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

Optimal result

Integrand size = 12, antiderivative size = 89 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {3 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{40 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{20 c}+\frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {3 b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{40 c^5} \] Output:

3/40*b*(1-1/c^2/x^2)^(1/2)*x^2/c^3+1/20*b*(1-1/c^2/x^2)^(1/2)*x^4/c+1/5*x^ 
5*(a+b*arccsc(c*x))+3/40*b*arctanh((1-1/c^2/x^2)^(1/2))/c^5
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a x^5}{5}+b \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}} \left (\frac {3 x^2}{40 c^3}+\frac {x^4}{20 c}\right )+\frac {1}{5} b x^5 \csc ^{-1}(c x)+\frac {3 b \log \left (x \left (1+\sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}\right )\right )}{40 c^5} \] Input:

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

Output:

(a*x^5)/5 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((3*x^2)/(40*c^3) + x^4/(20*c 
)) + (b*x^5*ArcCsc[c*x])/5 + (3*b*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2) 
])])/(40*c^5)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5744, 798, 52, 52, 73, 221}

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.

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

\(\Big \downarrow \) 5744

\(\displaystyle \frac {b \int \frac {x^3}{\sqrt {1-\frac {1}{c^2 x^2}}}dx}{5 c}+\frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {b \int \frac {x^6}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x^2}}{10 c}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {b \left (\frac {3 \int \frac {x^4}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x^2}}{4 c^2}-\frac {1}{2} x^4 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{10 c}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {b \left (\frac {3 \left (\frac {\int \frac {x^2}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x^2}}{2 c^2}-x^2 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{4 c^2}-\frac {1}{2} x^4 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{10 c}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {b \left (\frac {3 \left (x^2 \left (-\sqrt {1-\frac {1}{c^2 x^2}}\right )-\int \frac {1}{c^2-\frac {c^2}{x^4}}d\sqrt {1-\frac {1}{c^2 x^2}}\right )}{4 c^2}-\frac {1}{2} x^4 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{10 c}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {b \left (\frac {3 \left (x^2 \left (-\sqrt {1-\frac {1}{c^2 x^2}}\right )-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^2}\right )}{4 c^2}-\frac {1}{2} x^4 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{10 c}\)

Input:

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

Output:

(x^5*(a + b*ArcCsc[c*x]))/5 - (b*(-1/2*(Sqrt[1 - 1/(c^2*x^2)]*x^4) + (3*(- 
(Sqrt[1 - 1/(c^2*x^2)]*x^2) - ArcTanh[Sqrt[1 - 1/(c^2*x^2)]]/c^2))/(4*c^2) 
))/(10*c)
 

Defintions of rubi rules used

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 5744

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Sim 
p[(d*x)^(m + 1)*((a + b*ArcCsc[c*x])/(d*(m + 1))), x] + Simp[b*(d/(c*(m + 1 
)))   Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c, d, 
 m}, x] && NeQ[m, -1]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.58

method	result	size
parts	\(\frac {a \,x^{5}}{5}+\frac {b \,\operatorname {arccsc}\left (c x \right ) x^{5}}{5}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2}}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right )}{40 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{6} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\)	\(141\)
derivativedivides	\(\frac {\frac {c^{5} x^{5} a}{5}+\frac {b \,c^{5} x^{5} \operatorname {arccsc}\left (c x \right )}{5}+\frac {b \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{5}}\)	\(148\)
default	\(\frac {\frac {c^{5} x^{5} a}{5}+\frac {b \,c^{5} x^{5} \operatorname {arccsc}\left (c x \right )}{5}+\frac {b \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{5}}\)	\(148\)

Input:

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

Output:

1/5*a*x^5+1/5*b*arccsc(c*x)*x^5+1/20*b/c^3*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^ 
2)^(1/2)*x^2+3/40*b/c^5*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)+3/40*b/c^6 
*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*ln(c*x+(c^2*x^2-1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.19 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {8 \, a c^{5} x^{5} - 16 \, b c^{5} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 8 \, {\left (b c^{5} x^{5} - b c^{5}\right )} \operatorname {arccsc}\left (c x\right ) - 3 \, b \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (2 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} - 1}}{40 \, c^{5}} \] Input:

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

Output:

1/40*(8*a*c^5*x^5 - 16*b*c^5*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 8*(b*c^5*x 
^5 - b*c^5)*arccsc(c*x) - 3*b*log(-c*x + sqrt(c^2*x^2 - 1)) + (2*b*c^3*x^3 
 + 3*b*c*x)*sqrt(c^2*x^2 - 1))/c^5
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 4.35 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.97 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a x^{5}}{5} + \frac {b x^{5} \operatorname {acsc}{\left (c x \right )}}{5} + \frac {b \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} \] Input:

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

Output:

a*x**5/5 + b*x**5*acsc(c*x)/5 + b*Piecewise((c*x**5/(4*sqrt(c**2*x**2 - 1) 
) + x**3/(8*c*sqrt(c**2*x**2 - 1)) - 3*x/(8*c**3*sqrt(c**2*x**2 - 1)) + 3* 
acosh(c*x)/(8*c**4), Abs(c**2*x**2) > 1), (-I*c*x**5/(4*sqrt(-c**2*x**2 + 
1)) - I*x**3/(8*c*sqrt(-c**2*x**2 + 1)) + 3*I*x/(8*c**3*sqrt(-c**2*x**2 + 
1)) - 3*I*asin(c*x)/(8*c**4), True))/(5*c)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.48 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{5} \, a x^{5} + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arccsc}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b \] Input:

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

Output:

1/5*a*x^5 + 1/80*(16*x^5*arccsc(c*x) - (2*(3*(-1/(c^2*x^2) + 1)^(3/2) - 5* 
sqrt(-1/(c^2*x^2) + 1))/(c^4*(1/(c^2*x^2) - 1)^2 + 2*c^4*(1/(c^2*x^2) - 1) 
 + c^4) - 3*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 + 3*log(sqrt(-1/(c^2*x^2) 
+ 1) - 1)/c^4)/c)*b
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (75) = 150\).

Time = 0.37 (sec) , antiderivative size = 480, normalized size of antiderivative = 5.39 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{320} \, {\left (\frac {2 \, b x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {2 \, a x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}{c} + \frac {b x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c^{2}} + \frac {10 \, b x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {10 \, a x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{3}} + \frac {8 \, b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{4}} + \frac {20 \, b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {20 \, a x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{5}} + \frac {24 \, b \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {24 \, b \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{6}} + \frac {20 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{7} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {20 \, a}{c^{7} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {8 \, b}{c^{8} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {10 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {10 \, a}{c^{9} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} - \frac {b}{c^{10} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {2 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{11} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}} + \frac {2 \, a}{c^{11} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}\right )} c \] Input:

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

Output:

1/320*(2*b*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*arcsin(1/(c*x))/c + 2*a*x^5* 
(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c + b*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c^ 
2 + 10*b*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*arcsin(1/(c*x))/c^3 + 10*a*x^3 
*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^3 + 8*b*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1) 
^2/c^4 + 20*b*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c^5 + 20*a*x* 
(sqrt(-1/(c^2*x^2) + 1) + 1)/c^5 + 24*b*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^ 
6 - 24*b*log(1/(abs(c)*abs(x)))/c^6 + 20*b*arcsin(1/(c*x))/(c^7*x*(sqrt(-1 
/(c^2*x^2) + 1) + 1)) + 20*a/(c^7*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) - 8*b/(c 
^8*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 10*b*arcsin(1/(c*x))/(c^9*x^3*(sq 
rt(-1/(c^2*x^2) + 1) + 1)^3) + 10*a/(c^9*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^ 
3) - b/(c^10*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) + 2*b*arcsin(1/(c*x))/(c^ 
11*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 2*a/(c^11*x^5*(sqrt(-1/(c^2*x^2) 
+ 1) + 1)^5))*c
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\left (\int \mathit {acsc} \left (c x \right ) x^{4}d x \right ) b +\frac {a \,x^{5}}{5} \] Input:

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

Output:

(5*int(acsc(c*x)*x**4,x)*b + a*x**5)/5