\(\int \frac {(a+b \csc ^{-1}(c x))^2}{x^3} \, dx\) [21]

Optimal result

Integrand size = 14, antiderivative size = 88 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\frac {b^2}{4 x^2}+\frac {1}{2} a b c^2 \csc ^{-1}(c x)+\frac {1}{4} b^2 c^2 \csc ^{-1}(c x)^2-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5331, 4489, 3391} \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{2 x}+\frac {1}{2} a b c^2 \csc ^{-1}(c x)-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{4} b^2 c^2 \csc ^{-1}(c x)^2+\frac {b^2}{4 x^2} \]

[In]

Int[(a + b*ArcCsc[c*x])^2/x^3,x]

[Out]

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

Rule 3391

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 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)^2 \cos (x) \sin (x) \, dx,x,\csc ^{-1}(c x)\right )\right ) \\ & = -\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2}+\left (b c^2\right ) \text {Subst}\left (\int (a+b x) \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {b^2}{4 x^2}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{2} \left (b c^2\right ) \text {Subst}\left (\int (a+b x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {b^2}{4 x^2}+\frac {1}{2} a b c^2 \csc ^{-1}(c x)+\frac {1}{4} b^2 c^2 \csc ^{-1}(c x)^2-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\frac {-2 a^2+b^2-2 a b c \sqrt {1-\frac {1}{c^2 x^2}} x-2 b \left (2 a+b c \sqrt {1-\frac {1}{c^2 x^2}} x\right ) \csc ^{-1}(c x)+b^2 \left (-2+c^2 x^2\right ) \csc ^{-1}(c x)^2+2 a b c^2 x^2 \arcsin \left (\frac {1}{c x}\right )}{4 x^2} \]

[In]

Integrate[(a + b*ArcCsc[c*x])^2/x^3,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(183\) vs. \(2(76)=152\).

Time = 0.59 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.09

method	result	size
parts	\(-\frac {a^{2}}{2 x^{2}}+b^{2} c^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}\right )\)	\(184\)
derivativedivides	\(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+2 a b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3} c^{3}}\right )\right )\)	\(186\)
default	\(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+2 a b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3} c^{3}}\right )\right )\)	\(186\)

[In]

int((a+b*arccsc(c*x))^2/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*a^2/x^2+b^2*c^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)+2*a*b*c^2*(-1/2/c^2/x^2*arccsc(c*x)-1/4*(c^2*x^2-1)^(1/2)*(-arc
tan(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 = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\frac {{\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \operatorname {arccsc}\left (c x\right )^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left (a b c^{2} x^{2} - 2 \, a b\right )} \operatorname {arccsc}\left (c x\right ) - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (b^{2} \operatorname {arccsc}\left (c x\right ) + a b\right )}}{4 \, x^{2}} \]

[In]

integrate((a+b*arccsc(c*x))^2/x^3,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]

[In]

integrate((a+b*acsc(c*x))**2/x**3,x)

[Out]

Integral((a + b*acsc(c*x))**2/x**3, x)

Maxima [F]

\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]

[In]

integrate((a+b*arccsc(c*x))^2/x^3,x, algorithm="maxima")

[Out]

1/2*a*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) - 1/8*(4*(c^2*(log(c*x + 1) + log(c*x - 1) - 2*log(x))*log(c)^2 - 4*c^2*integrate(1/
2*x^2*log(c^2*x^2)/(c^2*x^5 - x^3), x)*log(c) + 8*c^2*integrate(1/2*x^2*log(x)/(c^2*x^5 - x^3), x)*log(c) - 4*
c^2*integrate(1/2*x^2*log(c^2*x^2)*log(x)/(c^2*x^5 - x^3), x) + 4*c^2*integrate(1/2*x^2*log(x)^2/(c^2*x^5 - x^
3), x) + 2*c^2*integrate(1/2*x^2*log(c^2*x^2)/(c^2*x^5 - x^3), x) - (c^2*log(c*x + 1) + c^2*log(c*x - 1) - 2*c
^2*log(x) + 1/x^2)*log(c)^2 + 4*integrate(1/2*log(c^2*x^2)/(c^2*x^5 - x^3), x)*log(c) - 8*integrate(1/2*log(x)
/(c^2*x^5 - x^3), x)*log(c) + 4*integrate(1/2*sqrt(c*x + 1)*sqrt(c*x - 1)*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1
)))/(c^2*x^5 - x^3), x) + 4*integrate(1/2*log(c^2*x^2)*log(x)/(c^2*x^5 - x^3), x) - 4*integrate(1/2*log(x)^2/(
c^2*x^5 - x^3), x) - 2*integrate(1/2*log(c^2*x^2)/(c^2*x^5 - x^3), x))*x^2 + 4*arctan2(1, sqrt(c*x + 1)*sqrt(c
*x - 1))^2 - log(c^2*x^2)^2)*b^2/x^2 - 1/2*a^2/x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (76) = 152\).

Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=-\frac {1}{8} \, {\left (4 \, b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2} + 8 \, a b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 2 \, b^{2} c \arcsin \left (\frac {1}{c x}\right )^{2} + 4 \, a^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 2 \, b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + 4 \, a b c \arcsin \left (\frac {1}{c x}\right ) - b^{2} c + \frac {4 \, b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {4 \, a b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x}\right )} c \]

[In]

integrate((a+b*arccsc(c*x))^2/x^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2}{x^3} \,d x \]

[In]

int((a + b*asin(1/(c*x)))^2/x^3,x)

[Out]

int((a + b*asin(1/(c*x)))^2/x^3, x)