3.56 \(\int (d+e x)^3 (a+b \sec ^{-1}(c x)) \, dx\)
Optimal. Leaf size=167 \[ \frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {b d e^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{2 c}-\frac {b e^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{12 c}-\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (9 c^2 d^2+e^2\right )}{6 c^3}-\frac {b d \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c^3}+\frac {b d^4 \csc ^{-1}(c x)}{4 e} \]
[Out]
1/4*b*d^4*arccsc(c*x)/e+1/4*(e*x+d)^4*(a+b*arcsec(c*x))/e-1/2*b*d*(2*c^2*d^2+e^2)*arctanh((1-1/c^2/x^2)^(1/2))
/c^3-1/6*b*e*(9*c^2*d^2+e^2)*x*(1-1/c^2/x^2)^(1/2)/c^3-1/2*b*d*e^2*x^2*(1-1/c^2/x^2)^(1/2)/c-1/12*b*e^3*x^3*(1
-1/c^2/x^2)^(1/2)/c
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Rubi [A] time = 0.40, antiderivative size = 167, normalized size of antiderivative = 1.00,
number of steps used = 11, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used
= {5226, 1568, 1475, 1807, 844, 216, 266, 63, 208} \[ \frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (9 c^2 d^2+e^2\right )}{6 c^3}-\frac {b d \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c^3}-\frac {b d e^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{2 c}-\frac {b e^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{12 c}+\frac {b d^4 \csc ^{-1}(c x)}{4 e} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^3*(a + b*ArcSec[c*x]),x]
[Out]
-(b*e*(9*c^2*d^2 + e^2)*Sqrt[1 - 1/(c^2*x^2)]*x)/(6*c^3) - (b*d*e^2*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(2*c) - (b*e^3*
Sqrt[1 - 1/(c^2*x^2)]*x^3)/(12*c) + (b*d^4*ArcCsc[c*x])/(4*e) + ((d + e*x)^4*(a + b*ArcSec[c*x]))/(4*e) - (b*d
*(2*c^2*d^2 + e^2)*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(2*c^3)
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 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 1475
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x
] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
Rule 1568
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Int[x^(m + mn*q
)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; FreeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (P
osQ[n2] || !IntegerQ[p])
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 5226
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + b
*ArcSec[c*x]))/(e*(m + 1)), x] - Dist[b/(c*e*(m + 1)), Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x],
x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {b \int \frac {(d+e x)^4}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{4 c e}\\ &=\frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {b \int \frac {\left (e+\frac {d}{x}\right )^4 x^2}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{4 c e}\\ &=\frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}+\frac {b \operatorname {Subst}\left (\int \frac {(e+d x)^4}{x^4 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c e}\\ &=-\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {b \operatorname {Subst}\left (\int \frac {-12 d e^3-2 e^2 \left (9 d^2+\frac {e^2}{c^2}\right ) x-12 d^3 e x^2-3 d^4 x^3}{x^3 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{12 c e}\\ &=-\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}-\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}+\frac {b \operatorname {Subst}\left (\int \frac {4 e^2 \left (9 d^2+\frac {e^2}{c^2}\right )+12 d e \left (2 d^2+\frac {e^2}{c^2}\right ) x+6 d^4 x^2}{x^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{24 c e}\\ &=-\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}-\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}-\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {b \operatorname {Subst}\left (\int \frac {-12 d e \left (2 d^2+\frac {e^2}{c^2}\right )-6 d^4 x}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{24 c e}\\ &=-\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}-\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}-\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}+\frac {\left (b d^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c e}+\frac {\left (b d \left (2 c^2 d^2+e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c^3}\\ &=-\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}-\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}-\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {b d^4 \csc ^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}+\frac {\left (b d \left (2 c^2 d^2+e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 c^3}\\ &=-\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}-\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}-\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {b d^4 \csc ^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {\left (b d \left (2 c^2 d^2+e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c}\\ &=-\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}-\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}-\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {b d^4 \csc ^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {b d \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 166, normalized size = 0.99 \[ \frac {3 a c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+3 b c^3 x \sec ^{-1}(c x) \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 \left (18 d^2+6 d e x+e^2 x^2\right )+2 e^2\right )-6 b d \left (2 c^2 d^2+e^2\right ) \log \left (x \left (\sqrt {1-\frac {1}{c^2 x^2}}+1\right )\right )}{12 c^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^3*(a + b*ArcSec[c*x]),x]
[Out]
(3*a*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(2*e^2 + c^2*(18*d^2 + 6*
d*e*x + e^2*x^2)) + 3*b*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcSec[c*x] - 6*b*d*(2*c^2*d^2 + e^2
)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(12*c^3)
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fricas [A] time = 0.66, size = 291, normalized size = 1.74 \[ \frac {3 \, a c^{4} e^{3} x^{4} + 12 \, a c^{4} d e^{2} x^{3} + 18 \, a c^{4} d^{2} e x^{2} + 12 \, a c^{4} d^{3} x + 3 \, {\left (b c^{4} e^{3} x^{4} + 4 \, b c^{4} d e^{2} x^{3} + 6 \, b c^{4} d^{2} e x^{2} + 4 \, b c^{4} d^{3} x - 4 \, b c^{4} d^{3} - 6 \, b c^{4} d^{2} e - 4 \, b c^{4} d e^{2} - b c^{4} e^{3}\right )} \operatorname {arcsec}\left (c x\right ) + 6 \, {\left (4 \, b c^{4} d^{3} + 6 \, b c^{4} d^{2} e + 4 \, b c^{4} d e^{2} + b c^{4} e^{3}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 6 \, {\left (2 \, b c^{3} d^{3} + b c d e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} e^{3} x^{2} + 6 \, b c^{2} d e^{2} x + 18 \, b c^{2} d^{2} e + 2 \, b e^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^3*(a+b*arcsec(c*x)),x, algorithm="fricas")
[Out]
1/12*(3*a*c^4*e^3*x^4 + 12*a*c^4*d*e^2*x^3 + 18*a*c^4*d^2*e*x^2 + 12*a*c^4*d^3*x + 3*(b*c^4*e^3*x^4 + 4*b*c^4*
d*e^2*x^3 + 6*b*c^4*d^2*e*x^2 + 4*b*c^4*d^3*x - 4*b*c^4*d^3 - 6*b*c^4*d^2*e - 4*b*c^4*d*e^2 - b*c^4*e^3)*arcse
c(c*x) + 6*(4*b*c^4*d^3 + 6*b*c^4*d^2*e + 4*b*c^4*d*e^2 + b*c^4*e^3)*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 6*(2*b
*c^3*d^3 + b*c*d*e^2)*log(-c*x + sqrt(c^2*x^2 - 1)) - (b*c^2*e^3*x^2 + 6*b*c^2*d*e^2*x + 18*b*c^2*d^2*e + 2*b*
e^3)*sqrt(c^2*x^2 - 1))/c^4
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giac [B] time = 2.94, size = 9404, normalized size = 56.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^3*(a+b*arcsec(c*x)),x, algorithm="giac")
[Out]
1/12*(12*b*c^3*d^3*arccos(1/(c*x))/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/
(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) - 12*b*
c^3*d^3*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*
(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(
c*x) + 1)^8) + 12*b*c^3*d^3*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(
c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(
c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) + 12*a*c^3*d^3/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2
*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) +
1)^8) + 24*b*c^3*d^3*(1/(c^2*x^2) - 1)*arccos(1/(c*x))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5
*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/
(c*x) + 1)^8)*(1/(c*x) + 1)^2) + 18*b*c^2*d^2*arccos(1/(c*x))*e/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2
+ 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) -
1)^4/(1/(c*x) + 1)^8) - 48*b*c^3*d^3*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^5 +
4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^
3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) + 48*b*c^3*d^3*(1/(c^2*x^2) - 1)
*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^
2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) +
1)^8)*(1/(c*x) + 1)^2) + 24*a*c^3*d^3*(1/(c^2*x^2) - 1)/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c
^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(
1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) + 18*a*c^2*d^2*e/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c
^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x)
+ 1)^8) - 72*b*c^3*d^3*(1/(c^2*x^2) - 1)^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^5 + 4*c^5*(1/(c^
2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) +
1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 72*b*c^3*d^3*(1/(c^2*x^2) - 1)^2*log(abs(s
qrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1
)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/
(c*x) + 1)^4) - 36*b*c^2*d^2*sqrt(-1/(c^2*x^2) + 1)*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*
(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(
c*x) + 1)^8)*(1/(c*x) + 1)) - 24*b*c^3*d^3*(1/(c^2*x^2) - 1)^3*arccos(1/(c*x))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)
/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5
*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 12*b*c*d*arccos(1/(c*x))*e^2/(c^5 + 4*c^5*(1/(c^2*x^2
) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6
+ c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) - 36*b*c^2*d^2*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))*e/((c^5 + 4*c^5
*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/
(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) - 48*b*c^3*d^3*(1/(c^2*x^2) - 1)^3*lo
g(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x
^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)
^8)*(1/(c*x) + 1)^6) - 6*b*c*d*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(c^5 + 4*c^5*(1/(c^2*x^2) -
1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c
^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) + 48*b*c^3*d^3*(1/(c^2*x^2) - 1)^3*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/
(c*x) - 1))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^
5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 6*b*c*d*e^
2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^
2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) +
1)^8) + 108*b*c^2*d^2*(-1/(c^2*x^2) + 1)^(3/2)*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(
c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x)
+ 1)^8)*(1/(c*x) + 1)^3) - 24*a*c^3*d^3*(1/(c^2*x^2) - 1)^3/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 +
6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)
^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) - 12*b*c^3*d^3*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))/((c^5 + 4*c^5*(1/(c^2*
x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1
)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) + 12*a*c*d*e^2/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/
(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*
(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) - 24*b*c*d*(1/(c^2*x^2) - 1)*arccos(1/(c*x))*e^2/((c^5 + 4*c^5*(1/(c^2*x^
2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^
6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) - 36*a*c^2*d^2*(1/(c^2*x^2) - 1)^2*e/((c^5 + 4*c
^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(
1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) - 12*b*c^3*d^3*(1/(c^2*x^2) - 1)^4*
log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2
*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) +
1)^8)*(1/(c*x) + 1)^8) - 24*b*c*d*(1/(c^2*x^2) - 1)*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^5 +
4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)
^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) + 12*b*c^3*d^3*(1/(c^2*x^2) - 1
)^4*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/
(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x
) + 1)^8)*(1/(c*x) + 1)^8) + 24*b*c*d*(1/(c^2*x^2) - 1)*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c
^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2)
- 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) - 12*b*c*d*sqrt(-1/(c^2*x^2
) + 1)*e^2/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5
*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)) - 108*b*c^2*d^2
*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*
x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1
)^8)*(1/(c*x) + 1)^5) - 12*a*c^3*d^3*(1/(c^2*x^2) - 1)^4/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c
^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(
1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) + 3*b*arccos(1/(c*x))*e^3/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*
c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/
(1/(c*x) + 1)^8) - 24*a*c*d*(1/(c^2*x^2) - 1)*e^2/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(
c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x)
+ 1)^8)*(1/(c*x) + 1)^2) + 18*b*c^2*d^2*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)
/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5
*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) - 36*b*c*d*(1/(c^2*x^2) - 1)^2*e^2*log(abs(sqrt(-1/(c^2
*x^2) + 1) + 1/(c*x) + 1))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x
) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^
4) + 36*b*c*d*(1/(c^2*x^2) - 1)^2*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^5 + 4*c^5*(1/(c^2*x^2
) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6
+ c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 12*b*c*d*(-1/(c^2*x^2) + 1)^(3/2)*e^2/((c^5 + 4
*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3
/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^3) - 36*b*c^2*d^2*(1/(c^2*x^2) - 1)^
3*sqrt(-1/(c^2*x^2) + 1)*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x
) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^
7) + 3*a*e^3/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^
5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) - 12*b*(1/(c^2*x^2) - 1)*arcc
os(1/(c*x))*e^3/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 +
4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) + 24*b*c
*d*(1/(c^2*x^2) - 1)^3*arccos(1/(c*x))*e^2/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2
) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8
)*(1/(c*x) + 1)^6) + 18*a*c^2*d^2*(1/(c^2*x^2) - 1)^4*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^
5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1
/(c*x) + 1)^8)*(1/(c*x) + 1)^8) - 24*b*c*d*(1/(c^2*x^2) - 1)^3*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) +
1))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^
2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 24*b*c*d*(1/(c^2*x
^2) - 1)^3*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2
+ 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1
)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) - 6*b*sqrt(-1/(c^2*x^2) + 1)*e^3/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x
) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2
*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)) + 12*b*c*d*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)*e^2/((c^5 +
4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)
^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^5) - 12*a*(1/(c^2*x^2) - 1)*e^3/((
c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2)
- 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) + 18*b*(1/(c^2*x^2) - 1)^2
*arccos(1/(c*x))*e^3/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)
^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 2
4*a*c*d*(1/(c^2*x^2) - 1)^3*e^2/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1
/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x)
+ 1)^6) - 12*b*c*d*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))*e^2/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6
*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4
/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) - 6*b*c*d*(1/(c^2*x^2) - 1)^4*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x)
+ 1))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(
c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) + 6*b*c*d*(1/(c^2*
x^2) - 1)^4*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2
+ 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) -
1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) + 10*b*(-1/(c^2*x^2) + 1)^(3/2)*e^3/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/
(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/
(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^3) + 12*b*c*d*(1/(c^2*x^2) - 1)^3*sqrt(-1/(c^2*x^2) + 1)*e^2/(
(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2
) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^7) + 18*a*(1/(c^2*x^2) - 1)^
2*e^3/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(
c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) - 12*b*(1/(c^2*x^2
) - 1)^3*arccos(1/(c*x))*e^3/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c
*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1
)^6) - 12*a*c*d*(1/(c^2*x^2) - 1)^4*e^2/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) -
1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(
1/(c*x) + 1)^8) - 10*b*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)*e^3/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x)
+ 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*
x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^5) - 12*a*(1/(c^2*x^2) - 1)^3*e^3/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(
1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(
1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 3*b*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))*e^3/((c^5 + 4*c
^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(
1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) - 6*b*(1/(c^2*x^2) - 1)^3*sqrt(-1/(
c^2*x^2) + 1)*e^3/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4
+ 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^7) + 3*a*
(1/(c^2*x^2) - 1)^4*e^3/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) +
1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8))
*c
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maple [B] time = 0.06, size = 486, normalized size = 2.91 \[ \frac {a \,e^{3} x^{4}}{4}+a \,e^{2} x^{3} d +\frac {3 a e \,x^{2} d^{2}}{2}+a x \,d^{3}+\frac {a \,d^{4}}{4 e}+\frac {b \,e^{3} \mathrm {arcsec}\left (c x \right ) x^{4}}{4}+b \,e^{2} \mathrm {arcsec}\left (c x \right ) x^{3} d +\frac {3 b e \,\mathrm {arcsec}\left (c x \right ) x^{2} d^{2}}{2}+b \,\mathrm {arcsec}\left (c x \right ) x \,d^{3}+\frac {b \,\mathrm {arcsec}\left (c x \right ) d^{4}}{4 e}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {b \,e^{3} x^{3}}{12 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,e^{3} x}{12 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,e^{2} d \,x^{2}}{2 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} d}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b e x \,d^{2}}{2 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b e \,d^{2}}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{3}}{6 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^3*(a+b*arcsec(c*x)),x)
[Out]
1/4*a*e^3*x^4+a*e^2*x^3*d+3/2*a*e*x^2*d^2+a*x*d^3+1/4*a/e*d^4+1/4*b*e^3*arcsec(c*x)*x^4+b*e^2*arcsec(c*x)*x^3*
d+3/2*b*e*arcsec(c*x)*x^2*d^2+b*arcsec(c*x)*x*d^3+1/4*b/e*arcsec(c*x)*d^4+1/4/c*b/e*(c^2*x^2-1)^(1/2)/((c^2*x^
2-1)/c^2/x^2)^(1/2)/x*d^4*arctan(1/(c^2*x^2-1)^(1/2))-1/c^2*b*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*
d^3*ln(c*x+(c^2*x^2-1)^(1/2))-1/12/c*b*e^3/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^3-1/12/c^3*b*e^3/((c^2*x^2-1)/c^2/x^2
)^(1/2)*x-1/2/c*b*e^2/((c^2*x^2-1)/c^2/x^2)^(1/2)*d*x^2+1/2/c^3*b*e^2/((c^2*x^2-1)/c^2/x^2)^(1/2)*d-3/2/c*b*e/
((c^2*x^2-1)/c^2/x^2)^(1/2)*x*d^2+3/2/c^3*b*e/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*d^2-1/2/c^4*b*e^2*(c^2*x^2-1)^(1/2
)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*d*ln(c*x+(c^2*x^2-1)^(1/2))+1/6/c^5*b*e^3/((c^2*x^2-1)/c^2/x^2)^(1/2)/x
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maxima [A] time = 0.33, size = 272, normalized size = 1.63 \[ \frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {3}{2} \, {\left (x^{2} \operatorname {arcsec}\left (c x\right ) - \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d^{2} e + \frac {1}{4} \, {\left (4 \, x^{3} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d e^{2} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arcsec}\left (c x\right ) - \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b e^{3} + a d^{3} x + \frac {{\left (2 \, c x \operatorname {arcsec}\left (c x\right ) - \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) + \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d^{3}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^3*(a+b*arcsec(c*x)),x, algorithm="maxima")
[Out]
1/4*a*e^3*x^4 + a*d*e^2*x^3 + 3/2*a*d^2*e*x^2 + 3/2*(x^2*arcsec(c*x) - x*sqrt(-1/(c^2*x^2) + 1)/c)*b*d^2*e + 1
/4*(4*x^3*arcsec(c*x) - (2*sqrt(-1/(c^2*x^2) + 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + log(sqrt(-1/(c^2*x^2) + 1) +
1)/c^2 - log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^2)/c)*b*d*e^2 + 1/12*(3*x^4*arcsec(c*x) - (c^2*x^3*(-1/(c^2*x^2) +
1)^(3/2) + 3*x*sqrt(-1/(c^2*x^2) + 1))/c^3)*b*e^3 + a*d^3*x + 1/2*(2*c*x*arcsec(c*x) - log(sqrt(-1/(c^2*x^2)
+ 1) + 1) + log(-sqrt(-1/(c^2*x^2) + 1) + 1))*b*d^3/c
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*acos(1/(c*x)))*(d + e*x)^3,x)
[Out]
int((a + b*acos(1/(c*x)))*(d + e*x)^3, x)
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sympy [A] time = 7.88, size = 362, normalized size = 2.17 \[ a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} + b d^{3} x \operatorname {asec}{\left (c x \right )} + \frac {3 b d^{2} e x^{2} \operatorname {asec}{\left (c x \right )}}{2} + b d e^{2} x^{3} \operatorname {asec}{\left (c x \right )} + \frac {b e^{3} x^{4} \operatorname {asec}{\left (c x \right )}}{4} - \frac {b d^{3} \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} - \frac {3 b d^{2} e \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} - \frac {b d e^{2} \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{c} - \frac {b e^{3} \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**3*(a+b*asec(c*x)),x)
[Out]
a*d**3*x + 3*a*d**2*e*x**2/2 + a*d*e**2*x**3 + a*e**3*x**4/4 + b*d**3*x*asec(c*x) + 3*b*d**2*e*x**2*asec(c*x)/
2 + b*d*e**2*x**3*asec(c*x) + b*e**3*x**4*asec(c*x)/4 - b*d**3*Piecewise((acosh(c*x), Abs(c**2*x**2) > 1), (-I
*asin(c*x), True))/c - 3*b*d**2*e*Piecewise((sqrt(c**2*x**2 - 1)/c, Abs(c**2*x**2) > 1), (I*sqrt(-c**2*x**2 +
1)/c, True))/(2*c) - b*d*e**2*Piecewise((x*sqrt(c**2*x**2 - 1)/(2*c) + acosh(c*x)/(2*c**2), Abs(c**2*x**2) > 1
), (-I*c*x**3/(2*sqrt(-c**2*x**2 + 1)) + I*x/(2*c*sqrt(-c**2*x**2 + 1)) - I*asin(c*x)/(2*c**2), True))/c - b*e
**3*Piecewise((x**2*sqrt(c**2*x**2 - 1)/(3*c) + 2*sqrt(c**2*x**2 - 1)/(3*c**3), Abs(c**2*x**2) > 1), (I*x**2*s
qrt(-c**2*x**2 + 1)/(3*c) + 2*I*sqrt(-c**2*x**2 + 1)/(3*c**3), True))/(4*c)
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