3.32 \(\int \frac {(a+b \csc ^{-1}(c x))^3}{x^5} \, dx\)

Optimal. Leaf size=208 \[ \frac {9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}+\frac {3}{32} c^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}-\frac {45}{256} b^3 c^4 \csc ^{-1}(c x)+\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{128 x^3}+\frac {45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{256 x} \]

[Out]

-45/256*b^3*c^4*arccsc(c*x)+3/32*b^2*(a+b*arccsc(c*x))/x^4+9/32*b^2*c^2*(a+b*arccsc(c*x))/x^2+3/32*c^4*(a+b*ar
ccsc(c*x))^3-1/4*(a+b*arccsc(c*x))^3/x^4+3/128*b^3*c*(1-1/c^2/x^2)^(1/2)/x^3+45/256*b^3*c^3*(1-1/c^2/x^2)^(1/2
)/x-3/16*b*c*(a+b*arccsc(c*x))^2*(1-1/c^2/x^2)^(1/2)/x^3-9/32*b*c^3*(a+b*arccsc(c*x))^2*(1-1/c^2/x^2)^(1/2)/x

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Rubi [A]  time = 0.18, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5223, 4404, 3311, 32, 2635, 8} \[ \frac {9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}-\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}+\frac {3}{32} c^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}+\frac {45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{256 x}+\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{128 x^3}-\frac {45}{256} b^3 c^4 \csc ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])^3/x^5,x]

[Out]

(3*b^3*c*Sqrt[1 - 1/(c^2*x^2)])/(128*x^3) + (45*b^3*c^3*Sqrt[1 - 1/(c^2*x^2)])/(256*x) - (45*b^3*c^4*ArcCsc[c*
x])/256 + (3*b^2*(a + b*ArcCsc[c*x]))/(32*x^4) + (9*b^2*c^2*(a + b*ArcCsc[c*x]))/(32*x^2) - (3*b*c*Sqrt[1 - 1/
(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/(16*x^3) - (9*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/(32*x) + (3
*c^4*(a + b*ArcCsc[c*x])^3)/32 - (a + b*ArcCsc[c*x])^3/(4*x^4)

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 2635

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] && Integer
Q[2*n]

Rule 3311

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 4404

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 5223

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*} \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^5} \, dx &=-\left (c^4 \operatorname {Subst}\left (\int (a+b x)^3 \cos (x) \sin ^3(x) \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}+\frac {1}{4} \left (3 b c^4\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sin ^4(x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}+\frac {1}{16} \left (9 b c^4\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right )-\frac {1}{32} \left (3 b^3 c^4\right ) \operatorname {Subst}\left (\int \sin ^4(x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{128 x^3}+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}+\frac {1}{32} \left (9 b c^4\right ) \operatorname {Subst}\left (\int (a+b x)^2 \, dx,x,\csc ^{-1}(c x)\right )-\frac {1}{128} \left (9 b^3 c^4\right ) \operatorname {Subst}\left (\int \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right )-\frac {1}{32} \left (9 b^3 c^4\right ) \operatorname {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}}}{128 x^3}+\frac {45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{256 x}+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}+\frac {3}{32} c^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}-\frac {1}{256} \left (9 b^3 c^4\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\csc ^{-1}(c x)\right )-\frac {1}{64} \left (9 b^3 c^4\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{128 x^3}+\frac {45 b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{256 x}-\frac {45}{256} b^3 c^4 \csc ^{-1}(c x)+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{32 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{32 x}+\frac {3}{32} c^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{4 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.36, size = 283, normalized size = 1.36 \[ \frac {-64 a^3+9 b c^4 x^4 \left (8 a^2-5 b^2\right ) \sin ^{-1}\left (\frac {1}{c x}\right )+24 b \csc ^{-1}(c x) \left (-8 a^2-2 a b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (3 c^2 x^2+2\right )+b^2 \left (3 c^2 x^2+1\right )\right )-48 a^2 b c x \sqrt {1-\frac {1}{c^2 x^2}}-72 a^2 b c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}}+72 a b^2 c^2 x^2-24 b^2 \csc ^{-1}(c x)^2 \left (a \left (8-3 c^4 x^4\right )+b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (3 c^2 x^2+2\right )\right )+24 a b^2+8 b^3 \left (3 c^4 x^4-8\right ) \csc ^{-1}(c x)^3+6 b^3 c x \sqrt {1-\frac {1}{c^2 x^2}}+45 b^3 c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{256 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsc[c*x])^3/x^5,x]

[Out]

(-64*a^3 + 24*a*b^2 - 48*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + 6*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x + 72*a*b^2*c^2*x^2
- 72*a^2*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 + 45*b^3*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 + 24*b*(-8*a^2 + b^2*(1 + 3*c^
2*x^2) - 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + 3*c^2*x^2))*ArcCsc[c*x] - 24*b^2*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2
+ 3*c^2*x^2) + a*(8 - 3*c^4*x^4))*ArcCsc[c*x]^2 + 8*b^3*(-8 + 3*c^4*x^4)*ArcCsc[c*x]^3 + 9*b*(8*a^2 - 5*b^2)*
c^4*x^4*ArcSin[1/(c*x)])/(256*x^4)

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fricas [A]  time = 0.93, size = 225, normalized size = 1.08 \[ \frac {72 \, a b^{2} c^{2} x^{2} + 8 \, {\left (3 \, b^{3} c^{4} x^{4} - 8 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right )^{3} - 64 \, a^{3} + 24 \, a b^{2} + 24 \, {\left (3 \, a b^{2} c^{4} x^{4} - 8 \, a b^{2}\right )} \operatorname {arccsc}\left (c x\right )^{2} + 3 \, {\left (3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c^{4} x^{4} + 24 \, b^{3} c^{2} x^{2} - 64 \, a^{2} b + 8 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right ) - 3 \, {\left (3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c^{2} x^{2} + 16 \, a^{2} b - 2 \, b^{3} + 8 \, {\left (3 \, b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right )^{2} + 16 \, {\left (3 \, a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{256 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))^3/x^5,x, algorithm="fricas")

[Out]

1/256*(72*a*b^2*c^2*x^2 + 8*(3*b^3*c^4*x^4 - 8*b^3)*arccsc(c*x)^3 - 64*a^3 + 24*a*b^2 + 24*(3*a*b^2*c^4*x^4 -
8*a*b^2)*arccsc(c*x)^2 + 3*(3*(8*a^2*b - 5*b^3)*c^4*x^4 + 24*b^3*c^2*x^2 - 64*a^2*b + 8*b^3)*arccsc(c*x) - 3*(
3*(8*a^2*b - 5*b^3)*c^2*x^2 + 16*a^2*b - 2*b^3 + 8*(3*b^3*c^2*x^2 + 2*b^3)*arccsc(c*x)^2 + 16*(3*a*b^2*c^2*x^2
+ 2*a*b^2)*arccsc(c*x))*sqrt(c^2*x^2 - 1))/x^4

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giac [B]  time = 0.18, size = 576, normalized size = 2.77 \[ -\frac {1}{256} \, {\left (64 \, b^{3} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )^{3} + 192 \, a b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )^{2} + 128 \, b^{3} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{3} + 192 \, a^{2} b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right ) - 24 \, b^{3} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right ) + 384 \, a b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2} + 40 \, b^{3} c^{3} \arcsin \left (\frac {1}{c x}\right )^{3} - 24 \, a b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 384 \, a^{2} b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) - 120 \, b^{3} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 120 \, a b^{2} c^{3} \arcsin \left (\frac {1}{c x}\right )^{2} - \frac {48 \, b^{3} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right )^{2}}{x} - 120 \, a b^{2} c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + 120 \, a^{2} b c^{3} \arcsin \left (\frac {1}{c x}\right ) - 51 \, b^{3} c^{3} \arcsin \left (\frac {1}{c x}\right ) - \frac {96 \, a b^{2} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {120 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )^{2}}{x} - 51 \, a b^{2} c^{3} - \frac {48 \, a^{2} b c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}{x} + \frac {6 \, b^{3} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}{x} + \frac {240 \, a b^{2} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {120 \, a^{2} b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} - \frac {51 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} + \frac {64 \, a^{3}}{c x^{4}}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))^3/x^5,x, algorithm="giac")

[Out]

-1/256*(64*b^3*c^3*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x))^3 + 192*a*b^2*c^3*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x))^2
+ 128*b^3*c^3*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))^3 + 192*a^2*b*c^3*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x)) - 24*b^
3*c^3*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x)) + 384*a*b^2*c^3*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))^2 + 40*b^3*c^3*arc
sin(1/(c*x))^3 - 24*a*b^2*c^3*(1/(c^2*x^2) - 1)^2 + 384*a^2*b*c^3*(1/(c^2*x^2) - 1)*arcsin(1/(c*x)) - 120*b^3*
c^3*(1/(c^2*x^2) - 1)*arcsin(1/(c*x)) + 120*a*b^2*c^3*arcsin(1/(c*x))^2 - 48*b^3*c^2*(-1/(c^2*x^2) + 1)^(3/2)*
arcsin(1/(c*x))^2/x - 120*a*b^2*c^3*(1/(c^2*x^2) - 1) + 120*a^2*b*c^3*arcsin(1/(c*x)) - 51*b^3*c^3*arcsin(1/(c
*x)) - 96*a*b^2*c^2*(-1/(c^2*x^2) + 1)^(3/2)*arcsin(1/(c*x))/x + 120*b^3*c^2*sqrt(-1/(c^2*x^2) + 1)*arcsin(1/(
c*x))^2/x - 51*a*b^2*c^3 - 48*a^2*b*c^2*(-1/(c^2*x^2) + 1)^(3/2)/x + 6*b^3*c^2*(-1/(c^2*x^2) + 1)^(3/2)/x + 24
0*a*b^2*c^2*sqrt(-1/(c^2*x^2) + 1)*arcsin(1/(c*x))/x + 120*a^2*b*c^2*sqrt(-1/(c^2*x^2) + 1)/x - 51*b^3*c^2*sqr
t(-1/(c^2*x^2) + 1)/x + 64*a^3/(c*x^4))*c

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maple [B]  time = 0.76, size = 472, normalized size = 2.27 \[ -\frac {a^{3}}{4 x^{4}}-\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{3}}{4 x^{4}}+\frac {3 c^{4} b^{3} \mathrm {arccsc}\left (c x \right )^{3}}{32}-\frac {9 c^{3} b^{3} \mathrm {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{32 x}-\frac {3 c \,b^{3} \mathrm {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{16 x^{3}}+\frac {3 b^{3} \mathrm {arccsc}\left (c x \right )}{32 x^{4}}+\frac {45 c^{3} b^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{256 x}+\frac {3 c \,b^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{128 x^{3}}-\frac {45 b^{3} c^{4} \mathrm {arccsc}\left (c x \right )}{256}+\frac {9 c^{2} b^{3} \mathrm {arccsc}\left (c x \right )}{32 x^{2}}-\frac {3 a \,b^{2} \mathrm {arccsc}\left (c x \right )^{2}}{4 x^{4}}+\frac {9 c^{4} a \,b^{2} \mathrm {arccsc}\left (c x \right )^{2}}{32}-\frac {9 c^{3} a \,b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{16 x}-\frac {3 c a \,b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 x^{3}}+\frac {3 a \,b^{2}}{32 x^{4}}+\frac {9 c^{2} a \,b^{2}}{32 x^{2}}-\frac {3 a^{2} b \,\mathrm {arccsc}\left (c x \right )}{4 x^{4}}+\frac {9 c^{3} a^{2} b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {9 c^{3} a^{2} b}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {3 c \,a^{2} b}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}}+\frac {3 a^{2} b}{16 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsc(c*x))^3/x^5,x)

[Out]

-1/4*a^3/x^4-1/4*b^3/x^4*arccsc(c*x)^3+3/32*c^4*b^3*arccsc(c*x)^3-9/32*c^3*b^3*arccsc(c*x)^2/x*((c^2*x^2-1)/c^
2/x^2)^(1/2)-3/16*c*b^3*arccsc(c*x)^2/x^3*((c^2*x^2-1)/c^2/x^2)^(1/2)+3/32*b^3/x^4*arccsc(c*x)+45/256*c^3*b^3*
((c^2*x^2-1)/c^2/x^2)^(1/2)/x+3/128*c*b^3/x^3*((c^2*x^2-1)/c^2/x^2)^(1/2)-45/256*b^3*c^4*arccsc(c*x)+9/32*c^2*
b^3/x^2*arccsc(c*x)-3/4*a*b^2/x^4*arccsc(c*x)^2+9/32*c^4*a*b^2*arccsc(c*x)^2-9/16*c^3*a*b^2*arccsc(c*x)/x*((c^
2*x^2-1)/c^2/x^2)^(1/2)-3/8*c*a*b^2*arccsc(c*x)/x^3*((c^2*x^2-1)/c^2/x^2)^(1/2)+3/32*a*b^2/x^4+9/32*c^2*a*b^2/
x^2-3/4*a^2*b/x^4*arccsc(c*x)+9/32*c^3*a^2*b*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*arctan(1/(c^2*x^2
-1)^(1/2))-9/32*c^3*a^2*b/((c^2*x^2-1)/c^2/x^2)^(1/2)/x+3/32*c*a^2*b/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^3+3/16/c*a^
2*b/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^5

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))^3/x^5,x, algorithm="maxima")

[Out]

-3/32*a^2*b*((3*c^5*arctan(c*x*sqrt(-1/(c^2*x^2) + 1)) + (3*c^8*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 5*c^6*x*sqrt(-1
/(c^2*x^2) + 1))/(c^4*x^4*(1/(c^2*x^2) - 1)^2 - 2*c^2*x^2*(1/(c^2*x^2) - 1) + 1))/c + 8*arccsc(c*x)/x^4) - 1/4
*a^3/x^4 - 1/16*(4*b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 3*b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1
))*log(c^2*x^2)^2 + 12*(2*(c^2*log(c*x + 1) + c^2*log(c*x - 1) - 2*c^2*log(x) + 1/x^2)*a*b^2*c^2*log(c)^2 + 64
*b^3*c^2*integrate(1/16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^7 - x^5), x)*log(c)^2 - 64*b^3*c^2*
integrate(1/16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^7 - x^5), x)*log(c) + 128*b^3*c
^2*integrate(1/16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^7 - x^5), x)*log(c) - 64*a*b^2*c^2
*integrate(1/16*x^2*log(c^2*x^2)/(c^2*x^7 - x^5), x)*log(c) + 128*a*b^2*c^2*integrate(1/16*x^2*log(x)/(c^2*x^7
- x^5), x)*log(c) - 64*b^3*c^2*integrate(1/16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)*log(x)
/(c^2*x^7 - x^5), x) + 64*b^3*c^2*integrate(1/16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^7
- x^5), x) - 64*a*b^2*c^2*integrate(1/16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^7 - x^5), x) +
16*b^3*c^2*integrate(1/16*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^7 - x^5), x) + 16*a*
b^2*c^2*integrate(1/16*x^2*log(c^2*x^2)^2/(c^2*x^7 - x^5), x) - 64*a*b^2*c^2*integrate(1/16*x^2*log(c^2*x^2)*l
og(x)/(c^2*x^7 - x^5), x) + 64*a*b^2*c^2*integrate(1/16*x^2*log(x)^2/(c^2*x^7 - x^5), x) - (2*c^4*log(c*x + 1)
+ 2*c^4*log(c*x - 1) - 4*c^4*log(x) + (2*c^2*x^2 + 1)/x^4)*a*b^2*log(c)^2 - 64*b^3*integrate(1/16*arctan(1/(s
qrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^7 - x^5), x)*log(c)^2 + 64*b^3*integrate(1/16*arctan(1/(sqrt(c*x + 1)*sqrt
(c*x - 1)))*log(c^2*x^2)/(c^2*x^7 - x^5), x)*log(c) - 128*b^3*integrate(1/16*arctan(1/(sqrt(c*x + 1)*sqrt(c*x
- 1)))*log(x)/(c^2*x^7 - x^5), x)*log(c) + 64*a*b^2*integrate(1/16*log(c^2*x^2)/(c^2*x^7 - x^5), x)*log(c) - 1
28*a*b^2*integrate(1/16*log(x)/(c^2*x^7 - x^5), x)*log(c) + 16*b^3*integrate(1/16*sqrt(c*x + 1)*sqrt(c*x - 1)*
arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^7 - x^5), x) - 4*b^3*integrate(1/16*sqrt(c*x + 1)*sqrt(c*x -
1)*log(c^2*x^2)^2/(c^2*x^7 - x^5), x) + 64*b^3*integrate(1/16*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*
x^2)*log(x)/(c^2*x^7 - x^5), x) - 64*b^3*integrate(1/16*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*
x^7 - x^5), x) + 64*a*b^2*integrate(1/16*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^7 - x^5), x) - 16*b^
3*integrate(1/16*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^7 - x^5), x) - 16*a*b^2*integrate
(1/16*log(c^2*x^2)^2/(c^2*x^7 - x^5), x) + 64*a*b^2*integrate(1/16*log(c^2*x^2)*log(x)/(c^2*x^7 - x^5), x) - 6
4*a*b^2*integrate(1/16*log(x)^2/(c^2*x^7 - x^5), x))*x^4)/x^4

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(1/(c*x)))^3/x^5,x)

[Out]

int((a + b*asin(1/(c*x)))^3/x^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsc(c*x))**3/x**5,x)

[Out]

Integral((a + b*acsc(c*x))**3/x**5, x)

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