3.1.82 \(\int (d+e x^2)^2 (a+b \sec ^{-1}(c x)) \, dx\) [82]

Optimal. Leaf size=191 \[ -\frac {b e \left (40 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+d^2 x \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{120 c^4 \sqrt {c^2 x^2}} \]

[Out]

d^2*x*(a+b*arcsec(c*x))+2/3*d*e*x^3*(a+b*arcsec(c*x))+1/5*e^2*x^5*(a+b*arcsec(c*x))-1/120*b*(120*c^4*d^2+40*c^
2*d*e+9*e^2)*x*arctanh(c*x/(c^2*x^2-1)^(1/2))/c^4/(c^2*x^2)^(1/2)-1/120*b*e*(40*c^2*d+9*e)*x^2*(c^2*x^2-1)^(1/
2)/c^3/(c^2*x^2)^(1/2)-1/20*b*e^2*x^4*(c^2*x^2-1)^(1/2)/c/(c^2*x^2)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {200, 5336, 12, 1173, 396, 223, 212} \begin {gather*} d^2 x \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sec ^{-1}(c x)\right )-\frac {b e^2 x^4 \sqrt {c^2 x^2-1}}{20 c \sqrt {c^2 x^2}}-\frac {b x \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{120 c^4 \sqrt {c^2 x^2}}-\frac {b e x^2 \sqrt {c^2 x^2-1} \left (40 c^2 d+9 e\right )}{120 c^3 \sqrt {c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^2*(a + b*ArcSec[c*x]),x]

[Out]

-1/120*(b*e*(40*c^2*d + 9*e)*x^2*Sqrt[-1 + c^2*x^2])/(c^3*Sqrt[c^2*x^2]) - (b*e^2*x^4*Sqrt[-1 + c^2*x^2])/(20*
c*Sqrt[c^2*x^2]) + d^2*x*(a + b*ArcSec[c*x]) + (2*d*e*x^3*(a + b*ArcSec[c*x]))/3 + (e^2*x^5*(a + b*ArcSec[c*x]
))/5 - (b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(120*c^4*Sqrt[c^2*x^2])

Rule 12

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

Rule 200

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 1173

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*x^(4*p - 1)*(
(d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1))), x] + Dist[1/(e*(4*p + 2*q + 1)), Int[(d + e*x^2)^q*ExpandToSum[e*(4*
p + 2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], x], x] /
; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] &&  !LtQ[
q, -1]

Rule 5336

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

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {15 d^2+10 d e x^2+3 e^2 x^4}{15 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=d^2 x \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {15 d^2+10 d e x^2+3 e^2 x^4}{\sqrt {-1+c^2 x^2}} \, dx}{15 \sqrt {c^2 x^2}}\\ &=-\frac {b e^2 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+d^2 x \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b x) \int \frac {60 c^2 d^2+e \left (40 c^2 d+9 e\right ) x^2}{\sqrt {-1+c^2 x^2}} \, dx}{60 c \sqrt {c^2 x^2}}\\ &=-\frac {b e \left (40 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+d^2 x \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {\left (b \left (-120 c^4 d^2-e \left (40 c^2 d+9 e\right )\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{120 c^3 \sqrt {c^2 x^2}}\\ &=-\frac {b e \left (40 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+d^2 x \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {\left (b \left (-120 c^4 d^2-e \left (40 c^2 d+9 e\right )\right ) x\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{120 c^3 \sqrt {c^2 x^2}}\\ &=-\frac {b e \left (40 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+d^2 x \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{120 c^4 \sqrt {c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 153, normalized size = 0.80 \begin {gather*} \frac {c^2 x \left (8 a c^3 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-b e \sqrt {1-\frac {1}{c^2 x^2}} x \left (9 e+c^2 \left (40 d+6 e x^2\right )\right )\right )+8 b c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \sec ^{-1}(c x)-b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{120 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^2*(a + b*ArcSec[c*x]),x]

[Out]

(c^2*x*(8*a*c^3*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(9*e + c^2*(40*d + 6*e*x^2)))
+ 8*b*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcSec[c*x] - b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*Log[(1 + Sqrt
[1 - 1/(c^2*x^2)])*x])/(120*c^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(358\) vs. \(2(169)=338\).
time = 0.14, size = 359, normalized size = 1.88

method	result	size
derivativedivides	\(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+b \,\mathrm {arcsec}\left (c x \right ) d^{2} c x +\frac {2 b c \,\mathrm {arcsec}\left (c x \right ) d e \,x^{3}}{3}+\frac {b c \,\mathrm {arcsec}\left (c x \right ) e^{2} x^{5}}{5}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {b \left (c^{2} x^{2}-1\right ) d e}{3 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{20 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {3 b \left (c^{2} x^{2}-1\right ) e^{2}}{40 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\)	\(359\)
default	\(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+b \,\mathrm {arcsec}\left (c x \right ) d^{2} c x +\frac {2 b c \,\mathrm {arcsec}\left (c x \right ) d e \,x^{3}}{3}+\frac {b c \,\mathrm {arcsec}\left (c x \right ) e^{2} x^{5}}{5}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {b \left (c^{2} x^{2}-1\right ) d e}{3 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{20 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {3 b \left (c^{2} x^{2}-1\right ) e^{2}}{40 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\)	\(359\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^2*(a+b*arcsec(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c*(a/c^4*(d^2*c^5*x+2/3*d*c^5*e*x^3+1/5*e^2*c^5*x^5)+b*arcsec(c*x)*d^2*c*x+2/3*b*c*arcsec(c*x)*d*e*x^3+1/5*b
*c*arcsec(c*x)*e^2*x^5-b*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x*d^2*ln(c*x+(c^2*x^2-1)^(1/2))-1/3*b
/c^2*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*d*e-1/20*b/c^2*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2*e^2-1/
3*b/c^3*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*d*e*ln(c*x+(c^2*x^2-1)^(1/2))-3/40*b/c^4*(c^2*x^2-1)/(
(c^2*x^2-1)/c^2/x^2)^(1/2)*e^2-3/40*b/c^5*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*e^2*ln(c*x+(c^2*x^2-
1)^(1/2)))

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Maxima [A]
time = 0.28, size = 296, normalized size = 1.55 \begin {gather*} \frac {1}{5} \, a x^{5} e^{2} + \frac {2}{3} \, a d x^{3} e + a d^{2} x + \frac {1}{6} \, {\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 + \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^{2}}{2 \, c} + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arcsec}\left (c x\right ) + \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arcsec(c*x)),x, algorithm="maxima")

[Out]

1/5*a*x^5*e^2 + 2/3*a*d*x^3*e + a*d^2*x + 1/6*(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 + 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^2/c + 1/80*(16*x^5*arc
sec(c*x) + (2*(3*(-1/(c^2*x^2) + 1)^(3/2) - 5*sqrt(-1/(c^2*x^2) + 1))/(c^4*(1/(c^2*x^2) - 1)^2 + 2*c^4*(1/(c^2
*x^2) - 1) + c^4) - 3*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 + 3*log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^4)/c)*b*e^2

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Fricas [A]
time = 3.49, size = 236, normalized size = 1.24 \begin {gather*} \frac {24 \, a c^{5} x^{5} e^{2} + 80 \, a c^{5} d x^{3} e + 120 \, a c^{5} d^{2} x + 8 \, {\left (15 \, b c^{5} d^{2} x - 15 \, b c^{5} d^{2} + 3 \, {\left (b c^{5} x^{5} - b c^{5}\right )} e^{2} + 10 \, {\left (b c^{5} d x^{3} - b c^{5} d\right )} e\right )} \operatorname {arcsec}\left (c x\right ) + 16 \, {\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (120 \, b c^{4} d^{2} + 40 \, b c^{2} d e + 9 \, b e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (40 \, b c^{3} d x e + 3 \, {\left (2 \, b c^{3} x^{3} + 3 \, b c x\right )} e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{120 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arcsec(c*x)),x, algorithm="fricas")

[Out]

1/120*(24*a*c^5*x^5*e^2 + 80*a*c^5*d*x^3*e + 120*a*c^5*d^2*x + 8*(15*b*c^5*d^2*x - 15*b*c^5*d^2 + 3*(b*c^5*x^5
- b*c^5)*e^2 + 10*(b*c^5*d*x^3 - b*c^5*d)*e)*arcsec(c*x) + 16*(15*b*c^5*d^2 + 10*b*c^5*d*e + 3*b*c^5*e^2)*arc
tan(-c*x + sqrt(c^2*x^2 - 1)) + (120*b*c^4*d^2 + 40*b*c^2*d*e + 9*b*e^2)*log(-c*x + sqrt(c^2*x^2 - 1)) - (40*b
*c^3*d*x*e + 3*(2*b*c^3*x^3 + 3*b*c*x)*e^2)*sqrt(c^2*x^2 - 1))/c^5

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Sympy [A]
time = 7.88, size = 355, normalized size = 1.86 \begin {gather*} a d^{2} x + \frac {2 a d e x^{3}}{3} + \frac {a e^{2} x^{5}}{5} + b d^{2} x \operatorname {asec}{\left (c x \right )} + \frac {2 b d e x^{3} \operatorname {asec}{\left (c x \right )}}{3} + \frac {b e^{2} x^{5} \operatorname {asec}{\left (c x \right )}}{5} - \frac {b d^{2} \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 {2 b d e \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 )}{3 c} - \frac {b e^{2} \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**2*(a+b*asec(c*x)),x)

[Out]

a*d**2*x + 2*a*d*e*x**3/3 + a*e**2*x**5/5 + b*d**2*x*asec(c*x) + 2*b*d*e*x**3*asec(c*x)/3 + b*e**2*x**5*asec(c
*x)/5 - b*d**2*Piecewise((acosh(c*x), Abs(c**2*x**2) > 1), (-I*asin(c*x), True))/c - 2*b*d*e*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))/(3*c) - b*e**2*Piecewise((c*x**5/(4*sqrt(c**2*x**2 - 1
)) + x**3/(8*c*sqrt(c**2*x**2 - 1)) - 3*x/(8*c**3*sqrt(c**2*x**2 - 1)) + 3*acosh(c*x)/(8*c**4), Abs(c**2*x**2)
> 1), (-I*c*x**5/(4*sqrt(-c**2*x**2 + 1)) - I*x**3/(8*c*sqrt(-c**2*x**2 + 1)) + 3*I*x/(8*c**3*sqrt(-c**2*x**2
+ 1)) - 3*I*asin(c*x)/(8*c**4), True))/(5*c)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 14166 vs. \(2 (169) = 338\).
time = 4.82, size = 14166, normalized size = 74.17 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arcsec(c*x)),x, algorithm="giac")

[Out]

1/120*(120*b*c^4*d^2*arccos(1/(c*x))/(c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)
^2/(1/(c*x) + 1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 +
c^6*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10) - 120*b*c^4*d^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(c^6
+ 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 10*c^6*(1/(c^2*x^2) -
1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10)
+ 120*b*c^4*d^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2
+ 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2)
- 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10) + 120*a*c^4*d^2/(c^6 + 5*c^6*(1/(c^2*x^2)
- 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6
+ 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10) + 360*b*c^4*d^2*(1/(c
^2*x^2) - 1)*arccos(1/(c*x))/((c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(
c*x) + 1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/
(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10)*(1/(c*x) + 1)^2) - 600*b*c^4*d^2*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c^2*x^2
) + 1) + 1/(c*x) + 1))/((c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) +
1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x
^2) - 1)^5/(1/(c*x) + 1)^10)*(1/(c*x) + 1)^2) + 600*b*c^4*d^2*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c^2*x^2) + 1)
- 1/(c*x) - 1))/((c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4
+ 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x^2) -
1)^5/(1/(c*x) + 1)^10)*(1/(c*x) + 1)^2) + 360*a*c^4*d^2*(1/(c^2*x^2) - 1)/((c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(
c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*
(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10)*(1/(c*x) + 1)^2) + 80*b*c^2*d*
e*arccos(1/(c*x))/(c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4
+ 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x^2) -
1)^5/(1/(c*x) + 1)^10) + 240*b*c^4*d^2*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))/((c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/
(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6
*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10)*(1/(c*x) + 1)^4) - 40*b*c^2*d
*e*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(
c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c
*x) + 1)^8 + c^6*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10) - 1200*b*c^4*d^2*(1/(c^2*x^2) - 1)^2*log(abs(sqrt(-1/(c
^2*x^2) + 1) + 1/(c*x) + 1))/((c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(
c*x) + 1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/
(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10)*(1/(c*x) + 1)^4) + 40*b*c^2*d*e*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) -
1))/(c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 10*c^6*(1/(c
^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x^2) - 1)^5/(1/(c*x)
+ 1)^10) + 1200*b*c^4*d^2*(1/(c^2*x^2) - 1)^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^6 + 5*c^6*(1/
(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c
*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10)*(1/(c*x) +
1)^4) + 80*a*c^2*d*e/(c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)
^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x^2)
- 1)^5/(1/(c*x) + 1)^10) + 240*a*c^4*d^2*(1/(c^2*x^2) - 1)^2/((c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2
+ 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2)
- 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10)*(1/(c*x) + 1)^4) - 80*b*c^2*d*e*(1/(c^2*x^
2) - 1)*arccos(1/(c*x))/((c^6 + 5*c^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(1/(c^2*x^2) - 1)^2/(1/(c*x)
+ 1)^4 + 10*c^6*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 5*c^6*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + c^6*(1/(c^2*
x^2) - 1)^5/(1/(c*x) + 1)^10)*(1/(c*x) + 1)^2) - 240*b*c^4*d^2*(1/(c^2*x^2) - 1)^3*arccos(1/(c*x))/((c^6 + 5*c
^6*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 10*c^6*(...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2)^2*(a + b*acos(1/(c*x))),x)

[Out]

int((d + e*x^2)^2*(a + b*acos(1/(c*x))), x)

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