3.31 \(\int \frac {(a+b \csc ^{-1}(c x))^3}{x^4} \, dx\)

Optimal. Leaf size=170 \[ \frac {4 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x}+\frac {2 b^2 \left (a+b \csc ^{-1}(c x)\right )}{9 x^3}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 x^2}-\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 x^3}-\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {14}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5223, 4404, 3311, 3296, 2638, 2633} \[ \frac {4 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x}+\frac {2 b^2 \left (a+b \csc ^{-1}(c x)\right )}{9 x^3}-\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 x^3}-\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {14}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[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*ArcCsc[c*x]))/(9
*x^3) + (4*b^2*c^2*(a + b*ArcCsc[c*x]))/(3*x) - (2*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/3 - (b*c
*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/(3*x^2) - (a + b*ArcCsc[c*x])^3/(3*x^3)

Rule 2633

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 2638

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

Rule 3296

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

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Mathematica [A]  time = 0.33, size = 204, normalized size = 1.20 \[ \frac {-9 a^3+3 b \csc ^{-1}(c x) \left (-9 a^2-6 a b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (2 c^2 x^2+1\right )+2 b^2 \left (6 c^2 x^2+1\right )\right )-9 a^2 b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (2 c^2 x^2+1\right )+6 a b^2 \left (6 c^2 x^2+1\right )-9 b^2 \csc ^{-1}(c x)^2 \left (3 a+b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (2 c^2 x^2+1\right )\right )+2 b^3 c x \sqrt {1-\frac {1}{c^2 x^2}} \left (20 c^2 x^2+1\right )-9 b^3 \csc ^{-1}(c x)^3}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsc[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))*ArcCsc[c*x] - 9*b^2*(3*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2))*ArcCsc[c*x]^2 - 9*b^3*ArcCsc[c*x
]^3)/(27*x^3)

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fricas [A]  time = 0.72, size = 173, normalized size = 1.02 \[ \frac {36 \, a b^{2} c^{2} x^{2} - 9 \, b^{3} \operatorname {arccsc}\left (c x\right )^{3} - 27 \, a b^{2} \operatorname {arccsc}\left (c x\right )^{2} - 9 \, a^{3} + 6 \, a b^{2} + 3 \, {\left (12 \, b^{3} c^{2} x^{2} - 9 \, a^{2} b + 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right ) - {\left (2 \, {\left (9 \, a^{2} b - 20 \, b^{3}\right )} c^{2} x^{2} + 9 \, a^{2} b - 2 \, b^{3} + 9 \, {\left (2 \, b^{3} c^{2} x^{2} + b^{3}\right )} \operatorname {arccsc}\left (c x\right )^{2} + 18 \, {\left (2 \, a b^{2} c^{2} x^{2} + a b^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{27 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(36*a*b^2*c^2*x^2 - 9*b^3*arccsc(c*x)^3 - 27*a*b^2*arccsc(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)*arccsc(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)*arccs
c(c*x)^2 + 18*(2*a*b^2*c^2*x^2 + a*b^2)*arccsc(c*x))*sqrt(c^2*x^2 - 1))/x^3

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giac [B]  time = 0.18, size = 428, normalized size = 2.52 \[ \frac {1}{27} \, {\left (9 \, b^{3} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right )^{2} + 18 \, a b^{2} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right ) - 27 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )^{2} - \frac {9 \, b^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{3}}{x} + 9 \, a^{2} b c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 2 \, b^{3} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 54 \, a b^{2} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right ) - \frac {27 \, a b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2}}{x} - \frac {9 \, b^{3} c \arcsin \left (\frac {1}{c x}\right )^{3}}{x} - 27 \, a^{2} b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 42 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {27 \, a^{2} b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {6 \, b^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - \frac {27 \, a b^{2} c \arcsin \left (\frac {1}{c x}\right )^{2}}{x} + \frac {6 \, a b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )}}{x} - \frac {27 \, a^{2} b c \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {42 \, b^{3} c \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {42 \, a b^{2} c}{x} - \frac {9 \, a^{3}}{c x^{3}}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(9*b^3*c^2*(-1/(c^2*x^2) + 1)^(3/2)*arcsin(1/(c*x))^2 + 18*a*b^2*c^2*(-1/(c^2*x^2) + 1)^(3/2)*arcsin(1/(c
*x)) - 27*b^3*c^2*sqrt(-1/(c^2*x^2) + 1)*arcsin(1/(c*x))^2 - 9*b^3*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))^3/x + 9
*a^2*b*c^2*(-1/(c^2*x^2) + 1)^(3/2) - 2*b^3*c^2*(-1/(c^2*x^2) + 1)^(3/2) - 54*a*b^2*c^2*sqrt(-1/(c^2*x^2) + 1)
*arcsin(1/(c*x)) - 27*a*b^2*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))^2/x - 9*b^3*c*arcsin(1/(c*x))^3/x - 27*a^2*b*c
^2*sqrt(-1/(c^2*x^2) + 1) + 42*b^3*c^2*sqrt(-1/(c^2*x^2) + 1) - 27*a^2*b*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))/x
+ 6*b^3*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))/x - 27*a*b^2*c*arcsin(1/(c*x))^2/x + 6*a*b^2*c*(1/(c^2*x^2) - 1)/
x - 27*a^2*b*c*arcsin(1/(c*x))/x + 42*b^3*c*arcsin(1/(c*x))/x + 42*a*b^2*c/x - 9*a^3/(c*x^3))*c

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maple [B]  time = 0.59, size = 299, normalized size = 1.76 \[ c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\mathrm {arccsc}\left (c x \right )^{3}}{3 c^{3} x^{3}}-\frac {\mathrm {arccsc}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}+\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\mathrm {arccsc}\left (c x \right )}{3 c x}+\frac {2 \,\mathrm {arccsc}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\mathrm {arccsc}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {2 \,\mathrm {arccsc}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsc(c*x))^3/x^4,x)

[Out]

c^3*(-1/3*a^3/c^3/x^3+b^3*(-1/3/c^3/x^3*arccsc(c*x)^3-1/3*arccsc(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*arccsc(c*x)+2/9/c^3/x^3*arccsc(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*arccsc(c*x)^2-2/9*arccsc(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*arccsc(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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \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 {arccsc}\left (c x\right )}{x^{3}}\right )} - \frac {a b^{2} \operatorname {arccsc}\left (c x\right )^{2}}{x^{3}} + \frac {-\frac {1}{4} \, {\left (12 \, x^{3} \int \frac {16 \, c^{4} x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - 8 \, {\left (18 \, c^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) \log \relax (c)^{2} - 12 \, c^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) \log \relax (c) + c^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} x^{2} + 144 \, \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) \log \relax (c)^{2} - 144 \, {\left (c^{2} x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} \log \relax (x)^{2} + \sqrt {c x + 1} \sqrt {c x - 1} {\left (36 \, \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} + 4 \, {\left (2 \, c^{4} x^{4} - c^{2} x^{2} - 1\right )} \log \left (c^{2} x^{2}\right ) + 3 \, \log \left (c^{2} x^{2}\right )^{2} - 48 \, \log \relax (c)^{2} - 96 \, \log \relax (c) \log \relax (x) - 48 \, \log \relax (x)^{2}\right )} - 96 \, \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) \log \relax (c) - 96 \, {\left ({\left (3 \, c^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) \log \relax (c) - c^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} x^{2} - 3 \, \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) \log \relax (c) + \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} \log \relax (x) - 8 \, \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{12 \, {\left (c^{2} x^{6} - x^{4}\right )}}\,{d x} + 16 \, \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )^{3} - 12 \, \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) \log \left (c^{2} x^{2}\right )^{2} + {\left (8 \, c^{2} x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} - {\left (2 \, c^{2} x^{2} + 1\right )} \log \left (c^{2} x^{2}\right )^{2} + 4 \, \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} b^{3}}{12 \, x^{3}} - \frac {a^{3}}{3 \, x^{3}} - \frac {2 \, {\left (6 \, c^{5} x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - 3 \, c^{3} x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - {\left (6 \, c^{3} x^{2} + c\right )} \sqrt {c x + 1} \sqrt {c x - 1} - 3 \, c \arctan \left (1, \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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*arccsc(c*x)/x^3) - a*b^2*arccsc
(c*x)^2/x^3 + 1/12*(12*x^3*integrate(-1/4*(12*c^2*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c)^2 - 12*ar
ctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c)^2 + 12*(c^2*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) - arctan2
(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*(4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x
- 1))^2 - log(c^2*x^2)^2) - 4*((3*c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c) - c^2*arctan2(1, sqrt(c*x
+ 1)*sqrt(c*x - 1)))*x^2 - 3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c) + 3*(c^2*x^2*arctan2(1, sqrt(c*x
+ 1)*sqrt(c*x - 1)) - arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x) + arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)
))*log(c^2*x^2) + 24*(c^2*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c) - arctan2(1, sqrt(c*x + 1)*sqrt(c
*x - 1))*log(c))*log(x))/(c^2*x^6 - x^4), x) - 4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 + 3*arctan2(1, sqrt
(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2)*b^3/x^3 - 1/3*a^3/x^3 - 2/9*(6*c^5*x^4*arctan2(1, sqrt(c*x + 1)*sqrt(
c*x - 1)) - 3*c^3*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) - (6*c^3*x^2 + c)*sqrt(c*x + 1)*sqrt(c*x - 1) -
3*c*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*a*b^2/(sqrt(c*x + 1)*sqrt(c*x - 1)*c*x^3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(1/(c*x)))^3/x^4,x)

[Out]

int((a + b*asin(1/(c*x)))^3/x^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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