\(\int \frac {(a+b \csc ^{-1}(c x))^3}{x^3} \, dx\) [30]

Optimal result

Integrand size = 14, antiderivative size = 125 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{8 x}-\frac {3}{8} b^3 c^2 \csc ^{-1}(c x)+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{4 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 x}+\frac {1}{4} c^2 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 x^2} \]

[Out]

-3/8*b^3*c^2*arccsc(c*x)+3/4*b^2*(a+b*arccsc(c*x))/x^2+1/4*c^2*(a+b*arccsc(c*x))^3-1/2*(a+b*arccsc(c*x))^3/x^2
+3/8*b^3*c*(1-1/c^2/x^2)^(1/2)/x-3/4*b*c*(a+b*arccsc(c*x))^2*(1-1/c^2/x^2)^(1/2)/x

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5331, 4489, 3392, 32, 2715, 8} \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{4 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 x}+\frac {1}{4} c^2 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 x^2}+\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{8 x}-\frac {3}{8} b^3 c^2 \csc ^{-1}(c x) \]

[In]

Int[(a + b*ArcCsc[c*x])^3/x^3,x]

[Out]

(3*b^3*c*Sqrt[1 - 1/(c^2*x^2)])/(8*x) - (3*b^3*c^2*ArcCsc[c*x])/8 + (3*b^2*(a + b*ArcCsc[c*x]))/(4*x^2) - (3*b
*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/(4*x) + (c^2*(a + b*ArcCsc[c*x])^3)/4 - (a + b*ArcCsc[c*x])^3/
(2*x^2)

Rule 8

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

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 4489

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

Rule 5331

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G
tQ[n, 0] || LtQ[m, -1])

Rubi steps \begin{align*} \text {integral}& = -\left (c^2 \text {Subst}\left (\int (a+b x)^3 \cos (x) \sin (x) \, dx,x,\csc ^{-1}(c x)\right )\right ) \\ & = -\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 x^2}+\frac {1}{2} \left (3 b c^2\right ) \text {Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{4 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 x^2}+\frac {1}{4} \left (3 b c^2\right ) \text {Subst}\left (\int (a+b x)^2 \, dx,x,\csc ^{-1}(c x)\right )-\frac {1}{4} \left (3 b^3 c^2\right ) \text {Subst}\left (\int \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{8 x}+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{4 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 x}+\frac {1}{4} c^2 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 x^2}-\frac {1}{8} \left (3 b^3 c^2\right ) \text {Subst}\left (\int 1 \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{8 x}-\frac {3}{8} b^3 c^2 \csc ^{-1}(c x)+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{4 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 x}+\frac {1}{4} c^2 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=\frac {-4 a^3+6 a b^2-6 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x+3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x+6 b \left (-2 a^2+b^2-2 a b c \sqrt {1-\frac {1}{c^2 x^2}} x\right ) \csc ^{-1}(c x)-6 b^2 \left (b c \sqrt {1-\frac {1}{c^2 x^2}} x+a \left (2-c^2 x^2\right )\right ) \csc ^{-1}(c x)^2+2 b^3 \left (-2+c^2 x^2\right ) \csc ^{-1}(c x)^3-3 b \left (-2 a^2+b^2\right ) c^2 x^2 \arcsin \left (\frac {1}{c x}\right )}{8 x^2} \]

[In]

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

[Out]

(-4*a^3 + 6*a*b^2 - 6*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + 3*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x + 6*b*(-2*a^2 + b^2 -
2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcCsc[c*x] - 6*b^2*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x + a*(2 - c^2*x^2))*ArcCsc[c*
x]^2 + 2*b^3*(-2 + c^2*x^2)*ArcCsc[c*x]^3 - 3*b*(-2*a^2 + b^2)*c^2*x^2*ArcSin[1/(c*x)])/(8*x^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(317\) vs. \(2(109)=218\).

Time = 1.07 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.54

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a^{3}}{2 c^{2} x^{2}}+b^{3} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{3}}{2 c^{2} x^{2}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{4 c x}-\frac {3 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{4 c^{2} x^{2}}+\frac {3 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 c x}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{8}+\frac {\operatorname {arccsc}\left (c x \right )^{3}}{2}\right )+3 a \,b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+3 a^{2} b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}\right )\right )\)	\(318\)
default	\(c^{2} \left (-\frac {a^{3}}{2 c^{2} x^{2}}+b^{3} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{3}}{2 c^{2} x^{2}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{4 c x}-\frac {3 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{4 c^{2} x^{2}}+\frac {3 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 c x}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{8}+\frac {\operatorname {arccsc}\left (c x \right )^{3}}{2}\right )+3 a \,b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+3 a^{2} b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}\right )\right )\)	\(318\)
parts	\(-\frac {a^{3}}{2 x^{2}}+b^{3} c^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{3}}{2 c^{2} x^{2}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{4 c x}-\frac {3 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{4 c^{2} x^{2}}+\frac {3 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 c x}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{8}+\frac {\operatorname {arccsc}\left (c x \right )^{3}}{2}\right )+3 a \,b^{2} c^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+3 a^{2} b \,c^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3} c^{3}}\right )\)	\(321\)

[In]

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

[Out]

c^2*(-1/2*a^3/c^2/x^2+b^3*(1/2*(c^2*x^2-1)/c^2/x^2*arccsc(c*x)^3-3/4*arccsc(c*x)^2*(arccsc(c*x)*c*x+((c^2*x^2-
1)/c^2/x^2)^(1/2))/c/x-3/4*(c^2*x^2-1)/c^2/x^2*arccsc(c*x)+3/8/c/x*((c^2*x^2-1)/c^2/x^2)^(1/2)+3/8*arccsc(c*x)
+1/2*arccsc(c*x)^3)+3*a*b^2*(1/2*(c^2*x^2-1)/c^2/x^2*arccsc(c*x)^2-1/2*arccsc(c*x)*(arccsc(c*x)*c*x+((c^2*x^2-
1)/c^2/x^2)^(1/2))/c/x+1/4*arccsc(c*x)^2+1/4/c^2/x^2)+3*a^2*b*(-1/2/c^2/x^2*arccsc(c*x)-1/4*(c^2*x^2-1)^(1/2)*
(-arctan(1/(c^2*x^2-1)^(1/2))*c^2*x^2+(c^2*x^2-1)^(1/2))/((c^2*x^2-1)/c^2/x^2)^(1/2)/c^3/x^3))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=\frac {2 \, {\left (b^{3} c^{2} x^{2} - 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right )^{3} - 4 \, a^{3} + 6 \, a b^{2} + 6 \, {\left (a b^{2} c^{2} x^{2} - 2 \, a b^{2}\right )} \operatorname {arccsc}\left (c x\right )^{2} + 3 \, {\left ({\left (2 \, a^{2} b - b^{3}\right )} c^{2} x^{2} - 4 \, a^{2} b + 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right ) - 3 \, {\left (2 \, b^{3} \operatorname {arccsc}\left (c x\right )^{2} + 4 \, a b^{2} \operatorname {arccsc}\left (c x\right ) + 2 \, a^{2} b - b^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{8 \, x^{2}} \]

[In]

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

[Out]

1/8*(2*(b^3*c^2*x^2 - 2*b^3)*arccsc(c*x)^3 - 4*a^3 + 6*a*b^2 + 6*(a*b^2*c^2*x^2 - 2*a*b^2)*arccsc(c*x)^2 + 3*(
(2*a^2*b - b^3)*c^2*x^2 - 4*a^2*b + 2*b^3)*arccsc(c*x) - 3*(2*b^3*arccsc(c*x)^2 + 4*a*b^2*arccsc(c*x) + 2*a^2*
b - b^3)*sqrt(c^2*x^2 - 1))/x^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

3/4*a^2*b*((c^4*x*sqrt(-1/(c^2*x^2) + 1)/(c^2*x^2*(1/(c^2*x^2) - 1) - 1) - c^3*arctan(c*x*sqrt(-1/(c^2*x^2) +
1)))/c - 2*arccsc(c*x)/x^2) - 1/2*a^3/x^2 - 1/8*(4*b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 3*b^3*arcta
n2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2 + 12*(a*b^2*c^2*(log(c*x + 1) + log(c*x - 1) - 2*log(x))*log
(c)^2 + 16*b^3*c^2*integrate(1/8*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^5 - x^3), x)*log(c)^2 - 16
*b^3*c^2*integrate(1/8*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^5 - x^3), x)*log(c) + 3
2*b^3*c^2*integrate(1/8*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^5 - x^3), x)*log(c) - 16*a*b
^2*c^2*integrate(1/8*x^2*log(c^2*x^2)/(c^2*x^5 - x^3), x)*log(c) + 32*a*b^2*c^2*integrate(1/8*x^2*log(x)/(c^2*
x^5 - x^3), x)*log(c) - 16*b^3*c^2*integrate(1/8*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)*log(
x)/(c^2*x^5 - x^3), x) + 16*b^3*c^2*integrate(1/8*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^
5 - x^3), x) - 16*a*b^2*c^2*integrate(1/8*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^5 - x^3), x) +
8*b^3*c^2*integrate(1/8*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^5 - x^3), x) + 4*a*b^2
*c^2*integrate(1/8*x^2*log(c^2*x^2)^2/(c^2*x^5 - x^3), x) - 16*a*b^2*c^2*integrate(1/8*x^2*log(c^2*x^2)*log(x)
/(c^2*x^5 - x^3), x) + 16*a*b^2*c^2*integrate(1/8*x^2*log(x)^2/(c^2*x^5 - x^3), x) - (c^2*log(c*x + 1) + c^2*l
og(c*x - 1) - 2*c^2*log(x) + 1/x^2)*a*b^2*log(c)^2 - 16*b^3*integrate(1/8*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1
)))/(c^2*x^5 - x^3), x)*log(c)^2 + 16*b^3*integrate(1/8*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(
c^2*x^5 - x^3), x)*log(c) - 32*b^3*integrate(1/8*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^5 - x^3
), x)*log(c) + 16*a*b^2*integrate(1/8*log(c^2*x^2)/(c^2*x^5 - x^3), x)*log(c) - 32*a*b^2*integrate(1/8*log(x)/
(c^2*x^5 - x^3), x)*log(c) + 8*b^3*integrate(1/8*sqrt(c*x + 1)*sqrt(c*x - 1)*arctan(1/(sqrt(c*x + 1)*sqrt(c*x
- 1)))^2/(c^2*x^5 - x^3), x) - 2*b^3*integrate(1/8*sqrt(c*x + 1)*sqrt(c*x - 1)*log(c^2*x^2)^2/(c^2*x^5 - x^3),
x) + 16*b^3*integrate(1/8*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)*log(x)/(c^2*x^5 - x^3), x) - 1
6*b^3*integrate(1/8*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^5 - x^3), x) + 16*a*b^2*integrate(
1/8*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^5 - x^3), x) - 8*b^3*integrate(1/8*arctan(1/(sqrt(c*x + 1
)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^5 - x^3), x) - 4*a*b^2*integrate(1/8*log(c^2*x^2)^2/(c^2*x^5 - x^3), x)
+ 16*a*b^2*integrate(1/8*log(c^2*x^2)*log(x)/(c^2*x^5 - x^3), x) - 16*a*b^2*integrate(1/8*log(x)^2/(c^2*x^5 -
x^3), x))*x^2)/x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (109) = 218\).

Time = 0.32 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.42 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=-\frac {1}{8} \, {\left (4 \, b^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{3} + 12 \, a b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2} + 2 \, b^{3} c \arcsin \left (\frac {1}{c x}\right )^{3} + 12 \, a^{2} b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) - 6 \, b^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 6 \, a b^{2} c \arcsin \left (\frac {1}{c x}\right )^{2} + 4 \, a^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 6 \, a b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + 6 \, a^{2} b c \arcsin \left (\frac {1}{c x}\right ) - 3 \, b^{3} c \arcsin \left (\frac {1}{c x}\right ) + \frac {6 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )^{2}}{x} - 3 \, a b^{2} c + \frac {12 \, a b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {6 \, a^{2} b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} - \frac {3 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x}\right )} c \]

[In]

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

[Out]

-1/8*(4*b^3*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))^3 + 12*a*b^2*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))^2 + 2*b^3*c*a
rcsin(1/(c*x))^3 + 12*a^2*b*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x)) - 6*b^3*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x)) +
6*a*b^2*c*arcsin(1/(c*x))^2 + 4*a^3*c*(1/(c^2*x^2) - 1) - 6*a*b^2*c*(1/(c^2*x^2) - 1) + 6*a^2*b*c*arcsin(1/(c*
x)) - 3*b^3*c*arcsin(1/(c*x)) + 6*b^3*sqrt(-1/(c^2*x^2) + 1)*arcsin(1/(c*x))^2/x - 3*a*b^2*c + 12*a*b^2*sqrt(-
1/(c^2*x^2) + 1)*arcsin(1/(c*x))/x + 6*a^2*b*sqrt(-1/(c^2*x^2) + 1)/x - 3*b^3*sqrt(-1/(c^2*x^2) + 1)/x)*c

Mupad [F(-1)]

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

[In]

int((a + b*asin(1/(c*x)))^3/x^3,x)

[Out]

int((a + b*asin(1/(c*x)))^3/x^3, x)