3.1.18 \(\int x^4 \sec ^{-1}(a+b x) \, dx\) [18]

Optimal. Leaf size=197 \[ \frac {a \left (20+53 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{30 b^5}+\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}-\frac {\left (9+58 a^2\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\left (3+40 a^2+40 a^4\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{40 b^5} \]

[Out]

1/5*a^5*arcsec(b*x+a)/b^5+1/5*x^5*arcsec(b*x+a)-1/40*(40*a^4+40*a^2+3)*arctanh((1-1/(b*x+a)^2)^(1/2))/b^5+1/30
*a*(53*a^2+20)*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^5+11/60*a*x^2*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^3-1/20*x^3*(b*x+a
)*(1-1/(b*x+a)^2)^(1/2)/b^2-1/120*(58*a^2+9)*(b*x+a)^2*(1-1/(b*x+a)^2)^(1/2)/b^5

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Rubi [A]
time = 0.18, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5366, 4511, 3867, 4141, 4133, 3855, 3852, 8} \begin {gather*} \frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}+\frac {\left (53 a^2+20\right ) a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{30 b^5}-\frac {\left (58 a^2+9\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}-\frac {\left (40 a^4+40 a^2+3\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{40 b^5}+\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}+\frac {1}{5} x^5 \sec ^{-1}(a+b x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*ArcSec[a + b*x],x]

[Out]

(a*(20 + 53*a^2)*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(30*b^5) + (11*a*x^2*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/
(60*b^3) - (x^3*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(20*b^2) - ((9 + 58*a^2)*(a + b*x)^2*Sqrt[1 - (a + b*x)^(-
2)])/(120*b^5) + (a^5*ArcSec[a + b*x])/(5*b^5) + (x^5*ArcSec[a + b*x])/5 - ((3 + 40*a^2 + 40*a^4)*ArcTanh[Sqrt
[1 - (a + b*x)^(-2)]])/(40*b^5)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3867

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*Cot[c + d*x]*((a + b*Csc[c + d*x])^(
n - 2)/(d*(n - 1))), x] + Dist[1/(n - 1), Int[(a + b*Csc[c + d*x])^(n - 3)*Simp[a^3*(n - 1) + (b*(b^2*(n - 2)
+ 3*a^2*(n - 1)))*Csc[c + d*x] + (a*b^2*(3*n - 4))*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ
[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]

Rule 4133

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*(Cot[e + f*x]/(2*f)), x] + Dist[1/2, Int[Simp[2*A*a + (2*B*a + b
*(2*A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x]

Rule 4141

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_.), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Dist[1/(m + 1), I
nt[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*A*(m + 1) + ((A*b + a*B)*(m + 1) + b*C*m)*Csc[e + f*x] + (b*B*(m + 1) +
a*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]

Rule 4511

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

Rule 5366

Int[((a_.) + ArcSec[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d^(m + 1),
Subst[Int[(a + b*x)^p*Sec[x]*Tan[x]*(d*e - c*f + f*Sec[x])^m, x], x, ArcSec[c + d*x]], x] /; FreeQ[{a, b, c,
d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int x^4 \sec ^{-1}(a+b x) \, dx &=\frac {\text {Subst}\left (\int x \sec (x) (-a+\sec (x))^4 \tan (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^5}\\ &=\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\text {Subst}\left (\int (-a+\sec (x))^5 \, dx,x,\sec ^{-1}(a+b x)\right )}{5 b^5}\\ &=-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\text {Subst}\left (\int (-a+\sec (x))^2 \left (-4 a^3+3 \left (1+4 a^2\right ) \sec (x)-11 a \sec ^2(x)\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{20 b^5}\\ &=\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\text {Subst}\left (\int (-a+\sec (x)) \left (12 a^4-a \left (31+48 a^2\right ) \sec (x)+\left (9+58 a^2\right ) \sec ^2(x)\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{60 b^5}\\ &=\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}-\frac {\left (9+58 a^2\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\text {Subst}\left (\int \left (-24 a^5+3 \left (3+40 a^2+40 a^4\right ) \sec (x)-4 a \left (20+53 a^2\right ) \sec ^2(x)\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{120 b^5}\\ &=\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}-\frac {\left (9+58 a^2\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)+\frac {\left (a \left (20+53 a^2\right )\right ) \text {Subst}\left (\int \sec ^2(x) \, dx,x,\sec ^{-1}(a+b x)\right )}{30 b^5}-\frac {\left (3+40 a^2+40 a^4\right ) \text {Subst}\left (\int \sec (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{40 b^5}\\ &=\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}-\frac {\left (9+58 a^2\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\left (3+40 a^2+40 a^4\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{40 b^5}-\frac {\left (a \left (20+53 a^2\right )\right ) \text {Subst}\left (\int 1 \, dx,x,-(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )}{30 b^5}\\ &=\frac {a \left (20+53 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{30 b^5}+\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}-\frac {\left (9+58 a^2\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\left (3+40 a^2+40 a^4\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{40 b^5}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 173, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (a^2 \left (71+154 a^2\right )+2 a \left (31+48 a^2\right ) b x-9 \left (1+4 a^2\right ) b^2 x^2+16 a b^3 x^3-6 b^4 x^4\right )+24 b^5 x^5 \sec ^{-1}(a+b x)-24 a^5 \text {ArcSin}\left (\frac {1}{a+b x}\right )-3 \left (3+40 a^2+40 a^4\right ) \log \left ((a+b x) \left (1+\sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{120 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcSec[a + b*x],x]

[Out]

(Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(a^2*(71 + 154*a^2) + 2*a*(31 + 48*a^2)*b*x - 9*(1 + 4*a^2)*
b^2*x^2 + 16*a*b^3*x^3 - 6*b^4*x^4) + 24*b^5*x^5*ArcSec[a + b*x] - 24*a^5*ArcSin[(a + b*x)^(-1)] - 3*(3 + 40*a
^2 + 40*a^4)*Log[(a + b*x)*(1 + Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])])/(120*b^5)

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Maple [A]
time = 0.14, size = 329, normalized size = 1.67

method	result	size
derivativedivides	\(\frac {-\frac {\mathrm {arcsec}\left (b x +a \right ) a^{5}}{5}+\mathrm {arcsec}\left (b x +a \right ) a^{4} \left (b x +a \right )-2 \,\mathrm {arcsec}\left (b x +a \right ) a^{3} \left (b x +a \right )^{2}+2 \,\mathrm {arcsec}\left (b x +a \right ) a^{2} \left (b x +a \right )^{3}-\mathrm {arcsec}\left (b x +a \right ) a \left (b x +a \right )^{4}+\frac {\mathrm {arcsec}\left (b x +a \right ) \left (b x +a \right )^{5}}{5}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (24 a^{5} \arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right )+120 a^{4} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-240 a^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-40 a \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+6 \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-80 a \sqrt {\left (b x +a \right )^{2}-1}+9 \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}+9 \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )\right )}{120 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{5}}\)	\(329\)
default	\(\frac {-\frac {\mathrm {arcsec}\left (b x +a \right ) a^{5}}{5}+\mathrm {arcsec}\left (b x +a \right ) a^{4} \left (b x +a \right )-2 \,\mathrm {arcsec}\left (b x +a \right ) a^{3} \left (b x +a \right )^{2}+2 \,\mathrm {arcsec}\left (b x +a \right ) a^{2} \left (b x +a \right )^{3}-\mathrm {arcsec}\left (b x +a \right ) a \left (b x +a \right )^{4}+\frac {\mathrm {arcsec}\left (b x +a \right ) \left (b x +a \right )^{5}}{5}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (24 a^{5} \arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right )+120 a^{4} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-240 a^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-40 a \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+6 \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-80 a \sqrt {\left (b x +a \right )^{2}-1}+9 \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}+9 \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )\right )}{120 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{5}}\)	\(329\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arcsec(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/b^5*(-1/5*arcsec(b*x+a)*a^5+arcsec(b*x+a)*a^4*(b*x+a)-2*arcsec(b*x+a)*a^3*(b*x+a)^2+2*arcsec(b*x+a)*a^2*(b*x
+a)^3-arcsec(b*x+a)*a*(b*x+a)^4+1/5*arcsec(b*x+a)*(b*x+a)^5-1/120*((b*x+a)^2-1)^(1/2)*(24*a^5*arctan(1/((b*x+a
)^2-1)^(1/2))+120*a^4*ln(b*x+a+((b*x+a)^2-1)^(1/2))-240*a^3*((b*x+a)^2-1)^(1/2)+120*a^2*(b*x+a)*((b*x+a)^2-1)^
(1/2)-40*a*(b*x+a)^2*((b*x+a)^2-1)^(1/2)+6*(b*x+a)^3*((b*x+a)^2-1)^(1/2)+120*a^2*ln(b*x+a+((b*x+a)^2-1)^(1/2))
-80*a*((b*x+a)^2-1)^(1/2)+9*(b*x+a)*((b*x+a)^2-1)^(1/2)+9*ln(b*x+a+((b*x+a)^2-1)^(1/2)))/(((b*x+a)^2-1)/(b*x+a
)^2)^(1/2)/(b*x+a))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*arctan(sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - integrate(1/5*(b^2*x^6 + a*b*x^5)*e^(1/2*log(b*x + a + 1
) + 1/2*log(b*x + a - 1))/(b^2*x^2 + 2*a*b*x + a^2 + (b^2*x^2 + 2*a*b*x + a^2 - 1)*e^(log(b*x + a + 1) + log(b
*x + a - 1)) - 1), x)

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Fricas [A]
time = 2.60, size = 152, normalized size = 0.77 \begin {gather*} \frac {24 \, b^{5} x^{5} \operatorname {arcsec}\left (b x + a\right ) + 48 \, a^{5} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 3 \, {\left (40 \, a^{4} + 40 \, a^{2} + 3\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (6 \, b^{3} x^{3} - 22 \, a b^{2} x^{2} - 154 \, a^{3} + {\left (58 \, a^{2} + 9\right )} b x - 71 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{120 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*b^5*x^5*arcsec(b*x + a) + 48*a^5*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + 3*(40*a^4 +
40*a^2 + 3)*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) - (6*b^3*x^3 - 22*a*b^2*x^2 - 154*a^3 + (58*a^2
+ 9)*b*x - 71*a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/b^5

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \operatorname {asec}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*asec(b*x+a),x)

[Out]

Integral(x**4*asec(a + b*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (173) = 346\).
time = 0.47, size = 409, normalized size = 2.08 \begin {gather*} -\frac {1}{960} \, b {\left (\frac {192 \, {\left (b x + a\right )}^{5} {\left (\frac {5 \, a}{b x + a} - \frac {10 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac {10 \, a^{3}}{{\left (b x + a\right )}^{3}} - \frac {5 \, a^{4}}{{\left (b x + a\right )}^{4}} - 1\right )} \arccos \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{6}} - \frac {3 \, {\left (b x + a\right )}^{4} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4} + 40 \, {\left (b x + a\right )}^{3} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 240 \, {\left (b x + a\right )}^{2} a^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 960 \, {\left (b x + a\right )} a^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 24 \, {\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 360 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 24 \, {\left (40 \, a^{4} + 40 \, a^{2} + 3\right )} \log \left (-{\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} {\left | b x + a \right |}\right ) - \frac {120 \, {\left (8 \, a^{3} + 3 \, a\right )} {\left (b x + a\right )}^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 24 \, {\left (10 \, a^{2} + 1\right )} {\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 40 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 3}{{\left (b x + a\right )}^{4} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4}}}{b^{6}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/960*b*(192*(b*x + a)^5*(5*a/(b*x + a) - 10*a^2/(b*x + a)^2 + 10*a^3/(b*x + a)^3 - 5*a^4/(b*x + a)^4 - 1)*ar
ccos(-1/((b*x + a)*(a/(b*x + a) - 1) - a))/b^6 - (3*(b*x + a)^4*(sqrt(-1/(b*x + a)^2 + 1) - 1)^4 + 40*(b*x + a
)^3*a*(sqrt(-1/(b*x + a)^2 + 1) - 1)^3 + 240*(b*x + a)^2*a^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 + 960*(b*x + a)*
a^3*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 24*(b*x + a)^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 + 360*(b*x + a)*a*(sqrt(-
1/(b*x + a)^2 + 1) - 1) + 24*(40*a^4 + 40*a^2 + 3)*log(-(sqrt(-1/(b*x + a)^2 + 1) - 1)*abs(b*x + a)) - (120*(8
*a^3 + 3*a)*(b*x + a)^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^3 + 24*(10*a^2 + 1)*(b*x + a)^2*(sqrt(-1/(b*x + a)^2 +
1) - 1)^2 + 40*(b*x + a)*a*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 3)/((b*x + a)^4*(sqrt(-1/(b*x + a)^2 + 1) - 1)^4))
/b^6)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,\mathrm {acos}\left (\frac {1}{a+b\,x}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*acos(1/(a + b*x)),x)

[Out]

int(x^4*acos(1/(a + b*x)), x)

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