3.69 \(\int x^4 (d+e x^2) (a+b \sec ^{-1}(c x)) \, dx\)

Optimal. Leaf size=206 \[ \frac {1}{5} d x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b e x^6 \sqrt {c^2 x^2-1}}{42 c \sqrt {c^2 x^2}}-\frac {b x \left (42 c^2 d+25 e\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{560 c^6 \sqrt {c^2 x^2}}-\frac {b x^2 \sqrt {c^2 x^2-1} \left (42 c^2 d+25 e\right )}{560 c^5 \sqrt {c^2 x^2}}-\frac {b x^4 \sqrt {c^2 x^2-1} \left (42 c^2 d+25 e\right )}{840 c^3 \sqrt {c^2 x^2}} \]

[Out]

1/5*d*x^5*(a+b*arcsec(c*x))+1/7*e*x^7*(a+b*arcsec(c*x))-1/560*b*(42*c^2*d+25*e)*x*arctanh(c*x/(c^2*x^2-1)^(1/2
))/c^6/(c^2*x^2)^(1/2)-1/560*b*(42*c^2*d+25*e)*x^2*(c^2*x^2-1)^(1/2)/c^5/(c^2*x^2)^(1/2)-1/840*b*(42*c^2*d+25*
e)*x^4*(c^2*x^2-1)^(1/2)/c^3/(c^2*x^2)^(1/2)-1/42*b*e*x^6*(c^2*x^2-1)^(1/2)/c/(c^2*x^2)^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {14, 5238, 12, 459, 321, 217, 206} \[ \frac {1}{5} d x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b x^4 \sqrt {c^2 x^2-1} \left (42 c^2 d+25 e\right )}{840 c^3 \sqrt {c^2 x^2}}-\frac {b x^2 \sqrt {c^2 x^2-1} \left (42 c^2 d+25 e\right )}{560 c^5 \sqrt {c^2 x^2}}-\frac {b x \left (42 c^2 d+25 e\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{560 c^6 \sqrt {c^2 x^2}}-\frac {b e x^6 \sqrt {c^2 x^2-1}}{42 c \sqrt {c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^2)*(a + b*ArcSec[c*x]),x]

[Out]

-(b*(42*c^2*d + 25*e)*x^2*Sqrt[-1 + c^2*x^2])/(560*c^5*Sqrt[c^2*x^2]) - (b*(42*c^2*d + 25*e)*x^4*Sqrt[-1 + c^2
*x^2])/(840*c^3*Sqrt[c^2*x^2]) - (b*e*x^6*Sqrt[-1 + c^2*x^2])/(42*c*Sqrt[c^2*x^2]) + (d*x^5*(a + b*ArcSec[c*x]
))/5 + (e*x^7*(a + b*ArcSec[c*x]))/7 - (b*(42*c^2*d + 25*e)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(560*c^6*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 206

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

Rule 217

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 321

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

Rule 459

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

Rule 5238

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSec[c*x], u, x] - Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

\begin {align*} \int x^4 \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^4 \left (7 d+5 e x^2\right )}{35 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {1}{5} d x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^4 \left (7 d+5 e x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{35 \sqrt {c^2 x^2}}\\ &=-\frac {b e x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sec ^{-1}(c x)\right )+\frac {\left (b c \left (-42 d-\frac {25 e}{c^2}\right ) x\right ) \int \frac {x^4}{\sqrt {-1+c^2 x^2}} \, dx}{210 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (42 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sec ^{-1}(c x)\right )+\frac {\left (b \left (-42 d-\frac {25 e}{c^2}\right ) x\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2}} \, dx}{280 c \sqrt {c^2 x^2}}\\ &=-\frac {b \left (42 c^2 d+25 e\right ) x^2 \sqrt {-1+c^2 x^2}}{560 c^5 \sqrt {c^2 x^2}}-\frac {b \left (42 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sec ^{-1}(c x)\right )+\frac {\left (b \left (-42 d-\frac {25 e}{c^2}\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{560 c^3 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (42 c^2 d+25 e\right ) x^2 \sqrt {-1+c^2 x^2}}{560 c^5 \sqrt {c^2 x^2}}-\frac {b \left (42 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sec ^{-1}(c x)\right )+\frac {\left (b \left (-42 d-\frac {25 e}{c^2}\right ) x\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{560 c^3 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (42 c^2 d+25 e\right ) x^2 \sqrt {-1+c^2 x^2}}{560 c^5 \sqrt {c^2 x^2}}-\frac {b \left (42 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \left (42 c^2 d+25 e\right ) x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{560 c^6 \sqrt {c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 141, normalized size = 0.68 \[ \frac {48 a c^7 x^5 \left (7 d+5 e x^2\right )+48 b c^7 x^5 \sec ^{-1}(c x) \left (7 d+5 e x^2\right )-3 b \left (42 c^2 d+25 e\right ) \log \left (x \left (\sqrt {1-\frac {1}{c^2 x^2}}+1\right )\right )-b c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (c^4 \left (84 d x^2+40 e x^4\right )+2 c^2 \left (63 d+25 e x^2\right )+75 e\right )}{1680 c^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^2)*(a + b*ArcSec[c*x]),x]

[Out]

(48*a*c^7*x^5*(7*d + 5*e*x^2) - b*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2*(75*e + 2*c^2*(63*d + 25*e*x^2) + c^4*(84*d*x^
2 + 40*e*x^4)) + 48*b*c^7*x^5*(7*d + 5*e*x^2)*ArcSec[c*x] - 3*b*(42*c^2*d + 25*e)*Log[(1 + Sqrt[1 - 1/(c^2*x^2
)])*x])/(1680*c^7)

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fricas [A]  time = 1.93, size = 192, normalized size = 0.93 \[ \frac {240 \, a c^{7} e x^{7} + 336 \, a c^{7} d x^{5} + 48 \, {\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5} - 7 \, b c^{7} d - 5 \, b c^{7} e\right )} \operatorname {arcsec}\left (c x\right ) + 96 \, {\left (7 \, b c^{7} d + 5 \, b c^{7} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 3 \, {\left (42 \, b c^{2} d + 25 \, b e\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (40 \, b c^{5} e x^{5} + 2 \, {\left (42 \, b c^{5} d + 25 \, b c^{3} e\right )} x^{3} + 3 \, {\left (42 \, b c^{3} d + 25 \, b c e\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{1680 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)*(a+b*arcsec(c*x)),x, algorithm="fricas")

[Out]

1/1680*(240*a*c^7*e*x^7 + 336*a*c^7*d*x^5 + 48*(5*b*c^7*e*x^7 + 7*b*c^7*d*x^5 - 7*b*c^7*d - 5*b*c^7*e)*arcsec(
c*x) + 96*(7*b*c^7*d + 5*b*c^7*e)*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 3*(42*b*c^2*d + 25*b*e)*log(-c*x + sqrt(c
^2*x^2 - 1)) - (40*b*c^5*e*x^5 + 2*(42*b*c^5*d + 25*b*c^3*e)*x^3 + 3*(42*b*c^3*d + 25*b*c*e)*x)*sqrt(c^2*x^2 -
1))/c^7

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)*(a+b*arcsec(c*x)),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.06, size = 338, normalized size = 1.64 \[ \frac {a e \,x^{7}}{7}+\frac {a \,x^{5} d}{5}+\frac {b \,\mathrm {arcsec}\left (c x \right ) e \,x^{7}}{7}+\frac {b \,\mathrm {arcsec}\left (c x \right ) x^{5} d}{5}-\frac {b \,x^{6} e}{42 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,x^{4} e}{168 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,x^{4} d}{20 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,x^{2} d}{40 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \,x^{2} e}{336 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b d}{40 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{6} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {5 b e}{112 c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{8} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x^2+d)*(a+b*arcsec(c*x)),x)

[Out]

1/7*a*e*x^7+1/5*a*x^5*d+1/7*b*arcsec(c*x)*e*x^7+1/5*b*arcsec(c*x)*x^5*d-1/42/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x
^6*e-1/168/c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^4*e-1/20/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^4*d-1/40/c^3*b/((c^2
*x^2-1)/c^2/x^2)^(1/2)*x^2*d-5/336/c^5*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2*e+3/40/c^5*b/((c^2*x^2-1)/c^2/x^2)^(1
/2)*d-3/40/c^6*b/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*(c^2*x^2-1)^(1/2)*d*ln(c*x+(c^2*x^2-1)^(1/2))+5/112/c^7*b/((c^2
*x^2-1)/c^2/x^2)^(1/2)*e-5/112/c^8*b*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*e*ln(c*x+(c^2*x^2-1)^(1/2
))

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maxima [A]  time = 0.33, size = 296, normalized size = 1.44 \[ \frac {1}{7} \, a e x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arcsec}\left (c x\right ) + \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, {\left (15 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)*(a+b*arcsec(c*x)),x, algorithm="maxima")

[Out]

1/7*a*e*x^7 + 1/5*a*d*x^5 + 1/80*(16*x^5*arcsec(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*d + 1/672*(96*x^7*arcsec(c*x) - (2*(15*(-1/(c^2*x^2) + 1)^(5/2) - 40*(-1
/(c^2*x^2) + 1)^(3/2) + 33*sqrt(-1/(c^2*x^2) + 1))/(c^6*(1/(c^2*x^2) - 1)^3 + 3*c^6*(1/(c^2*x^2) - 1)^2 + 3*c^
6*(1/(c^2*x^2) - 1) + c^6) + 15*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^6 - 15*log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^6)/
c)*b*e

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d + e*x^2)*(a + b*acos(1/(c*x))),x)

[Out]

int(x^4*(d + e*x^2)*(a + b*acos(1/(c*x))), x)

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sympy [A]  time = 11.68, size = 408, normalized size = 1.98 \[ \frac {a d x^{5}}{5} + \frac {a e x^{7}}{7} + \frac {b d x^{5} \operatorname {asec}{\left (c x \right )}}{5} + \frac {b e x^{7} \operatorname {asec}{\left (c x \right )}}{7} - \frac {b d \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} - \frac {b e \left (\begin {cases} \frac {c x^{7}}{6 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{5}}{24 c \sqrt {c^{2} x^{2} - 1}} + \frac {5 x^{3}}{48 c^{3} \sqrt {c^{2} x^{2} - 1}} - \frac {5 x}{16 c^{5} \sqrt {c^{2} x^{2} - 1}} + \frac {5 \operatorname {acosh}{\left (c x \right )}}{16 c^{6}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{7}}{6 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{5}}{24 c \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i x^{3}}{48 c^{3} \sqrt {- c^{2} x^{2} + 1}} + \frac {5 i x}{16 c^{5} \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i \operatorname {asin}{\left (c x \right )}}{16 c^{6}} & \text {otherwise} \end {cases}\right )}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x**2+d)*(a+b*asec(c*x)),x)

[Out]

a*d*x**5/5 + a*e*x**7/7 + b*d*x**5*asec(c*x)/5 + b*e*x**7*asec(c*x)/7 - b*d*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) - b*e*Piecewise((c*x**7/(6*sqrt(c**2*x**2 - 1)) + x**5/(24*
c*sqrt(c**2*x**2 - 1)) + 5*x**3/(48*c**3*sqrt(c**2*x**2 - 1)) - 5*x/(16*c**5*sqrt(c**2*x**2 - 1)) + 5*acosh(c*
x)/(16*c**6), Abs(c**2*x**2) > 1), (-I*c*x**7/(6*sqrt(-c**2*x**2 + 1)) - I*x**5/(24*c*sqrt(-c**2*x**2 + 1)) -
5*I*x**3/(48*c**3*sqrt(-c**2*x**2 + 1)) + 5*I*x/(16*c**5*sqrt(-c**2*x**2 + 1)) - 5*I*asin(c*x)/(16*c**6), True
))/(7*c)

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