\(\int \frac {(a+b \sec ^{-1}(c x))^3}{x^4} \, dx\) [31]

Optimal result

Integrand size = 14, antiderivative size = 170 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=-\frac {14}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}+\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x}+\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3} \]

[Out]

2/27*b^3*c^3*(1-1/c^2/x^2)^(3/2)+2/9*b^2*(a+b*arcsec(c*x))/x^3+4/3*b^2*c^2*(a+b*arcsec(c*x))/x-1/3*(a+b*arcsec
(c*x))^3/x^3-14/9*b^3*c^3*(1-1/c^2/x^2)^(1/2)+2/3*b*c^3*(a+b*arcsec(c*x))^2*(1-1/c^2/x^2)^(1/2)+1/3*b*c*(a+b*a
rcsec(c*x))^2*(1-1/c^2/x^2)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5330, 4490, 3392, 3377, 2717, 2713} \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=\frac {4 b^2 c^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x}+\frac {2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}+\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}+\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {14}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}} \]

[In]

Int[(a + b*ArcSec[c*x])^3/x^4,x]

[Out]

(-14*b^3*c^3*Sqrt[1 - 1/(c^2*x^2)])/9 + (2*b^3*c^3*(1 - 1/(c^2*x^2))^(3/2))/27 + (2*b^2*(a + b*ArcSec[c*x]))/(
9*x^3) + (4*b^2*c^2*(a + b*ArcSec[c*x]))/(3*x) + (2*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x])^2)/3 + (b*
c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x])^2)/(3*x^2) - (a + b*ArcSec[c*x])^3/(3*x^3)

Rule 2713

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

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 4490

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

Rule 5330

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
ec[x]^(m + 1)*Tan[x], x], x, ArcSec[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n,
0] || LtQ[m, -1])

Rubi steps \begin{align*} \text {integral}& = c^3 \text {Subst}\left (\int (a+b x)^3 \cos ^2(x) \sin (x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = -\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}+\left (b c^3\right ) \text {Subst}\left (\int (a+b x)^2 \cos ^3(x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}+\frac {1}{3} \left (2 b c^3\right ) \text {Subst}\left (\int (a+b x)^2 \cos (x) \, dx,x,\sec ^{-1}(c x)\right )-\frac {1}{9} \left (2 b^3 c^3\right ) \text {Subst}\left (\int \cos ^3(x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}-\frac {1}{3} \left (4 b^2 c^3\right ) \text {Subst}\left (\int (a+b x) \sin (x) \, dx,x,\sec ^{-1}(c x)\right )+\frac {1}{9} \left (2 b^3 c^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sqrt {1-\frac {1}{c^2 x^2}}\right ) \\ & = -\frac {2}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}+\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x}+\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}-\frac {1}{3} \left (4 b^3 c^3\right ) \text {Subst}\left (\int \cos (x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = -\frac {14}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}+\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x}+\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=\frac {-9 a^3+9 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )+6 a b^2 \left (1+6 c^2 x^2\right )-2 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+20 c^2 x^2\right )+3 b \left (-9 a^2+6 a b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )+2 b^2 \left (1+6 c^2 x^2\right )\right ) \sec ^{-1}(c x)+9 b^2 \left (-3 a+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )\right ) \sec ^{-1}(c x)^2-9 b^3 \sec ^{-1}(c x)^3}{27 x^3} \]

[In]

Integrate[(a + b*ArcSec[c*x])^3/x^4,x]

[Out]

(-9*a^3 + 9*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) + 6*a*b^2*(1 + 6*c^2*x^2) - 2*b^3*c*Sqrt[1 - 1/(c^
2*x^2)]*x*(1 + 20*c^2*x^2) + 3*b*(-9*a^2 + 6*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) + 2*b^2*(1 + 6*c^2*
x^2))*ArcSec[c*x] + 9*b^2*(-3*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2))*ArcSec[c*x]^2 - 9*b^3*ArcSec[c*
x]^3)/(27*x^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(298\) vs. \(2(148)=296\).

Time = 1.16 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.76

method	result	size
derivativedivides	\(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {\operatorname {arcsec}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}-\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\operatorname {arcsec}\left (c x \right )}{3 c x}+\frac {2 \,\operatorname {arcsec}\left (c x \right )}{9 c^{3} x^{3}}-\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsec}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\)	\(299\)
default	\(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {\operatorname {arcsec}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}-\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\operatorname {arcsec}\left (c x \right )}{3 c x}+\frac {2 \,\operatorname {arcsec}\left (c x \right )}{9 c^{3} x^{3}}-\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsec}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\)	\(299\)
parts	\(-\frac {a^{3}}{3 x^{3}}+b^{3} c^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {\operatorname {arcsec}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}-\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\operatorname {arcsec}\left (c x \right )}{3 c x}+\frac {2 \,\operatorname {arcsec}\left (c x \right )}{9 c^{3} x^{3}}-\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} c^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsec}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \,c^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\)	\(301\)

[In]

int((a+b*arcsec(c*x))^3/x^4,x,method=_RETURNVERBOSE)

[Out]

c^3*(-1/3*a^3/c^3/x^3+b^3*(-1/3/c^3/x^3*arcsec(c*x)^3+1/3*arcsec(c*x)^2*(2*c^2*x^2+1)/c^2/x^2*((c^2*x^2-1)/c^2
/x^2)^(1/2)-4/3*((c^2*x^2-1)/c^2/x^2)^(1/2)+4/3/c/x*arcsec(c*x)+2/9/c^3/x^3*arcsec(c*x)-2/27*(2*c^2*x^2+1)/c^2
/x^2*((c^2*x^2-1)/c^2/x^2)^(1/2))+3*a*b^2*(-1/3/c^3/x^3*arcsec(c*x)^2+2/9*arcsec(c*x)*(2*c^2*x^2+1)/c^2/x^2*((
c^2*x^2-1)/c^2/x^2)^(1/2)+2/27/c^3/x^3+4/9/c/x)+3*a^2*b*(-1/3/c^3/x^3*arcsec(c*x)+1/9*(c^2*x^2-1)*(2*c^2*x^2+1
)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c^4/x^4))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=\frac {36 \, a b^{2} c^{2} x^{2} - 9 \, b^{3} \operatorname {arcsec}\left (c x\right )^{3} - 27 \, a b^{2} \operatorname {arcsec}\left (c x\right )^{2} - 9 \, a^{3} + 6 \, a b^{2} + 3 \, {\left (12 \, b^{3} c^{2} x^{2} - 9 \, a^{2} b + 2 \, b^{3}\right )} \operatorname {arcsec}\left (c x\right ) + {\left (2 \, {\left (9 \, a^{2} b - 20 \, b^{3}\right )} c^{2} x^{2} + 9 \, a^{2} b - 2 \, b^{3} + 9 \, {\left (2 \, b^{3} c^{2} x^{2} + b^{3}\right )} \operatorname {arcsec}\left (c x\right )^{2} + 18 \, {\left (2 \, a b^{2} c^{2} x^{2} + a b^{2}\right )} \operatorname {arcsec}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{27 \, x^{3}} \]

[In]

integrate((a+b*arcsec(c*x))^3/x^4,x, algorithm="fricas")

[Out]

1/27*(36*a*b^2*c^2*x^2 - 9*b^3*arcsec(c*x)^3 - 27*a*b^2*arcsec(c*x)^2 - 9*a^3 + 6*a*b^2 + 3*(12*b^3*c^2*x^2 -
9*a^2*b + 2*b^3)*arcsec(c*x) + (2*(9*a^2*b - 20*b^3)*c^2*x^2 + 9*a^2*b - 2*b^3 + 9*(2*b^3*c^2*x^2 + b^3)*arcse
c(c*x)^2 + 18*(2*a*b^2*c^2*x^2 + a*b^2)*arcsec(c*x))*sqrt(c^2*x^2 - 1))/x^3

Sympy [F]

\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=\int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \]

[In]

integrate((a+b*asec(c*x))**3/x**4,x)

[Out]

Integral((a + b*asec(c*x))**3/x**4, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (148) = 296\).

Time = 0.70 (sec) , antiderivative size = 575, normalized size of antiderivative = 3.38 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=-\frac {1}{216} \, {\left (\frac {72 \, {\left (c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )} \operatorname {arcsec}\left (c x\right )^{2}}{c} + \frac {72 \, c^{4} {\left (\frac {c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {2 \, \sqrt {c^{2} x^{2} - 1} c}{x} - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}}{c} - \frac {c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {2 \, \sqrt {c^{2} x^{2} - 1} c}{x} - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}}{c} - \frac {4 \, \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\right )} + c^{2} {\left (\frac {9 \, c^{4} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {16 \, \sqrt {c^{2} x^{2} - 1} c^{3}}{x} - \frac {9 \, \sqrt {c^{2} x^{2} - 1} c^{2}}{x^{2}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} c}{x^{3}} - \frac {6 \, \sqrt {c^{2} x^{2} - 1}}{x^{4}}}{c} - \frac {9 \, c^{4} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {16 \, \sqrt {c^{2} x^{2} - 1} c^{3}}{x} - \frac {9 \, \sqrt {c^{2} x^{2} - 1} c^{2}}{x^{2}} - \frac {8 \, \sqrt {c^{2} x^{2} - 1} c}{x^{3}} - \frac {6 \, \sqrt {c^{2} x^{2} - 1}}{x^{4}}}{c} - \frac {48 \, \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )}{x^{3}}\right )}}{c^{2}}\right )} b^{3} - \frac {1}{3} \, a^{2} b {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {3 \, \operatorname {arcsec}\left (c x\right )}{x^{3}}\right )} - \frac {b^{3} \operatorname {arcsec}\left (c x\right )^{3}}{3 \, x^{3}} - \frac {a b^{2} \operatorname {arcsec}\left (c x\right )^{2}}{x^{3}} - \frac {a^{3}}{3 \, x^{3}} + \frac {2 \, {\left ({\left (6 \, c^{3} x^{2} + c\right )} \sqrt {c x + 1} \sqrt {c x - 1} + 3 \, {\left (2 \, c^{5} x^{4} - c^{3} x^{2} - c\right )} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} a b^{2}}{9 \, \sqrt {c x + 1} \sqrt {c x - 1} c x^{3}} \]

[In]

integrate((a+b*arcsec(c*x))^3/x^4,x, algorithm="maxima")

[Out]

-1/216*(72*(c^4*(-1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(-1/(c^2*x^2) + 1))*arcsec(c*x)^2/c + (72*c^4*((c^2*arcsi
n(1/(c*abs(x))) + 2*sqrt(c^2*x^2 - 1)*c/x - sqrt(c^2*x^2 - 1)/x^2)/c - (c^2*arcsin(1/(c*abs(x))) - 2*sqrt(c^2*
x^2 - 1)*c/x - sqrt(c^2*x^2 - 1)/x^2)/c - 4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/x) + c^2*((9*c^4*arcsin(1/(c*a
bs(x))) + 16*sqrt(c^2*x^2 - 1)*c^3/x - 9*sqrt(c^2*x^2 - 1)*c^2/x^2 + 8*sqrt(c^2*x^2 - 1)*c/x^3 - 6*sqrt(c^2*x^
2 - 1)/x^4)/c - (9*c^4*arcsin(1/(c*abs(x))) - 16*sqrt(c^2*x^2 - 1)*c^3/x - 9*sqrt(c^2*x^2 - 1)*c^2/x^2 - 8*sqr
t(c^2*x^2 - 1)*c/x^3 - 6*sqrt(c^2*x^2 - 1)/x^4)/c - 48*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/x^3))/c^2)*b^3 - 1/
3*a^2*b*((c^4*(-1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(-1/(c^2*x^2) + 1))/c + 3*arcsec(c*x)/x^3) - 1/3*b^3*arcsec
(c*x)^3/x^3 - a*b^2*arcsec(c*x)^2/x^3 - 1/3*a^3/x^3 + 2/9*((6*c^3*x^2 + c)*sqrt(c*x + 1)*sqrt(c*x - 1) + 3*(2*
c^5*x^4 - c^3*x^2 - c)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)))*a*b^2/(sqrt(c*x + 1)*sqrt(c*x - 1)*c*x^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (148) = 296\).

Time = 0.32 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=\frac {1}{27} \, {\left (18 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )^{2} + 36 \, a b^{2} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right ) + 18 \, a^{2} b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 40 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {36 \, b^{3} c \arccos \left (\frac {1}{c x}\right )}{x} + \frac {9 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )^{2}}{x^{2}} + \frac {36 \, a b^{2} c}{x} + \frac {18 \, a b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )}{x^{2}} - \frac {9 \, b^{3} \arccos \left (\frac {1}{c x}\right )^{3}}{c x^{3}} + \frac {9 \, a^{2} b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{2}} - \frac {2 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{2}} - \frac {27 \, a b^{2} \arccos \left (\frac {1}{c x}\right )^{2}}{c x^{3}} - \frac {27 \, a^{2} b \arccos \left (\frac {1}{c x}\right )}{c x^{3}} + \frac {6 \, b^{3} \arccos \left (\frac {1}{c x}\right )}{c x^{3}} - \frac {9 \, a^{3}}{c x^{3}} + \frac {6 \, a b^{2}}{c x^{3}}\right )} c \]

[In]

integrate((a+b*arcsec(c*x))^3/x^4,x, algorithm="giac")

[Out]

1/27*(18*b^3*c^2*sqrt(-1/(c^2*x^2) + 1)*arccos(1/(c*x))^2 + 36*a*b^2*c^2*sqrt(-1/(c^2*x^2) + 1)*arccos(1/(c*x)
) + 18*a^2*b*c^2*sqrt(-1/(c^2*x^2) + 1) - 40*b^3*c^2*sqrt(-1/(c^2*x^2) + 1) + 36*b^3*c*arccos(1/(c*x))/x + 9*b
^3*sqrt(-1/(c^2*x^2) + 1)*arccos(1/(c*x))^2/x^2 + 36*a*b^2*c/x + 18*a*b^2*sqrt(-1/(c^2*x^2) + 1)*arccos(1/(c*x
))/x^2 - 9*b^3*arccos(1/(c*x))^3/(c*x^3) + 9*a^2*b*sqrt(-1/(c^2*x^2) + 1)/x^2 - 2*b^3*sqrt(-1/(c^2*x^2) + 1)/x
^2 - 27*a*b^2*arccos(1/(c*x))^2/(c*x^3) - 27*a^2*b*arccos(1/(c*x))/(c*x^3) + 6*b^3*arccos(1/(c*x))/(c*x^3) - 9
*a^3/(c*x^3) + 6*a*b^2/(c*x^3))*c

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3}{x^4} \,d x \]

[In]

int((a + b*acos(1/(c*x)))^3/x^4,x)

[Out]

int((a + b*acos(1/(c*x)))^3/x^4, x)