\(\int (d+e x^2)^2 (a+b \csc ^{-1}(c x)) \, dx\) [89]

Optimal result

Integrand size = 18, antiderivative size = 191 \[ \int \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b e \left (40 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}+\frac {b e^2 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{120 c^4 \sqrt {c^2 x^2}} \] Output:

1/120*b*e*(40*c^2*d+9*e)*x^2*(c^2*x^2-1)^(1/2)/c^3/(c^2*x^2)^(1/2)+1/20*b* 
e^2*x^4*(c^2*x^2-1)^(1/2)/c/(c^2*x^2)^(1/2)+d^2*x*(a+b*arccsc(c*x))+2/3*d* 
e*x^3*(a+b*arccsc(c*x))+1/5*e^2*x^5*(a+b*arccsc(c*x))+1/120*b*(120*c^4*d^2 
+40*c^2*d*e+9*e^2)*x*arctanh(c*x/(c^2*x^2-1)^(1/2))/c^4/(c^2*x^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.79 \[ \int \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {c^2 x \left (8 a c^3 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b e \sqrt {1-\frac {1}{c^2 x^2}} x \left (9 e+c^2 \left (40 d+6 e x^2\right )\right )\right )+8 b c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \csc ^{-1}(c x)+b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{120 c^5} \] Input:

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

Output:

(c^2*x*(8*a*c^3*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 - 1/(c^2*x^ 
2)]*x*(9*e + c^2*(40*d + 6*e*x^2))) + 8*b*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e 
^2*x^4)*ArcCsc[c*x] + b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*Log[(1 + Sqrt[1 
 - 1/(c^2*x^2)])*x])/(120*c^5)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5752, 27, 1473, 299, 224, 219}

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.

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

\(\Big \downarrow \) 5752

\(\displaystyle \frac {b c x \int \frac {3 e^2 x^4+10 d e x^2+15 d^2}{15 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c x \int \frac {3 e^2 x^4+10 d e x^2+15 d^2}{\sqrt {c^2 x^2-1}}dx}{15 \sqrt {c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 1473

\(\displaystyle \frac {b c x \left (\frac {\int \frac {60 c^2 d^2+e \left (40 d c^2+9 e\right ) x^2}{\sqrt {c^2 x^2-1}}dx}{4 c^2}+\frac {3 e^2 x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b c x \left (\frac {\frac {\left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \int \frac {1}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (40 c^2 d+9 e\right )}{2 c^2}}{4 c^2}+\frac {3 e^2 x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {b c x \left (\frac {\frac {\left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (40 c^2 d+9 e\right )}{2 c^2}}{4 c^2}+\frac {3 e^2 x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 219

\(\displaystyle d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b c x \left (\frac {\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (120 c^4 d^2+40 c^2 d e+9 e^2\right )}{2 c^3}+\frac {e x \sqrt {c^2 x^2-1} \left (40 c^2 d+9 e\right )}{2 c^2}}{4 c^2}+\frac {3 e^2 x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {c^2 x^2}}\)

Input:

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

Output:

d^2*x*(a + b*ArcCsc[c*x]) + (2*d*e*x^3*(a + b*ArcCsc[c*x]))/3 + (e^2*x^5*( 
a + b*ArcCsc[c*x]))/5 + (b*c*x*((3*e^2*x^3*Sqrt[-1 + c^2*x^2])/(4*c^2) + ( 
(e*(40*c^2*d + 9*e)*x*Sqrt[-1 + c^2*x^2])/(2*c^2) + ((120*c^4*d^2 + 40*c^2 
*d*e + 9*e^2)*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(2*c^3))/(4*c^2)))/(15*Sq 
rt[c^2*x^2])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 1473

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> Simp[c^p*x^(4*p - 1)*((d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1))) 
, x] + Simp[1/(e*(4*p + 2*q + 1))   Int[(d + e*x^2)^q*ExpandToSum[e*(4*p + 
2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 
 2*q + 1)*x^(4*p), x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] &&  !LtQ[q, -1]
 

rule 5752

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symb 
ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCsc[c*x])   u, 
x] + Simp[b*c*(x/Sqrt[c^2*x^2])   Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 
 1]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1 
/2, 0])
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.77

method	result	size
parts	\(a \left (\frac {1}{5} e^{2} x^{5}+\frac {2}{3} d e \,x^{3}+d^{2} x \right )+\frac {b \,\operatorname {arccsc}\left (c x \right ) e^{2} x^{5}}{5}+\frac {2 b \,\operatorname {arccsc}\left (c x \right ) d e \,x^{3}}{3}+b \,\operatorname {arccsc}\left (c x \right ) d^{2} x +\frac {b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d e}{3 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right ) e^{2}}{40 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3 c^{4} x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{6} x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}\)	\(339\)
derivativedivides	\(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+b \,\operatorname {arccsc}\left (c x \right ) d^{2} c x +\frac {2 b c \,\operatorname {arccsc}\left (c x \right ) d e \,x^{3}}{3}+\frac {b c \,\operatorname {arccsc}\left (c x \right ) e^{2} x^{5}}{5}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b \left (c^{2} x^{2}-1\right ) d e}{3 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{20 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {3 b \left (c^{2} x^{2}-1\right ) e^{2}}{40 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\)	\(358\)
default	\(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+b \,\operatorname {arccsc}\left (c x \right ) d^{2} c x +\frac {2 b c \,\operatorname {arccsc}\left (c x \right ) d e \,x^{3}}{3}+\frac {b c \,\operatorname {arccsc}\left (c x \right ) e^{2} x^{5}}{5}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b \left (c^{2} x^{2}-1\right ) d e}{3 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{20 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {3 b \left (c^{2} x^{2}-1\right ) e^{2}}{40 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\)	\(358\)

Input:

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

Output:

a*(1/5*e^2*x^5+2/3*d*e*x^3+d^2*x)+1/5*b*arccsc(c*x)*e^2*x^5+2/3*b*arccsc(c 
*x)*d*e*x^3+b*arccsc(c*x)*d^2*x+1/20*b/c^3*(c^2*x^2-1)*x^2/((c^2*x^2-1)/c^ 
2/x^2)^(1/2)*e^2+1/3*b/c^3*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*d*e+b/c 
^2*(c^2*x^2-1)^(1/2)/x/((c^2*x^2-1)/c^2/x^2)^(1/2)*d^2*ln(c*x+(c^2*x^2-1)^ 
(1/2))+3/40*b/c^5*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*e^2+1/3*b/c^4*(c 
^2*x^2-1)^(1/2)/x/((c^2*x^2-1)/c^2/x^2)^(1/2)*d*e*ln(c*x+(c^2*x^2-1)^(1/2) 
)+3/40*b/c^6*(c^2*x^2-1)^(1/2)/x/((c^2*x^2-1)/c^2/x^2)^(1/2)*e^2*ln(c*x+(c 
^2*x^2-1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.24 \[ \int \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {24 \, a c^{5} e^{2} x^{5} + 80 \, a c^{5} d e x^{3} + 120 \, a c^{5} d^{2} x + 8 \, {\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x - 15 \, b c^{5} d^{2} - 10 \, b c^{5} d e - 3 \, b c^{5} e^{2}\right )} \operatorname {arccsc}\left (c x\right ) - 16 \, {\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (120 \, b c^{4} d^{2} + 40 \, b c^{2} d e + 9 \, b e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (6 \, b c^{3} e^{2} x^{3} + {\left (40 \, b c^{3} d e + 9 \, b c e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{120 \, c^{5}} \] Input:

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

Output:

1/120*(24*a*c^5*e^2*x^5 + 80*a*c^5*d*e*x^3 + 120*a*c^5*d^2*x + 8*(3*b*c^5* 
e^2*x^5 + 10*b*c^5*d*e*x^3 + 15*b*c^5*d^2*x - 15*b*c^5*d^2 - 10*b*c^5*d*e 
- 3*b*c^5*e^2)*arccsc(c*x) - 16*(15*b*c^5*d^2 + 10*b*c^5*d*e + 3*b*c^5*e^2 
)*arctan(-c*x + sqrt(c^2*x^2 - 1)) - (120*b*c^4*d^2 + 40*b*c^2*d*e + 9*b*e 
^2)*log(-c*x + sqrt(c^2*x^2 - 1)) + (6*b*c^3*e^2*x^3 + (40*b*c^3*d*e + 9*b 
*c*e^2)*x)*sqrt(c^2*x^2 - 1))/c^5
 

Sympy [A] (verification not implemented)

Time = 6.67 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.86 \[ \int \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=a d^{2} x + \frac {2 a d e x^{3}}{3} + \frac {a e^{2} x^{5}}{5} + b d^{2} x \operatorname {acsc}{\left (c x \right )} + \frac {2 b d e x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b e^{2} x^{5} \operatorname {acsc}{\left (c x \right )}}{5} + \frac {b d^{2} \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} + \frac {2 b d e \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} + \frac {b e^{2} \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} \] Input:

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

Output:

a*d**2*x + 2*a*d*e*x**3/3 + a*e**2*x**5/5 + b*d**2*x*acsc(c*x) + 2*b*d*e*x 
**3*acsc(c*x)/3 + b*e**2*x**5*acsc(c*x)/5 + b*d**2*Piecewise((acosh(c*x), 
Abs(c**2*x**2) > 1), (-I*asin(c*x), True))/c + 2*b*d*e*Piecewise((x*sqrt(c 
**2*x**2 - 1)/(2*c) + acosh(c*x)/(2*c**2), Abs(c**2*x**2) > 1), (-I*c*x**3 
/(2*sqrt(-c**2*x**2 + 1)) + I*x/(2*c*sqrt(-c**2*x**2 + 1)) - I*asin(c*x)/( 
2*c**2), True))/(3*c) + b*e**2*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*aco 
sh(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 = 0.05 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.55 \[ \int \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + \frac {1}{6} \, {\left (4 \, x^{3} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d e + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arccsc}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d^{2}}{2 \, c} \] Input:

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

Output:

1/5*a*e^2*x^5 + 2/3*a*d*e*x^3 + 1/6*(4*x^3*arccsc(c*x) + (2*sqrt(-1/(c^2*x 
^2) + 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + log(sqrt(-1/(c^2*x^2) + 1) + 1)/c 
^2 - log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^2)/c)*b*d*e + 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*e^2 + a*d^2*x 
 + 1/2*(2*c*x*arccsc(c*x) + log(sqrt(-1/(c^2*x^2) + 1) + 1) - log(-sqrt(-1 
/(c^2*x^2) + 1) + 1))*b*d^2/c
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1033 vs. \(2 (169) = 338\).

Time = 3.20 (sec) , antiderivative size = 1033, normalized size of antiderivative = 5.41 \[ \int \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/960*(6*b*e^2*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*arcsin(1/(c*x))/c + 6*a* 
e^2*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c + 3*b*e^2*x^4*(sqrt(-1/(c^2*x^2) 
+ 1) + 1)^4/c^2 + 80*b*d*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*arcsin(1/(c* 
x))/c + 80*a*d*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c + 30*b*e^2*x^3*(sqrt 
(-1/(c^2*x^2) + 1) + 1)^3*arcsin(1/(c*x))/c^3 + 30*a*e^2*x^3*(sqrt(-1/(c^2 
*x^2) + 1) + 1)^3/c^3 + 80*b*d*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^2 + 
480*b*d^2*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c + 480*a*d^2*x*( 
sqrt(-1/(c^2*x^2) + 1) + 1)/c + 24*b*e^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^ 
2/c^4 + 240*b*d*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c^3 + 240 
*a*d*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^3 + 960*b*d^2*log(sqrt(-1/(c^2*x^2 
) + 1) + 1)/c^2 - 960*b*d^2*log(1/(abs(c)*abs(x)))/c^2 + 60*b*e^2*x*(sqrt( 
-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c^5 + 60*a*e^2*x*(sqrt(-1/(c^2*x^2) 
 + 1) + 1)/c^5 + 320*b*d*e*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 - 320*b*d*e 
*log(1/(abs(c)*abs(x)))/c^4 + 480*b*d^2*arcsin(1/(c*x))/(c^3*x*(sqrt(-1/(c 
^2*x^2) + 1) + 1)) + 480*a*d^2/(c^3*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 72*b 
*e^2*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^6 - 72*b*e^2*log(1/(abs(c)*abs(x))) 
/c^6 + 240*b*d*e*arcsin(1/(c*x))/(c^5*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 24 
0*a*d*e/(c^5*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 60*b*e^2*arcsin(1/(c*x))/(c 
^7*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 60*a*e^2/(c^7*x*(sqrt(-1/(c^2*x^2) + 
1) + 1)) - 80*b*d*e/(c^6*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) - 24*b*e^2...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(15*int(acsc(c*x),x)*b*d**2 + 15*int(acsc(c*x)*x**4,x)*b*e**2 + 30*int(acs 
c(c*x)*x**2,x)*b*d*e + 15*a*d**2*x + 10*a*d*e*x**3 + 3*a*e**2*x**5)/15