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

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

[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)

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Rubi [A]  time = 0.149797, antiderivative size = 170, normalized size of antiderivative = 1., 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 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])

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 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 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 2638

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

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]

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*}

Mathematica [A]  time = 0.299556, size = 204, normalized size = 1.2 \[ \frac{3 b \csc ^{-1}(c x) \left (-9 a^2-6 a b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 x^2+1\right )+2 b^2 \left (6 c^2 x^2+1\right )\right )-9 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 x^2+1\right )-9 a^3+6 a b^2 \left (6 c^2 x^2+1\right )-9 b^2 \csc ^{-1}(c x)^2 \left (3 a+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 x^2+1\right )\right )+2 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}} \left (20 c^2 x^2+1\right )-9 b^3 \csc ^{-1}(c x)^3}{27 x^3} \]

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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Maple [B]  time = 0.261, size = 299, normalized size = 1.8 \begin{align*}{c}^{3} \left ( -{\frac{{a}^{3}}{3\,{c}^{3}{x}^{3}}}+{b}^{3} \left ( -{\frac{ \left ({\rm arccsc} \left (cx\right ) \right ) ^{3}}{3\,{c}^{3}{x}^{3}}}-{\frac{ \left ({\rm arccsc} \left (cx\right ) \right ) ^{2} \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{3\,{c}^{2}{x}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{4}{3}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{4\,{\rm arccsc} \left (cx\right )}{3\,cx}}+{\frac{2\,{\rm arccsc} \left (cx\right )}{9\,{c}^{3}{x}^{3}}}+{\frac{4\,{c}^{2}{x}^{2}+2}{27\,{c}^{2}{x}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}} \right ) +3\,a{b}^{2} \left ( -1/3\,{\frac{ \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{{c}^{3}{x}^{3}}}-2/9\,{\frac{{\rm arccsc} \left (cx\right ) \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{{c}^{2}{x}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{2}{27\,{c}^{3}{x}^{3}}}+4/9\,{\frac{1}{cx}} \right ) +3\,{a}^{2}b \left ( -1/3\,{\frac{{\rm arccsc} \left (cx\right )}{{c}^{3}{x}^{3}}}-1/9\,{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{{c}^{4}{x}^{4}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [A]  time = 2.3525, size = 401, normalized size = 2.36 \begin{align*} \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}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acsc}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )}^{3}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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