70 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cosh ^{-1}(a x)-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3}{a^6 \sqrt{-1+a^2 x}}+\frac{d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \sqrt{-1+a^2 x}}{a^6}+\frac{3 d^2 \left (7 a^2 c+5 d\right ) \left (-1+a^2 x\right )^{3/2}}{a^6}+\frac{5 d^3 \left (-1+a^2 x\right )^{5/2}}{a^6}\right ) \, dx,x,x^2\right )}{70 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \left (1-a^2 x^2\right )}{35 a^7 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \left (1-a^2 x^2\right )^2}{105 a^7 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 d^2 \left (7 a^2 c+5 d\right ) \left (1-a^2 x^2\right )^3}{175 a^7 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{d^3 \left (1-a^2 x^2\right )^4}{49 a^7 \sqrt{-1+a x} \sqrt{1+a x}}+c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cosh ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.195587, size = 154, normalized size = 0.58 \[ \frac{1}{35} x \cosh ^{-1}(a x) \left (35 c^2 d x^2+35 c^3+21 c d^2 x^4+5 d^3 x^6\right )-\frac{\sqrt{a x-1} \sqrt{a x+1} \left (a^6 \left (1225 c^2 d x^2+3675 c^3+441 c d^2 x^4+75 d^3 x^6\right )+2 a^4 d \left (1225 c^2+294 c d x^2+45 d^2 x^4\right )+24 a^2 d^2 \left (49 c+5 d x^2\right )+240 d^3\right )}{3675 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^3*ArcCosh[a*x],x]

[Out]

-(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(240*d^3 + 24*a^2*d^2*(49*c + 5*d*x^2) + 2*a^4*d*(1225*c^2 + 294*c*d*x^2 + 45*d
^2*x^4) + a^6*(3675*c^3 + 1225*c^2*d*x^2 + 441*c*d^2*x^4 + 75*d^3*x^6)))/(3675*a^7) + (x*(35*c^3 + 35*c^2*d*x^
2 + 21*c*d^2*x^4 + 5*d^3*x^6)*ArcCosh[a*x])/35

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 176, normalized size = 0.7 \begin{align*}{\frac{1}{a} \left ({\frac{a{\rm arccosh} \left (ax\right ){d}^{3}{x}^{7}}{7}}+{\frac{3\,a{\rm arccosh} \left (ax\right )c{d}^{2}{x}^{5}}{5}}+a{\rm arccosh} \left (ax\right ){c}^{2}d{x}^{3}+{\rm arccosh} \left (ax\right ){c}^{3}ax-{\frac{75\,{a}^{6}{d}^{3}{x}^{6}+441\,{a}^{6}c{d}^{2}{x}^{4}+1225\,{a}^{6}{c}^{2}d{x}^{2}+90\,{a}^{4}{d}^{3}{x}^{4}+3675\,{a}^{6}{c}^{3}+588\,{a}^{4}c{d}^{2}{x}^{2}+2450\,{a}^{4}{c}^{2}d+120\,{a}^{2}{d}^{3}{x}^{2}+1176\,{a}^{2}c{d}^{2}+240\,{d}^{3}}{3675\,{a}^{6}}\sqrt{ax-1}\sqrt{ax+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^3*arccosh(a*x),x)

[Out]

1/a*(1/7*a*arccosh(a*x)*d^3*x^7+3/5*a*arccosh(a*x)*c*d^2*x^5+a*arccosh(a*x)*c^2*d*x^3+arccosh(a*x)*c^3*a*x-1/3
675/a^6*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(75*a^6*d^3*x^6+441*a^6*c*d^2*x^4+1225*a^6*c^2*d*x^2+90*a^4*d^3*x^4+3675*a
^6*c^3+588*a^4*c*d^2*x^2+2450*a^4*c^2*d+120*a^2*d^3*x^2+1176*a^2*c*d^2+240*d^3))

________________________________________________________________________________________

Maxima [A]  time = 1.11726, size = 347, normalized size = 1.3 \begin{align*} -\frac{1}{3675} \,{\left (\frac{75 \, \sqrt{a^{2} x^{2} - 1} d^{3} x^{6}}{a^{2}} + \frac{441 \, \sqrt{a^{2} x^{2} - 1} c d^{2} x^{4}}{a^{2}} + \frac{1225 \, \sqrt{a^{2} x^{2} - 1} c^{2} d x^{2}}{a^{2}} + \frac{90 \, \sqrt{a^{2} x^{2} - 1} d^{3} x^{4}}{a^{4}} + \frac{3675 \, \sqrt{a^{2} x^{2} - 1} c^{3}}{a^{2}} + \frac{588 \, \sqrt{a^{2} x^{2} - 1} c d^{2} x^{2}}{a^{4}} + \frac{2450 \, \sqrt{a^{2} x^{2} - 1} c^{2} d}{a^{4}} + \frac{120 \, \sqrt{a^{2} x^{2} - 1} d^{3} x^{2}}{a^{6}} + \frac{1176 \, \sqrt{a^{2} x^{2} - 1} c d^{2}}{a^{6}} + \frac{240 \, \sqrt{a^{2} x^{2} - 1} d^{3}}{a^{8}}\right )} a + \frac{1}{35} \,{\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \operatorname{arcosh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3*arccosh(a*x),x, algorithm="maxima")

[Out]

-1/3675*(75*sqrt(a^2*x^2 - 1)*d^3*x^6/a^2 + 441*sqrt(a^2*x^2 - 1)*c*d^2*x^4/a^2 + 1225*sqrt(a^2*x^2 - 1)*c^2*d
*x^2/a^2 + 90*sqrt(a^2*x^2 - 1)*d^3*x^4/a^4 + 3675*sqrt(a^2*x^2 - 1)*c^3/a^2 + 588*sqrt(a^2*x^2 - 1)*c*d^2*x^2
/a^4 + 2450*sqrt(a^2*x^2 - 1)*c^2*d/a^4 + 120*sqrt(a^2*x^2 - 1)*d^3*x^2/a^6 + 1176*sqrt(a^2*x^2 - 1)*c*d^2/a^6
+ 240*sqrt(a^2*x^2 - 1)*d^3/a^8)*a + 1/35*(5*d^3*x^7 + 21*c*d^2*x^5 + 35*c^2*d*x^3 + 35*c^3*x)*arccosh(a*x)

________________________________________________________________________________________

Fricas [A]  time = 2.22104, size = 406, normalized size = 1.52 \begin{align*} \frac{105 \,{\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) -{\left (75 \, a^{6} d^{3} x^{6} + 3675 \, a^{6} c^{3} + 2450 \, a^{4} c^{2} d + 1176 \, a^{2} c d^{2} + 9 \,{\left (49 \, a^{6} c d^{2} + 10 \, a^{4} d^{3}\right )} x^{4} + 240 \, d^{3} +{\left (1225 \, a^{6} c^{2} d + 588 \, a^{4} c d^{2} + 120 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt{a^{2} x^{2} - 1}}{3675 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3*arccosh(a*x),x, algorithm="fricas")

[Out]

1/3675*(105*(5*a^7*d^3*x^7 + 21*a^7*c*d^2*x^5 + 35*a^7*c^2*d*x^3 + 35*a^7*c^3*x)*log(a*x + sqrt(a^2*x^2 - 1))
- (75*a^6*d^3*x^6 + 3675*a^6*c^3 + 2450*a^4*c^2*d + 1176*a^2*c*d^2 + 9*(49*a^6*c*d^2 + 10*a^4*d^3)*x^4 + 240*d
^3 + (1225*a^6*c^2*d + 588*a^4*c*d^2 + 120*a^2*d^3)*x^2)*sqrt(a^2*x^2 - 1))/a^7

________________________________________________________________________________________

Sympy [A]  time = 9.39132, size = 328, normalized size = 1.23 \begin{align*} \begin{cases} c^{3} x \operatorname{acosh}{\left (a x \right )} + c^{2} d x^{3} \operatorname{acosh}{\left (a x \right )} + \frac{3 c d^{2} x^{5} \operatorname{acosh}{\left (a x \right )}}{5} + \frac{d^{3} x^{7} \operatorname{acosh}{\left (a x \right )}}{7} - \frac{c^{3} \sqrt{a^{2} x^{2} - 1}}{a} - \frac{c^{2} d x^{2} \sqrt{a^{2} x^{2} - 1}}{3 a} - \frac{3 c d^{2} x^{4} \sqrt{a^{2} x^{2} - 1}}{25 a} - \frac{d^{3} x^{6} \sqrt{a^{2} x^{2} - 1}}{49 a} - \frac{2 c^{2} d \sqrt{a^{2} x^{2} - 1}}{3 a^{3}} - \frac{4 c d^{2} x^{2} \sqrt{a^{2} x^{2} - 1}}{25 a^{3}} - \frac{6 d^{3} x^{4} \sqrt{a^{2} x^{2} - 1}}{245 a^{3}} - \frac{8 c d^{2} \sqrt{a^{2} x^{2} - 1}}{25 a^{5}} - \frac{8 d^{3} x^{2} \sqrt{a^{2} x^{2} - 1}}{245 a^{5}} - \frac{16 d^{3} \sqrt{a^{2} x^{2} - 1}}{245 a^{7}} & \text{for}\: a \neq 0 \\\frac{i \pi \left (c^{3} x + c^{2} d x^{3} + \frac{3 c d^{2} x^{5}}{5} + \frac{d^{3} x^{7}}{7}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**3*acosh(a*x),x)

[Out]

Piecewise((c**3*x*acosh(a*x) + c**2*d*x**3*acosh(a*x) + 3*c*d**2*x**5*acosh(a*x)/5 + d**3*x**7*acosh(a*x)/7 -
c**3*sqrt(a**2*x**2 - 1)/a - c**2*d*x**2*sqrt(a**2*x**2 - 1)/(3*a) - 3*c*d**2*x**4*sqrt(a**2*x**2 - 1)/(25*a)
- d**3*x**6*sqrt(a**2*x**2 - 1)/(49*a) - 2*c**2*d*sqrt(a**2*x**2 - 1)/(3*a**3) - 4*c*d**2*x**2*sqrt(a**2*x**2
- 1)/(25*a**3) - 6*d**3*x**4*sqrt(a**2*x**2 - 1)/(245*a**3) - 8*c*d**2*sqrt(a**2*x**2 - 1)/(25*a**5) - 8*d**3*
x**2*sqrt(a**2*x**2 - 1)/(245*a**5) - 16*d**3*sqrt(a**2*x**2 - 1)/(245*a**7), Ne(a, 0)), (I*pi*(c**3*x + c**2*
d*x**3 + 3*c*d**2*x**5/5 + d**3*x**7/7)/2, True))

________________________________________________________________________________________

Giac [A]  time = 1.13379, size = 325, normalized size = 1.22 \begin{align*} \frac{1}{35} \,{\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{3675 \, \sqrt{a^{2} x^{2} - 1} a^{6} c^{3} + 1225 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} a^{4} c^{2} d + 3675 \, \sqrt{a^{2} x^{2} - 1} a^{4} c^{2} d + 441 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} a^{2} c d^{2} + 1470 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} a^{2} c d^{2} + 75 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{7}{2}} d^{3} + 2205 \, \sqrt{a^{2} x^{2} - 1} a^{2} c d^{2} + 315 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} d^{3} + 525 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} d^{3} + 525 \, \sqrt{a^{2} x^{2} - 1} d^{3}}{3675 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3*arccosh(a*x),x, algorithm="giac")

[Out]

1/35*(5*d^3*x^7 + 21*c*d^2*x^5 + 35*c^2*d*x^3 + 35*c^3*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 1/3675*(3675*sqrt(a^2
*x^2 - 1)*a^6*c^3 + 1225*(a^2*x^2 - 1)^(3/2)*a^4*c^2*d + 3675*sqrt(a^2*x^2 - 1)*a^4*c^2*d + 441*(a^2*x^2 - 1)^
(5/2)*a^2*c*d^2 + 1470*(a^2*x^2 - 1)^(3/2)*a^2*c*d^2 + 75*(a^2*x^2 - 1)^(7/2)*d^3 + 2205*sqrt(a^2*x^2 - 1)*a^2
*c*d^2 + 315*(a^2*x^2 - 1)^(5/2)*d^3 + 525*(a^2*x^2 - 1)^(3/2)*d^3 + 525*sqrt(a^2*x^2 - 1)*d^3)/a^7