\(\int \frac {x^4 (a+b \csc ^{-1}(c x))}{(d+e x^2)^3} \, dx\) [115]

Optimal result

Integrand size = 21, antiderivative size = 1144 \[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Output:

-1/16*b*c*(-d)^(1/2)*(1-1/c^2/x^2)^(1/2)/e^(3/2)/(c^2*d+e)/((-d)^(1/2)*e^( 
1/2)-d/x)-1/16*b*c*(-d)^(1/2)*(1-1/c^2/x^2)^(1/2)/e^(3/2)/(c^2*d+e)/((-d)^ 
(1/2)*e^(1/2)+d/x)+1/16*(-d)^(1/2)*(a+b*arccsc(c*x))/e^(3/2)/((-d)^(1/2)*e 
^(1/2)-d/x)^2+3/16*(a+b*arccsc(c*x))/e^2/((-d)^(1/2)*e^(1/2)-d/x)-1/16*(-d 
)^(1/2)*(a+b*arccsc(c*x))/e^(3/2)/((-d)^(1/2)*e^(1/2)+d/x)^2-3/16*(a+b*arc 
csc(c*x))/e^2/((-d)^(1/2)*e^(1/2)+d/x)-1/16*b*arctanh((c^2*d-(-d)^(1/2)*e^ 
(1/2)/x)/c/d^(1/2)/(c^2*d+e)^(1/2)/(1-1/c^2/x^2)^(1/2))/d^(1/2)/e/(c^2*d+e 
)^(3/2)-3/16*b*arctanh((c^2*d-(-d)^(1/2)*e^(1/2)/x)/c/d^(1/2)/(c^2*d+e)^(1 
/2)/(1-1/c^2/x^2)^(1/2))/d^(1/2)/e^2/(c^2*d+e)^(1/2)-1/16*b*arctanh((c^2*d 
+(-d)^(1/2)*e^(1/2)/x)/c/d^(1/2)/(c^2*d+e)^(1/2)/(1-1/c^2/x^2)^(1/2))/d^(1 
/2)/e/(c^2*d+e)^(3/2)-3/16*b*arctanh((c^2*d+(-d)^(1/2)*e^(1/2)/x)/c/d^(1/2 
)/(c^2*d+e)^(1/2)/(1-1/c^2/x^2)^(1/2))/d^(1/2)/e^2/(c^2*d+e)^(1/2)-3/16*(a 
+b*arccsc(c*x))*ln(1-I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)-( 
c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(5/2)+3/16*(a+b*arccsc(c*x))*ln(1+I*c*(-d)^( 
1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^( 
5/2)-3/16*(a+b*arccsc(c*x))*ln(1-I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2) 
)/(e^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(5/2)+3/16*(a+b*arccsc(c*x))*ln( 
1+I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/(- 
d)^(1/2)/e^(5/2)-3/16*I*b*polylog(2,-I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^( 
1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(5/2)+3/16*I*b*polylog(2,...
 

Mathematica [A] (warning: unable to verify)

Time = 6.07 (sec) , antiderivative size = 2067, normalized size of antiderivative = 1.81 \[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \] Input:

Integrate[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3,x]
 

Output:

(a*d*x)/(4*e^2*(d + e*x^2)^2) - (5*a*x)/(8*e^2*(d + e*x^2)) + (3*a*ArcTan[ 
(Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*e^(5/2)) + b*((5*(-(ArcCsc[c*x]/((-I)*Sqr 
t[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e 
]*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)]) 
*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sq 
rt[d]))/(16*e^2) + (5*(-(ArcCsc[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcS 
in[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] 
 + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt 
[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(16*e^2) + ((I/16)*Sqr 
t[d]*((I*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*((-I)*Sqr 
t[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - 
ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2*d 
 + e]*(I*Sqrt[e] + c*(c*Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x 
))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/e^ 
2 - ((I/16)*Sqrt[d]*(((-I)*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^ 
2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt 
[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(-4*d*Sqrt[ 
e]*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 
 1/(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e 
)^(3/2))))/e^2 - (3*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*Arc...
 

Rubi [A] (verified)

Time = 2.18 (sec) , antiderivative size = 1208, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5764, 5172, 2009}

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.

\(\displaystyle \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 5764

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

\(\Big \downarrow \) 5172

\(\displaystyle -\int \left (-\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) d^3}{8 (-d)^{3/2} e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^3}-\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) d^3}{8 (-d)^{3/2} e^{3/2} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^3}-\frac {3 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) d}{8 e^2 \left (-\frac {d^2}{x^2}-e d\right )}-\frac {3 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) d}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {3 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) d}{16 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}\right )d\frac {1}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \sqrt {-d} \sqrt {1-\frac {1}{c^2 x^2}} c}{16 e^{3/2} \left (d c^2+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b \sqrt {-d} \sqrt {1-\frac {1}{c^2 x^2}} c}{16 e^{3/2} \left (d c^2+e\right ) \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {3 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {3 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{16 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {\sqrt {-d} \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {\sqrt {-d} \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{16 e^{3/2} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}-\frac {3 b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 \sqrt {d} e^2 \sqrt {d c^2+e}}-\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 \sqrt {d} e \left (d c^2+e\right )^{3/2}}-\frac {3 b \text {arctanh}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 \sqrt {d} e^2 \sqrt {d c^2+e}}-\frac {b \text {arctanh}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 \sqrt {d} e \left (d c^2+e\right )^{3/2}}-\frac {3 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (\frac {i \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (\frac {i \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}\)

Input:

Int[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3,x]
 

Output:

-1/16*(b*c*Sqrt[-d]*Sqrt[1 - 1/(c^2*x^2)])/(e^(3/2)*(c^2*d + e)*(Sqrt[-d]* 
Sqrt[e] - d/x)) - (b*c*Sqrt[-d]*Sqrt[1 - 1/(c^2*x^2)])/(16*e^(3/2)*(c^2*d 
+ e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[-d]*(a + b*ArcSin[1/(c*x)]))/(16*e^ 
(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (3*(a + b*ArcSin[1/(c*x)]))/(16*e^2*(S 
qrt[-d]*Sqrt[e] - d/x)) - (Sqrt[-d]*(a + b*ArcSin[1/(c*x)]))/(16*e^(3/2)*( 
Sqrt[-d]*Sqrt[e] + d/x)^2) - (3*(a + b*ArcSin[1/(c*x)]))/(16*e^2*(Sqrt[-d] 
*Sqrt[e] + d/x)) - (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sq 
rt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*Sqrt[d]*e*(c^2*d + e)^(3/2)) - 
(3*b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqr 
t[1 - 1/(c^2*x^2)])])/(16*Sqrt[d]*e^2*Sqrt[c^2*d + e]) - (b*ArcTanh[(c^2*d 
 + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)]) 
])/(16*Sqrt[d]*e*(c^2*d + e)^(3/2)) - (3*b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt 
[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*Sqrt[d]*e^ 
2*Sqrt[c^2*d + e]) - (3*(a + b*ArcSin[1/(c*x)])*Log[1 - (I*c*Sqrt[-d]*E^(I 
*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + ( 
3*(a + b*ArcSin[1/(c*x)])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sq 
rt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcSin[1/(c*x 
)])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e 
])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*ArcSin[1/(c*x)])*Log[1 + (I*c*Sqrt[ 
-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5172

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G 
tQ[p, 0] || IGtQ[n, 0])
 

rule 5764

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcSin[x/c])^n/x^( 
m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] 
&& IntegerQ[m] && IntegerQ[p]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 77.12 (sec) , antiderivative size = 1804, normalized size of antiderivative = 1.58

method	result	size
parts	\(\text {Expression too large to display}\)	\(1804\)
derivativedivides	\(\text {Expression too large to display}\)	\(1827\)
default	\(\text {Expression too large to display}\)	\(1827\)

Input:

int(x^4*(a+b*arccsc(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

a*((-5/8/e*x^3-3/8*d/e^2*x)/(e*x^2+d)^2+3/8/e^2/(d*e)^(1/2)*arctan(e*x/(d* 
e)^(1/2)))+b/c^5*(-1/8*x*c^7*(3*d^2*c^4*arccsc(c*x)+5*c^4*d*e*arccsc(c*x)* 
x^2+((c^2*x^2-1)/c^2/x^2)^(1/2)*c^3*d*e*x+((c^2*x^2-1)/c^2/x^2)^(1/2)*e^2* 
c^3*x^3+3*c^2*d*e*arccsc(c*x)+5*e^2*arccsc(c*x)*c^2*x^2)/e^2/(c^2*d+e)/(c^ 
2*e*x^2+c^2*d)^2+3/8*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*(-(e*(c^2 
*d+e))^(1/2)*c^2*d+2*c^2*d*e-2*(e*(c^2*d+e))^(1/2)*e+2*e^2)*c^3*arctanh(c* 
d*(I/c/x+(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)) 
/(c^2*d+e)^2/e^2/d^2-3/8*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*(c^2 
*d+2*(e*(c^2*d+e))^(1/2)+2*e)*c^3*arctan(c*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/( 
(-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/(c^2*d+e)/e^2/d^2-3/16/(c^2*d 
+e)/e*c^6*sum(1/_R1/(_R1^2*c^2*d-c^2*d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-( 
1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=R 
ootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))-3/16/(c^2*d+e)/e^2*c^8*d*sum( 
1/_R1/(_R1^2*c^2*d-c^2*d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-(1-1/c^2/x^2)^( 
1/2))/_R1)+dilog((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z 
^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))-3/16/(c^2*d+e)/e*c^6*sum(_R1/(_R1^2*c^2*d-c 
^2*d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R 
1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z 
^2+c^2*d))+1/2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*((e*(c^2*d+e)) 
^(1/2)*c^2*d+2*c^2*d*e+2*(e*(c^2*d+e))^(1/2)*e+2*e^2)*c*arctan(c*d*(I/c...
 

Fricas [F]

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

integrate(x^4*(a+b*arccsc(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
 

Output:

integral((b*x^4*arccsc(c*x) + a*x^4)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 
+ d^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**4*(a+b*acsc(c*x))/(e*x**2+d)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^4*(a+b*arccsc(c*x))/(e*x^2+d)^3,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4*(a+b*arccsc(c*x))/(e*x^2+d)^3,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. \[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:

int((x^4*(a + b*asin(1/(c*x))))/(d + e*x^2)^3,x)
 

Output:

int((x^4*(a + b*asin(1/(c*x))))/(d + e*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,d^{2}+6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d e \,x^{2}+3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{2} x^{4}+8 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{4}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{3} e^{3}+16 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{4}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{2} e^{4} x^{2}+8 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{4}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b d \,e^{5} x^{4}-3 a \,d^{2} e x -5 a d \,e^{2} x^{3}}{8 d \,e^{3} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int(x^4*(a+b*acsc(c*x))/(e*x^2+d)^3,x)
                                                                                    
                                                                                    
 

Output:

(3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d**2 + 6*sqrt(e)*sqrt(d 
)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e*x**2 + 3*sqrt(e)*sqrt(d)*atan((e*x)/ 
(sqrt(e)*sqrt(d)))*a*e**2*x**4 + 8*int((acsc(c*x)*x**4)/(d**3 + 3*d**2*e*x 
**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**3*e**3 + 16*int((acsc(c*x)*x**4)/ 
(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**2*e**4*x**2 + 8 
*int((acsc(c*x)*x**4)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x 
)*b*d*e**5*x**4 - 3*a*d**2*e*x - 5*a*d*e**2*x**3)/(8*d*e**3*(d**2 + 2*d*e* 
x**2 + e**2*x**4))