\(\int \frac {x (a+b \csc ^{-1}(c x))}{(d+e x^2)^2} \, dx\) [105]

Optimal result

Integrand size = 19, antiderivative size = 134 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {-a-b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {b c x \arctan \left (\sqrt {-1+c^2 x^2}\right )}{2 d e \sqrt {c^2 x^2}}+\frac {b c x \arctan \left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 d+e} \sqrt {c^2 x^2}} \] Output:

1/2*(-a-b*arccsc(c*x))/e/(e*x^2+d)-1/2*b*c*x*arctan((c^2*x^2-1)^(1/2))/d/e 
/(c^2*x^2)^(1/2)+1/2*b*c*x*arctan(e^(1/2)*(c^2*x^2-1)^(1/2)/(c^2*d+e)^(1/2 
))/d/e^(1/2)/(c^2*d+e)^(1/2)/(c^2*x^2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.13 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {\frac {2 a}{d+e x^2}+\frac {2 b \csc ^{-1}(c x)}{d+e x^2}-\frac {2 b \arcsin \left (\frac {1}{c x}\right )}{d}+\frac {b \sqrt {e} \log \left (\frac {4 i d e-4 c d \sqrt {e} \left (c \sqrt {d}+i \sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x}{b \sqrt {-c^2 d-e} \left (\sqrt {d}-i \sqrt {e} x\right )}\right )}{d \sqrt {-c^2 d-e}}+\frac {b \sqrt {e} \log \left (\frac {4 i \left (-d e+c d \sqrt {e} \left (i c \sqrt {d}+\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \sqrt {-c^2 d-e}}}{4 e} \] Input:

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

Output:

-1/4*((2*a)/(d + e*x^2) + (2*b*ArcCsc[c*x])/(d + e*x^2) - (2*b*ArcSin[1/(c 
*x)])/d + (b*Sqrt[e]*Log[((4*I)*d*e - 4*c*d*Sqrt[e]*(c*Sqrt[d] + I*Sqrt[-( 
c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x)/(b*Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*S 
qrt[e]*x))])/(d*Sqrt[-(c^2*d) - e]) + (b*Sqrt[e]*Log[((4*I)*(-(d*e) + c*d* 
Sqrt[e]*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*Sq 
rt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/(d*Sqrt[-(c^2*d) - e]))/e
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5760, 354, 97, 73, 218}

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.

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

\(\Big \downarrow \) 5760

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 97

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

\(\displaystyle -\frac {a+b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {b c x \left (\frac {2 \arctan \left (\sqrt {c^2 x^2-1}\right )}{d}-\frac {2 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{d \sqrt {c^2 d+e}}\right )}{4 e \sqrt {c^2 x^2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

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 97

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[b/(b*c - a*d)   Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c 
 - a*d)   Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, 
 x] &&  !IntegerQ[p]
 

rule 218

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

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 5760

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCsc[c*x])/(2*e*(p + 1))), x 
] + Simp[b*c*(x/(2*e*(p + 1)*Sqrt[c^2*x^2]))   Int[(d + e*x^2)^(p + 1)/(x*S 
qrt[c^2*x^2 - 1]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(262\) vs. \(2(112)=224\).

Time = 7.19 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.96

method	result	size
parts	\(-\frac {a}{2 e \left (e \,x^{2}+d \right )}+\frac {b \left (-\frac {c^{4} \operatorname {arccsc}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {c \sqrt {c^{2} x^{2}-1}\, \left (2 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\)	\(263\)
derivativedivides	\(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\)	\(276\)
default	\(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\)	\(276\)

Input:

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

Output:

-1/2*a/e/(e*x^2+d)+b/c^2*(-1/2*c^4/e/(c^2*e*x^2+c^2*d)*arccsc(c*x)+1/4*c/e 
*(c^2*x^2-1)^(1/2)*(2*arctan(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)-ln( 
-2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*e 
*x+(-c^2*d*e)^(1/2)))-ln(-2*(-(c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e+(-c 
^2*d*e)^(1/2)*c*x+e)/(c*e*x+(-c^2*d*e)^(1/2))))/((c^2*x^2-1)/c^2/x^2)^(1/2 
)/x/d/(-(c^2*d+e)/e)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.90 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\left [-\frac {2 \, a c^{2} d^{2} + 2 \, a d e + \sqrt {-c^{2} d e - e^{2}} {\left (b e x^{2} + b d\right )} \log \left (\frac {c^{2} e x^{2} - c^{2} d - 2 \, \sqrt {-c^{2} d e - e^{2}} \sqrt {c^{2} x^{2} - 1} - 2 \, e}{e x^{2} + d}\right ) + 2 \, {\left (b c^{2} d^{2} + b d e\right )} \operatorname {arccsc}\left (c x\right ) + 4 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{4 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} + a d e + \sqrt {c^{2} d e + e^{2}} {\left (b e x^{2} + b d\right )} \arctan \left (\frac {\sqrt {c^{2} d e + e^{2}} \sqrt {c^{2} x^{2} - 1}}{c^{2} e x^{2} - e}\right ) + {\left (b c^{2} d^{2} + b d e\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[-1/4*(2*a*c^2*d^2 + 2*a*d*e + sqrt(-c^2*d*e - e^2)*(b*e*x^2 + b*d)*log((c 
^2*e*x^2 - c^2*d - 2*sqrt(-c^2*d*e - e^2)*sqrt(c^2*x^2 - 1) - 2*e)/(e*x^2 
+ d)) + 2*(b*c^2*d^2 + b*d*e)*arccsc(c*x) + 4*(b*c^2*d^2 + b*d*e + (b*c^2* 
d*e + b*e^2)*x^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^3*e + d^2*e^2 + 
 (c^2*d^2*e^2 + d*e^3)*x^2), -1/2*(a*c^2*d^2 + a*d*e + sqrt(c^2*d*e + e^2) 
*(b*e*x^2 + b*d)*arctan(sqrt(c^2*d*e + e^2)*sqrt(c^2*x^2 - 1)/(c^2*e*x^2 - 
 e)) + (b*c^2*d^2 + b*d*e)*arccsc(c*x) + 2*(b*c^2*d^2 + b*d*e + (b*c^2*d*e 
 + b*e^2)*x^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^3*e + d^2*e^2 + (c 
^2*d^2*e^2 + d*e^3)*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*(2*(c^2*e^2*x^2 + c^2*d*e)*integrate(1/2*x*e^(1/2*log(c*x + 1) + 1/2* 
log(c*x - 1))/(c^2*e^2*x^4 + (c^2*d*e - e^2)*x^2 - d*e + (c^2*e^2*x^4 + (c 
^2*d*e - e^2)*x^2 - d*e)*e^(log(c*x + 1) + log(c*x - 1))), x) + arctan2(1, 
 sqrt(c*x + 1)*sqrt(c*x - 1)))*b/(e^2*x^2 + d*e) - 1/2*a/(e^2*x^2 + d*e)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*int((acsc(c*x)*x)/(d**2 + 2*d*e*x**2 + e**2*x**4),x)*b*d**2 + 2*int((ac 
sc(c*x)*x)/(d**2 + 2*d*e*x**2 + e**2*x**4),x)*b*d*e*x**2 + a*x**2)/(2*d*(d 
 + e*x**2))