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

Optimal result

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

[Out]

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)+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)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {14, 5347, 12, 470, 327, 223, 212} \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{5} d x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (42 c^2 d+25 e\right )}{560 c^6 \sqrt {c^2 x^2}}+\frac {b e x^6 \sqrt {c^2 x^2-1}}{42 c \sqrt {c^2 x^2}}+\frac {b x^2 \sqrt {c^2 x^2-1} \left (42 c^2 d+25 e\right )}{560 c^5 \sqrt {c^2 x^2}}+\frac {b x^4 \sqrt {c^2 x^2-1} \left (42 c^2 d+25 e\right )}{840 c^3 \sqrt {c^2 x^2}} \]

[In]

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

[Out]

(b*(42*c^2*d + 25*e)*x^2*Sqrt[-1 + c^2*x^2])/(560*c^5*Sqrt[c^2*x^2]) + (b*(42*c^2*d + 25*e)*x^4*Sqrt[-1 + c^2*
x^2])/(840*c^3*Sqrt[c^2*x^2]) + (b*e*x^6*Sqrt[-1 + c^2*x^2])/(42*c*Sqrt[c^2*x^2]) + (d*x^5*(a + b*ArcCsc[c*x])
)/5 + (e*x^7*(a + b*ArcCsc[c*x]))/7 + (b*(42*c^2*d + 25*e)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(560*c^6*Sqrt[
c^2*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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 327

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

Rule 470

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

Rule 5347

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]}, Dist[a + b*ArcCsc[c*x], u, x] + Dist[b*c*(x/Sqrt[c^2*x^2]), Int[SimplifyI
ntegrand[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])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps \begin{align*} \text {integral}& = \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) \int \frac {x^4 \left (7 d+5 e x^2\right )}{35 \sqrt {-1+c^2 x^2}} \, 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 )+\frac {(b c x) \int \frac {x^4 \left (7 d+5 e x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{35 \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 {\left (b c \left (-42 d-\frac {25 e}{c^2}\right ) x\right ) \int \frac {x^4}{\sqrt {-1+c^2 x^2}} \, dx}{210 \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 {\left (b \left (-42 d-\frac {25 e}{c^2}\right ) x\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2}} \, dx}{280 c \sqrt {c^2 x^2}} \\ & = \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 {\left (b \left (-42 d-\frac {25 e}{c^2}\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{560 c^3 \sqrt {c^2 x^2}} \\ & = \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 {\left (b \left (-42 d-\frac {25 e}{c^2}\right ) x\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{560 c^3 \sqrt {c^2 x^2}} \\ & = \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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.68 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {48 a c^7 x^5 \left (7 d+5 e x^2\right )+b c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (75 e+2 c^2 \left (63 d+25 e x^2\right )+c^4 \left (84 d x^2+40 e x^4\right )\right )+48 b c^7 x^5 \left (7 d+5 e x^2\right ) \csc ^{-1}(c x)+3 b \left (42 c^2 d+25 e\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{1680 c^7} \]

[In]

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

[Out]

(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)

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.59

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

[In]

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

[Out]

a*(1/7*e*x^7+1/5*d*x^5)+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)

none

Time = 0.37 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.93 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {240 \, a c^{7} e x^{7} + 336 \, a c^{7} d x^{5} + 48 \, {\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5} - 7 \, b c^{7} d - 5 \, b c^{7} e\right )} \operatorname {arccsc}\left (c x\right ) - 96 \, {\left (7 \, b c^{7} d + 5 \, b c^{7} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 3 \, {\left (42 \, b c^{2} d + 25 \, b e\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (40 \, b c^{5} e x^{5} + 2 \, {\left (42 \, b c^{5} d + 25 \, b c^{3} e\right )} x^{3} + 3 \, {\left (42 \, b c^{3} d + 25 \, b c e\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{1680 \, c^{7}} \]

[In]

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

[Out]

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)*arctan(-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 = 10.79 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.98 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d x^{5}}{5} + \frac {a e x^{7}}{7} + \frac {b d x^{5} \operatorname {acsc}{\left (c x \right )}}{5} + \frac {b e x^{7} \operatorname {acsc}{\left (c x \right )}}{7} + \frac {b d \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} + \frac {b e \left (\begin {cases} \frac {c x^{7}}{6 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{5}}{24 c \sqrt {c^{2} x^{2} - 1}} + \frac {5 x^{3}}{48 c^{3} \sqrt {c^{2} x^{2} - 1}} - \frac {5 x}{16 c^{5} \sqrt {c^{2} x^{2} - 1}} + \frac {5 \operatorname {acosh}{\left (c x \right )}}{16 c^{6}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{7}}{6 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{5}}{24 c \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i x^{3}}{48 c^{3} \sqrt {- c^{2} x^{2} + 1}} + \frac {5 i x}{16 c^{5} \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i \operatorname {asin}{\left (c x \right )}}{16 c^{6}} & \text {otherwise} \end {cases}\right )}{7 c} \]

[In]

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

[Out]

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), True))/(5*c) + b*e*Piecewise((c*x**7/(6*sqrt(c**2*x**2 - 1)) + x**5/(24*
c*sqrt(c**2*x**2 - 1)) + 5*x**3/(48*c**3*sqrt(c**2*x**2 - 1)) - 5*x/(16*c**5*sqrt(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)

none

Time = 0.20 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.44 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{7} \, a e x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arccsc}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, {\left (15 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e \]

[In]

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

[Out]

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(
sqrt(-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(sqrt(-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 = 1.66 (sec) , antiderivative size = 1166, normalized size of antiderivative = 5.66 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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(sqrt(-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*(sqrt(-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*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 420*b*d*arcsin(1/(c*x))/(c^9*
x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 420*a*d/(c^9*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 315*b*e*arcsin(1/(c*x
))/(c^11*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 315*a*e/(c^11*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) - 42*b*d/(c^1
0*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) - 45*b*e/(c^12*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) + 84*b*d*arcsin(1/(c*
x))/(c^11*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 84*a*d/(c^11*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 105*b*e*arc
sin(1/(c*x))/(c^13*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 105*a*e/(c^13*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) - 5
*b*e/(c^14*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6) + 15*b*e*arcsin(1/(c*x))/(c^15*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)
^7) + 15*a*e/(c^15*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))*c

Mupad [F(-1)]

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

[In]

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

[Out]

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