\(\int \frac {(a+b \csc ^{-1}(c x))^2}{x^4} \, dx\) [22]

Optimal result

Integrand size = 14, antiderivative size = 102 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^4} \, dx=\frac {2 b^2}{27 x^3}+\frac {4 b^2 c^2}{9 x}-\frac {4}{9} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac {2 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{9 x^2}-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{3 x^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^4} \, dx=-\frac {9 a^2+6 a b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )-2 b^2 \left (1+6 c^2 x^2\right )+6 b \left (3 a+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )\right ) \csc ^{-1}(c x)+9 b^2 \csc ^{-1}(c x)^2}{27 x^3} \] Input:

Integrate[(a + b*ArcCsc[c*x])^2/x^4,x]
 

Output:

-1/27*(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) + 6*b*(3*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2))*Arc 
Csc[c*x] + 9*b^2*ArcCsc[c*x]^2)/x^3
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5746, 4904, 3042, 3791, 3042, 3777, 3042, 3117}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^4} \, dx\)

\(\Big \downarrow \) 5746

\(\displaystyle -c^3 \int \frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{c^2 x^2}d\csc ^{-1}(c x)\)

\(\Big \downarrow \) 4904

\(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \int \frac {a+b \csc ^{-1}(c x)}{c^3 x^3}d\csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \int \left (a+b \csc ^{-1}(c x)\right ) \sin \left (\csc ^{-1}(c x)\right )^3d\csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (\frac {2}{3} \int \frac {a+b \csc ^{-1}(c x)}{c x}d\csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{3 c^2 x^2}+\frac {b}{9 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (\frac {2}{3} \int \left (a+b \csc ^{-1}(c x)\right ) \sin \left (\csc ^{-1}(c x)\right )d\csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{3 c^2 x^2}+\frac {b}{9 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (\frac {2}{3} \left (b \int \sqrt {1-\frac {1}{c^2 x^2}}d\csc ^{-1}(c x)-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )\right )-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{3 c^2 x^2}+\frac {b}{9 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (\frac {2}{3} \left (b \int \sin \left (\csc ^{-1}(c x)+\frac {\pi }{2}\right )d\csc ^{-1}(c x)-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )\right )-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{3 c^2 x^2}+\frac {b}{9 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3117

\(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{3 c^2 x^2}+\frac {2}{3} \left (\frac {b}{c x}-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )\right )+\frac {b}{9 c^3 x^3}\right )\right )\)

Input:

Int[(a + b*ArcCsc[c*x])^2/x^4,x]
 

Output:

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

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791

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] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 4904

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] - Simp[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 5746

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

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.50

method	result	size
parts	\(-\frac {a^{2}}{3 x^{3}}+b^{2} c^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+2 a b \,c^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\)	\(153\)
derivativedivides	\(c^{3} \left (-\frac {a^{2}}{3 c^{3} x^{3}}+b^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+2 a b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\)	\(154\)
default	\(c^{3} \left (-\frac {a^{2}}{3 c^{3} x^{3}}+b^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+2 a b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\)	\(154\)

Input:

int((a+b*arccsc(c*x))^2/x^4,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^4} \, dx=\frac {12 \, b^{2} c^{2} x^{2} - 9 \, b^{2} \operatorname {arccsc}\left (c x\right )^{2} - 18 \, a b \operatorname {arccsc}\left (c x\right ) - 9 \, a^{2} + 2 \, b^{2} - 6 \, {\left (2 \, a b c^{2} x^{2} + a b + {\left (2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{27 \, x^{3}} \] Input:

integrate((a+b*arccsc(c*x))^2/x^4,x, algorithm="fricas")
 

Output:

1/27*(12*b^2*c^2*x^2 - 9*b^2*arccsc(c*x)^2 - 18*a*b*arccsc(c*x) - 9*a^2 + 
2*b^2 - 6*(2*a*b*c^2*x^2 + a*b + (2*b^2*c^2*x^2 + b^2)*arccsc(c*x))*sqrt(c 
^2*x^2 - 1))/x^3
 

Sympy [F]

\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^4} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \] Input:

integrate((a+b*acsc(c*x))**2/x**4,x)
 

Output:

Integral((a + b*acsc(c*x))**2/x**4, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (88) = 176\).

Time = 0.35 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^4} \, dx=\frac {2}{9} \, a b {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arccsc}\left (c x\right )}{x^{3}}\right )} - \frac {b^{2} \operatorname {arccsc}\left (c x\right )^{2}}{3 \, x^{3}} - \frac {a^{2}}{3 \, x^{3}} - \frac {2 \, {\left (6 \, c^{5} x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - 3 \, c^{3} x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - {\left (6 \, c^{3} x^{2} + c\right )} \sqrt {c x + 1} \sqrt {c x - 1} - 3 \, c \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} b^{2}}{27 \, \sqrt {c x + 1} \sqrt {c x - 1} c x^{3}} \] Input:

integrate((a+b*arccsc(c*x))^2/x^4,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (88) = 176\).

Time = 0.14 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.20 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^4} \, dx=\frac {1}{27} \, {\left (6 \, b^{2} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right ) + 6 \, a b c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 18 \, b^{2} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right ) - \frac {9 \, b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2}}{x} - 18 \, a b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {18 \, a b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - \frac {9 \, b^{2} c \arcsin \left (\frac {1}{c x}\right )^{2}}{x} + \frac {2 \, b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )}}{x} - \frac {18 \, a b c \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {14 \, b^{2} c}{x} - \frac {9 \, a^{2}}{c x^{3}}\right )} c \] Input:

integrate((a+b*arccsc(c*x))^2/x^4,x, algorithm="giac")
 

Output:

1/27*(6*b^2*c^2*(-1/(c^2*x^2) + 1)^(3/2)*arcsin(1/(c*x)) + 6*a*b*c^2*(-1/( 
c^2*x^2) + 1)^(3/2) - 18*b^2*c^2*sqrt(-1/(c^2*x^2) + 1)*arcsin(1/(c*x)) - 
9*b^2*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))^2/x - 18*a*b*c^2*sqrt(-1/(c^2*x^ 
2) + 1) - 18*a*b*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))/x - 9*b^2*c*arcsin(1/ 
(c*x))^2/x + 2*b^2*c*(1/(c^2*x^2) - 1)/x - 18*a*b*c*arcsin(1/(c*x))/x + 14 
*b^2*c/x - 9*a^2/(c*x^3))*c
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2}{x^4} \,d x \] Input:

int((a + b*asin(1/(c*x)))^2/x^4,x)
 

Output:

int((a + b*asin(1/(c*x)))^2/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^4} \, dx=\frac {6 \left (\int \frac {\mathit {acsc} \left (c x \right )}{x^{4}}d x \right ) a b \,x^{3}+3 \left (\int \frac {\mathit {acsc} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}-a^{2}}{3 x^{3}} \] Input:

int((a+b*acsc(c*x))^2/x^4,x)
 

Output:

(6*int(acsc(c*x)/x**4,x)*a*b*x**3 + 3*int(acsc(c*x)**2/x**4,x)*b**2*x**3 - 
 a**2)/(3*x**3)