3.1.69 \(\int x^4 (d+e x^2) (a+b \sec ^{-1}(c x)) \, dx\) [69]

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

[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 = 0.09, 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, 5346, 12, 470, 327, 223, 212} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/560*(b*(42*c^2*d + 25*e)*x^2*Sqrt[-1 + c^2*x^2])/(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*Sq
rt[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 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 327

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 470

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 5346

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 ) \text {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 = 0.17, size = 141, normalized size = 0.68 \begin {gather*} \frac {48 a c^7 x^5 \left (7 d+5 e x^2\right )-b c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (75 e+2 c^2 \left (63 d+25 e x^2\right )+c^4 \left (84 d x^2+40 e x^4\right )\right )+48 b c^7 x^5 \left (7 d+5 e x^2\right ) \sec ^{-1}(c x)-3 b \left (42 c^2 d+25 e\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{1680 c^7} \end {gather*}

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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Maple [A]
time = 0.22, size = 341, normalized size = 1.66

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \,\mathrm {arcsec}\left (c x \right ) d \,c^{5} x^{5}}{5}+\frac {b \,c^{5} \mathrm {arcsec}\left (c x \right ) e \,x^{7}}{7}-\frac {b \left (c^{2} x^{2}-1\right ) c^{2} x^{2} d}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,c^{2} \left (c^{2} x^{2}-1\right ) x^{4} e}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \left (c^{2} x^{2}-1\right ) d}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e}{168 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {5 b \left (c^{2} x^{2}-1\right ) e}{112 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{5}}\)	\(341\)
default	\(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \,\mathrm {arcsec}\left (c x \right ) d \,c^{5} x^{5}}{5}+\frac {b \,c^{5} \mathrm {arcsec}\left (c x \right ) e \,x^{7}}{7}-\frac {b \left (c^{2} x^{2}-1\right ) c^{2} x^{2} d}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,c^{2} \left (c^{2} x^{2}-1\right ) x^{4} e}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \left (c^{2} x^{2}-1\right ) d}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e}{168 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {5 b \left (c^{2} x^{2}-1\right ) e}{112 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{5}}\)	\(341\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^5*(a/c^2*(1/5*d*c^7*x^5+1/7*e*c^7*x^7)+1/5*b*arcsec(c*x)*d*c^5*x^5+1/7*b*c^5*arcsec(c*x)*e*x^7-1/20*b*(c^2
*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*c^2*x^2*d-1/42*b*c^2*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^4*e-3/40*b*
(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*d-5/168*b*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2*e-3/40*b*(c^2*x^
2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x*d*ln(c*x+(c^2*x^2-1)^(1/2))-5/112*b/c^2*(c^2*x^2-1)/((c^2*x^2-1)/c^
2/x^2)^(1/2)*e-5/112*b/c^3*(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 = 0.27, size = 298, normalized size = 1.45 \begin {gather*} \frac {1}{7} \, a x^{7} e + \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 \end {gather*}

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*x^7*e + 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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Fricas [A]
time = 4.56, size = 197, normalized size = 0.96 \begin {gather*} \frac {240 \, a c^{7} x^{7} e + 336 \, a c^{7} d x^{5} + 48 \, {\left (7 \, b c^{7} d x^{5} - 7 \, b c^{7} d + 5 \, {\left (b c^{7} x^{7} - b c^{7}\right )} 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 (84 \, b c^{5} d x^{3} + 126 \, b c^{3} d x + 5 \, {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} e\right )} \sqrt {c^{2} x^{2} - 1}}{1680 \, c^{7}} \end {gather*}

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*x^7*e + 336*a*c^7*d*x^5 + 48*(7*b*c^7*d*x^5 - 7*b*c^7*d + 5*(b*c^7*x^7 - 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)) - (84*b*c^5*d*x^3 + 126*b*c^3*d*x + 5*(8*b*c^5*x^5 + 10*b*c^3*x^3 + 15*b*c*x)*e)*sqrt(c^2*x^2 - 1))
/c^7

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Sympy [A]
time = 13.35, size = 408, normalized size = 1.98 \begin {gather*} \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} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 17474 vs. \(2 (178) = 356\).
time = 2.81, size = 17474, normalized size = 84.83 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/1680*(336*b*c^2*d*arccos(1/(c*x))/(c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^
2/(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 +
21*c^8*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) -
1)^7/(1/(c*x) + 1)^14) - 126*b*c^2*d*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(c^8 + 7*c^8*(1/(c^2*x^2)
- 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6
+ 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^
2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14) + 126*b*c^2*d*log(abs(sqrt(-1/(c^2*x^2)
+ 1) - 1/(c*x) - 1))/(c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1
)^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8*(1/(c^2
*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x)
+ 1)^14) + 336*a*c^2*d/(c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) +
1)^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8*(1/(c^
2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x)
+ 1)^14) - 1008*b*c^2*d*(1/(c^2*x^2) - 1)*arccos(1/(c*x))/((c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2
1*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) -
1)^4/(1/(c*x) + 1)^8 + 21*c^8*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^
12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14)*(1/(c*x) + 1)^2) - 882*b*c^2*d*(1/(c^2*x^2) - 1)*log(abs(sqrt(-
1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/
(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21
*c^8*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)
^7/(1/(c*x) + 1)^14)*(1/(c*x) + 1)^2) + 882*b*c^2*d*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x)
- 1))/((c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(
1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8*(1/(c^2*x^2) - 1)^5/(
1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14)*(1/(c
*x) + 1)^2) - 420*b*c^2*d*sqrt(-1/(c^2*x^2) + 1)/((c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(
c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(
c*x) + 1)^8 + 21*c^8*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(
1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14)*(1/(c*x) + 1)) - 1008*a*c^2*d*(1/(c^2*x^2) - 1)/((c^8 + 7*c^8*(1/(c^2*x^2
) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)
^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*
x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14)*(1/(c*x) + 1)^2) + 240*b*e*arccos(1/(
c*x))/(c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(1/
(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8*(1/(c^2*x^2) - 1)^5/(1/
(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14) + 336*b
*c^2*d*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))/((c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^
2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) +
1)^8 + 21*c^8*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2
*x^2) - 1)^7/(1/(c*x) + 1)^14)*(1/(c*x) + 1)^4) - 75*b*e*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(c^8 +
7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) -
1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^1
0 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14) - 2646*b*c^2*d*(1/(
c^2*x^2) - 1)^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2
+ 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^
2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) +
1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14)*(1/(c*x) + 1)^4) + 75*b*e*log(abs(sqrt(-1/(c^2*x^2) + 1) -
1/(c*x) - 1))/(c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,\left (e\,x^2+d\right )\,\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(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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