\(\int x^4 (d+e x^2) (a+b \csc ^{-1}(c x)) \, dx\) [76]

Optimal result

Integrand size = 19, antiderivative size = 206 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (42 c^2 d+25 e\right ) x^2 \sqrt {-1+c^2 x^2}}{560 c^5 \sqrt {c^2 x^2}}+\frac {b \left (42 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}+\frac {b e x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (42 c^2 d+25 e\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{560 c^6 \sqrt {c^2 x^2}} \] Output:

1/560*b*(42*c^2*d+25*e)*x^2*(c^2*x^2-1)^(1/2)/c^5/(c^2*x^2)^(1/2)+1/840*b* 
(42*c^2*d+25*e)*x^4*(c^2*x^2-1)^(1/2)/c^3/(c^2*x^2)^(1/2)+1/42*b*e*x^6*(c^ 
2*x^2-1)^(1/2)/c/(c^2*x^2)^(1/2)+1/5*d*x^5*(a+b*arccsc(c*x))+1/7*e*x^7*(a+ 
b*arccsc(c*x))+1/560*b*(42*c^2*d+25*e)*x*arctanh(c*x/(c^2*x^2-1)^(1/2))/c^ 
6/(c^2*x^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.68 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {48 a c^7 x^5 \left (7 d+5 e x^2\right )+b c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (75 e+2 c^2 \left (63 d+25 e x^2\right )+c^4 \left (84 d x^2+40 e x^4\right )\right )+48 b c^7 x^5 \left (7 d+5 e x^2\right ) \csc ^{-1}(c x)+3 b \left (42 c^2 d+25 e\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{1680 c^7} \] Input:

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

Output:

(48*a*c^7*x^5*(7*d + 5*e*x^2) + b*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2*(75*e + 2* 
c^2*(63*d + 25*e*x^2) + c^4*(84*d*x^2 + 40*e*x^4)) + 48*b*c^7*x^5*(7*d + 5 
*e*x^2)*ArcCsc[c*x] + 3*b*(42*c^2*d + 25*e)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)] 
)*x])/(1680*c^7)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.81, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5762, 27, 363, 262, 262, 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.

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

\(\Big \downarrow \) 5762

\(\displaystyle \frac {b c x \int \frac {x^4 \left (5 e x^2+7 d\right )}{35 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c x \int \frac {x^4 \left (5 e x^2+7 d\right )}{\sqrt {c^2 x^2-1}}dx}{35 \sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {b c x \left (\frac {1}{6} \left (\frac {25 e}{c^2}+42 d\right ) \int \frac {x^4}{\sqrt {c^2 x^2-1}}dx+\frac {5 e x^5 \sqrt {c^2 x^2-1}}{6 c^2}\right )}{35 \sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {b c x \left (\frac {1}{6} \left (\frac {25 e}{c^2}+42 d\right ) \left (\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2-1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )+\frac {5 e x^5 \sqrt {c^2 x^2-1}}{6 c^2}\right )}{35 \sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {b c x \left (\frac {1}{6} \left (\frac {25 e}{c^2}+42 d\right ) \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )+\frac {5 e x^5 \sqrt {c^2 x^2-1}}{6 c^2}\right )}{35 \sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {b c x \left (\frac {1}{6} \left (\frac {25 e}{c^2}+42 d\right ) \left (\frac {3 \left (\frac {\int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )+\frac {5 e x^5 \sqrt {c^2 x^2-1}}{6 c^2}\right )}{35 \sqrt {c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \csc ^{-1}(c x)\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{5} d x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac {b c x \left (\frac {1}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right ) \left (\frac {25 e}{c^2}+42 d\right )+\frac {5 e x^5 \sqrt {c^2 x^2-1}}{6 c^2}\right )}{35 \sqrt {c^2 x^2}}\)

Input:

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

Output:

(d*x^5*(a + b*ArcCsc[c*x]))/5 + (e*x^7*(a + b*ArcCsc[c*x]))/7 + (b*c*x*((5 
*e*x^5*Sqrt[-1 + c^2*x^2])/(6*c^2) + ((42*d + (25*e)/c^2)*((x^3*Sqrt[-1 + 
c^2*x^2])/(4*c^2) + (3*((x*Sqrt[-1 + c^2*x^2])/(2*c^2) + ArcTanh[(c*x)/Sqr 
t[-1 + c^2*x^2]]/(2*c^3)))/(4*c^2)))/6))/(35*Sqrt[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 262

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

rule 363

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

rule 5762

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x 
_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Sim 
p[(a + b*ArcCsc[c*x])   u, x] + Simp[b*c*(x/Sqrt[c^2*x^2])   Int[SimplifyIn 
tegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, 
 p}, x] && ((IGtQ[p, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) | 
| (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (ILtQ[(m 
 + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.59

method	result	size
parts	\(a \left (\frac {1}{7} e \,x^{7}+\frac {1}{5} x^{5} d \right )+\frac {b \,\operatorname {arccsc}\left (c x \right ) e \,x^{7}}{7}+\frac {b \,\operatorname {arccsc}\left (c x \right ) x^{5} d}{5}+\frac {b \left (c^{2} x^{2}-1\right ) x^{4} e}{42 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2} d}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e}{168 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right ) d}{40 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \left (c^{2} x^{2}-1\right ) e}{112 c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{6} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {5 b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{8} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\)	\(328\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d \,c^{5} x^{5}}{5}+\frac {b \,c^{5} \operatorname {arccsc}\left (c x \right ) e \,x^{7}}{7}+\frac {b \left (c^{2} x^{2}-1\right ) c^{2} x^{2} d}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,c^{2} \left (c^{2} x^{2}-1\right ) x^{4} e}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right ) d}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e}{168 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {5 b \left (c^{2} x^{2}-1\right ) e}{112 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{5}}\)	\(341\)
default	\(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d \,c^{5} x^{5}}{5}+\frac {b \,c^{5} \operatorname {arccsc}\left (c x \right ) e \,x^{7}}{7}+\frac {b \left (c^{2} x^{2}-1\right ) c^{2} x^{2} d}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,c^{2} \left (c^{2} x^{2}-1\right ) x^{4} e}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right ) d}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e}{168 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {5 b \left (c^{2} x^{2}-1\right ) e}{112 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{5}}\)	\(341\)

Input:

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

Output:

a*(1/7*e*x^7+1/5*x^5*d)+1/7*b*arccsc(c*x)*e*x^7+1/5*b*arccsc(c*x)*x^5*d+1/ 
42*b/c^3*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^4*e+1/20*b/c^3*(c^2*x^2 
-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2*d+5/168*b/c^5*(c^2*x^2-1)/((c^2*x^2-1) 
/c^2/x^2)^(1/2)*x^2*e+3/40*b/c^5*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*d 
+5/112*b/c^7*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*e+3/40*b/c^6*(c^2*x^2 
-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*d*ln(c*x+(c^2*x^2-1)^(1/2))+5/112* 
b/c^8*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*e*ln(c*x+(c^2*x^2-1) 
^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.93 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {240 \, a c^{7} e x^{7} + 336 \, a c^{7} d x^{5} + 48 \, {\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5} - 7 \, b c^{7} d - 5 \, b c^{7} e\right )} \operatorname {arccsc}\left (c x\right ) - 96 \, {\left (7 \, b c^{7} d + 5 \, b c^{7} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 3 \, {\left (42 \, b c^{2} d + 25 \, b e\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (40 \, b c^{5} e x^{5} + 2 \, {\left (42 \, b c^{5} d + 25 \, b c^{3} e\right )} x^{3} + 3 \, {\left (42 \, b c^{3} d + 25 \, b c e\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{1680 \, c^{7}} \] Input:

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

Output:

1/1680*(240*a*c^7*e*x^7 + 336*a*c^7*d*x^5 + 48*(5*b*c^7*e*x^7 + 7*b*c^7*d* 
x^5 - 7*b*c^7*d - 5*b*c^7*e)*arccsc(c*x) - 96*(7*b*c^7*d + 5*b*c^7*e)*arct 
an(-c*x + sqrt(c^2*x^2 - 1)) - 3*(42*b*c^2*d + 25*b*e)*log(-c*x + sqrt(c^2 
*x^2 - 1)) + (40*b*c^5*e*x^5 + 2*(42*b*c^5*d + 25*b*c^3*e)*x^3 + 3*(42*b*c 
^3*d + 25*b*c*e)*x)*sqrt(c^2*x^2 - 1))/c^7
 

Sympy [A] (verification not implemented)

Time = 11.37 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.98 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d x^{5}}{5} + \frac {a e x^{7}}{7} + \frac {b d x^{5} \operatorname {acsc}{\left (c x \right )}}{5} + \frac {b e x^{7} \operatorname {acsc}{\left (c x \right )}}{7} + \frac {b d \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} + \frac {b e \left (\begin {cases} \frac {c x^{7}}{6 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{5}}{24 c \sqrt {c^{2} x^{2} - 1}} + \frac {5 x^{3}}{48 c^{3} \sqrt {c^{2} x^{2} - 1}} - \frac {5 x}{16 c^{5} \sqrt {c^{2} x^{2} - 1}} + \frac {5 \operatorname {acosh}{\left (c x \right )}}{16 c^{6}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{7}}{6 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{5}}{24 c \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i x^{3}}{48 c^{3} \sqrt {- c^{2} x^{2} + 1}} + \frac {5 i x}{16 c^{5} \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i \operatorname {asin}{\left (c x \right )}}{16 c^{6}} & \text {otherwise} \end {cases}\right )}{7 c} \] Input:

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

Output:

a*d*x**5/5 + a*e*x**7/7 + b*d*x**5*acsc(c*x)/5 + b*e*x**7*acsc(c*x)/7 + b* 
d*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), Tr 
ue))/(5*c) + b*e*Piecewise((c*x**7/(6*sqrt(c**2*x**2 - 1)) + x**5/(24*c*sq 
rt(c**2*x**2 - 1)) + 5*x**3/(48*c**3*sqrt(c**2*x**2 - 1)) - 5*x/(16*c**5*s 
qrt(c**2*x**2 - 1)) + 5*acosh(c*x)/(16*c**6), Abs(c**2*x**2) > 1), (-I*c*x 
**7/(6*sqrt(-c**2*x**2 + 1)) - I*x**5/(24*c*sqrt(-c**2*x**2 + 1)) - 5*I*x* 
*3/(48*c**3*sqrt(-c**2*x**2 + 1)) + 5*I*x/(16*c**5*sqrt(-c**2*x**2 + 1)) - 
 5*I*asin(c*x)/(16*c**6), True))/(7*c)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.44 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{7} \, a e x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arccsc}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, {\left (15 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e \] Input:

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

Output:

1/7*a*e*x^7 + 1/5*a*d*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(sq 
rt(-1/(c^2*x^2) + 1) - 1)/c^4)/c)*b*d + 1/672*(96*x^7*arccsc(c*x) + (2*(15 
*(-1/(c^2*x^2) + 1)^(5/2) - 40*(-1/(c^2*x^2) + 1)^(3/2) + 33*sqrt(-1/(c^2* 
x^2) + 1))/(c^6*(1/(c^2*x^2) - 1)^3 + 3*c^6*(1/(c^2*x^2) - 1)^2 + 3*c^6*(1 
/(c^2*x^2) - 1) + c^6) + 15*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^6 - 15*log(s 
qrt(-1/(c^2*x^2) + 1) - 1)/c^6)/c)*b*e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1166 vs. \(2 (178) = 356\).

Time = 1.45 (sec) , antiderivative size = 1166, normalized size of antiderivative = 5.66 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/13440*(15*b*e*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7*arcsin(1/(c*x))/c + 15* 
a*e*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7/c + 5*b*e*x^6*(sqrt(-1/(c^2*x^2) + 
1) + 1)^6/c^2 + 84*b*d*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*arcsin(1/(c*x))/ 
c + 84*a*d*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c + 105*b*e*x^5*(sqrt(-1/(c^ 
2*x^2) + 1) + 1)^5*arcsin(1/(c*x))/c^3 + 105*a*e*x^5*(sqrt(-1/(c^2*x^2) + 
1) + 1)^5/c^3 + 42*b*d*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c^2 + 45*b*e*x^4 
*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c^4 + 420*b*d*x^3*(sqrt(-1/(c^2*x^2) + 1) 
+ 1)^3*arcsin(1/(c*x))/c^3 + 420*a*d*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^ 
3 + 315*b*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*arcsin(1/(c*x))/c^5 + 315*a 
*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^5 + 336*b*d*x^2*(sqrt(-1/(c^2*x^2) 
 + 1) + 1)^2/c^4 + 225*b*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^6 + 840*b* 
d*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c^5 + 840*a*d*x*(sqrt(-1/ 
(c^2*x^2) + 1) + 1)/c^5 + 525*b*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/ 
(c*x))/c^7 + 525*a*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^7 + 1008*b*d*log(sqr 
t(-1/(c^2*x^2) + 1) + 1)/c^6 - 1008*b*d*log(1/(abs(c)*abs(x)))/c^6 + 600*b 
*e*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^8 - 600*b*e*log(1/(abs(c)*abs(x)))/c^ 
8 + 840*b*d*arcsin(1/(c*x))/(c^7*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 840*a*d 
/(c^7*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 525*b*e*arcsin(1/(c*x))/(c^9*x*(sq 
rt(-1/(c^2*x^2) + 1) + 1)) + 525*a*e/(c^9*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) 
- 336*b*d/(c^8*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) - 225*b*e/(c^10*x^2*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\left (\int \mathit {acsc} \left (c x \right ) x^{6}d x \right ) b e +\left (\int \mathit {acsc} \left (c x \right ) x^{4}d x \right ) b d +\frac {a d \,x^{5}}{5}+\frac {a e \,x^{7}}{7} \] Input:

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

Output:

(35*int(acsc(c*x)*x**6,x)*b*e + 35*int(acsc(c*x)*x**4,x)*b*d + 7*a*d*x**5 
+ 5*a*e*x**7)/35