3.1.77 \(\int x^2 (d+e x^2) (a+b \csc ^{-1}(c x)) \, dx\) [77]

Optimal. Leaf size=161 \[ \frac {b \left (20 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}+\frac {b e x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (20 c^2 d+9 e\right ) x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{120 c^4 \sqrt {c^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {14, 5347, 12, 470, 327, 223, 212} \begin {gather*} \frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e x^4 \sqrt {c^2 x^2-1}}{20 c \sqrt {c^2 x^2}}+\frac {b x \left (20 c^2 d+9 e\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{120 c^4 \sqrt {c^2 x^2}}+\frac {b x^2 \sqrt {c^2 x^2-1} \left (20 c^2 d+9 e\right )}{120 c^3 \sqrt {c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(20*c^2*d + 9*e)*x^2*Sqrt[-1 + c^2*x^2])/(120*c^3*Sqrt[c^2*x^2]) + (b*e*x^4*Sqrt[-1 + c^2*x^2])/(20*c*Sqrt[
c^2*x^2]) + (d*x^3*(a + b*ArcCsc[c*x]))/3 + (e*x^5*(a + b*ArcCsc[c*x]))/5 + (b*(20*c^2*d + 9*e)*x*ArcTanh[(c*x
)/Sqrt[-1 + c^2*x^2]])/(120*c^4*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*} \int x^2 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^2 \left (5 d+3 e x^2\right )}{15 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^2 \left (5 d+3 e x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{15 \sqrt {c^2 x^2}}\\ &=\frac {b e x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (b c \left (-20 d-\frac {9 e}{c^2}\right ) x\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2}} \, dx}{60 \sqrt {c^2 x^2}}\\ &=\frac {b \left (20 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 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (b \left (-20 d-\frac {9 e}{c^2}\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{120 c \sqrt {c^2 x^2}}\\ &=\frac {b \left (20 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 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (b \left (-20 d-\frac {9 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 )}{120 c \sqrt {c^2 x^2}}\\ &=\frac {b \left (20 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 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (20 c^2 d+9 e\right ) x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{120 c^4 \sqrt {c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 121, normalized size = 0.75 \begin {gather*} \frac {c^2 x^2 \left (8 a c^3 x \left (5 d+3 e x^2\right )+b \sqrt {1-\frac {1}{c^2 x^2}} \left (9 e+c^2 \left (20 d+6 e x^2\right )\right )\right )+8 b c^5 x^3 \left (5 d+3 e x^2\right ) \csc ^{-1}(c x)+b \left (20 c^2 d+9 e\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{120 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x^2*(8*a*c^3*x*(5*d + 3*e*x^2) + b*Sqrt[1 - 1/(c^2*x^2)]*(9*e + c^2*(20*d + 6*e*x^2))) + 8*b*c^5*x^3*(5*d
+ 3*e*x^2)*ArcCsc[c*x] + b*(20*c^2*d + 9*e)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(120*c^5)

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Maple [A]
time = 0.42, size = 267, normalized size = 1.66

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \,\mathrm {arccsc}\left (c x \right ) d \,c^{3} x^{3}}{3}+\frac {b \,c^{3} \mathrm {arccsc}\left (c x \right ) e \,x^{5}}{5}+\frac {b \left (c^{2} x^{2}-1\right ) d}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2} e}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b \left (c^{2} x^{2}-1\right ) e}{40 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{3}}\)	\(267\)
default	\(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \,\mathrm {arccsc}\left (c x \right ) d \,c^{3} x^{3}}{3}+\frac {b \,c^{3} \mathrm {arccsc}\left (c x \right ) e \,x^{5}}{5}+\frac {b \left (c^{2} x^{2}-1\right ) d}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2} e}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b \left (c^{2} x^{2}-1\right ) e}{40 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{3}}\)	\(267\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 234, normalized size = 1.45 \begin {gather*} \frac {1}{5} \, a x^{5} e + \frac {1}{3} \, a d x^{3} + \frac {1}{12} \, {\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 + \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*x^5*e + 1/3*a*d*x^3 + 1/12*(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 + 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

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Fricas [A]
time = 0.41, size = 177, normalized size = 1.10 \begin {gather*} \frac {24 \, a c^{5} x^{5} e + 40 \, a c^{5} d x^{3} + 8 \, {\left (5 \, b c^{5} d x^{3} - 5 \, b c^{5} d + 3 \, {\left (b c^{5} x^{5} - b c^{5}\right )} e\right )} \operatorname {arccsc}\left (c x\right ) - 16 \, {\left (5 \, b c^{5} d + 3 \, b c^{5} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (20 \, b c^{2} d + 9 \, b e\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (20 \, b c^{3} d x + 3 \, {\left (2 \, b c^{3} x^{3} + 3 \, b c x\right )} e\right )} \sqrt {c^{2} x^{2} - 1}}{120 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*a*c^5*x^5*e + 40*a*c^5*d*x^3 + 8*(5*b*c^5*d*x^3 - 5*b*c^5*d + 3*(b*c^5*x^5 - b*c^5)*e)*arccsc(c*x) -
16*(5*b*c^5*d + 3*b*c^5*e)*arctan(-c*x + sqrt(c^2*x^2 - 1)) - (20*b*c^2*d + 9*b*e)*log(-c*x + sqrt(c^2*x^2 -
1)) + (20*b*c^3*d*x + 3*(2*b*c^3*x^3 + 3*b*c*x)*e)*sqrt(c^2*x^2 - 1))/c^5

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Sympy [A]
time = 6.30, size = 294, normalized size = 1.83 \begin {gather*} \frac {a d x^{3}}{3} + \frac {a e x^{5}}{5} + \frac {b d x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b e x^{5} \operatorname {acsc}{\left (c x \right )}}{5} + \frac {b d \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 \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d*x**3/3 + a*e*x**5/5 + b*d*x**3*acsc(c*x)/3 + b*e*x**5*acsc(c*x)/5 + b*d*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*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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 822 vs. \(2 (139) = 278\).
time = 1.28, size = 822, normalized size = 5.11 \begin {gather*} \frac {1}{960} \, {\left (\frac {6 \, b e x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {6 \, a e x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}{c} + \frac {3 \, b e x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c^{2}} + \frac {40 \, b d x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {40 \, a d x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c} + \frac {30 \, b e x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {30 \, a e x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{3}} + \frac {40 \, b d x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{2}} + \frac {24 \, b e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{4}} + \frac {120 \, b d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {120 \, a d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{3}} + \frac {60 \, b e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {60 \, a e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{5}} + \frac {160 \, b d \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} - \frac {160 \, b d \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{4}} + \frac {72 \, b e \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {72 \, b e \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{6}} + \frac {120 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {120 \, a d}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {60 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{7} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {60 \, a e}{c^{7} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {40 \, b d}{c^{6} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} - \frac {24 \, b e}{c^{8} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {40 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {40 \, a d}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {30 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {30 \, a e}{c^{9} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} - \frac {3 \, b e}{c^{10} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {6 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{11} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}} + \frac {6 \, a e}{c^{11} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/960*(6*b*e*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*arcsin(1/(c*x))/c + 6*a*e*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c
+ 3*b*e*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c^2 + 40*b*d*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*arcsin(1/(c*x))/c
+ 40*a*d*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c + 30*b*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*arcsin(1/(c*x))/c^3
+ 30*a*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^3 + 40*b*d*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^2 + 24*b*e*x^2*(
sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^4 + 120*b*d*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c^3 + 120*a*d*x*(sq
rt(-1/(c^2*x^2) + 1) + 1)/c^3 + 60*b*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c^5 + 60*a*e*x*(sqrt(-1/
(c^2*x^2) + 1) + 1)/c^5 + 160*b*d*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 - 160*b*d*log(1/(abs(c)*abs(x)))/c^4 + 7
2*b*e*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^6 - 72*b*e*log(1/(abs(c)*abs(x)))/c^6 + 120*b*d*arcsin(1/(c*x))/(c^5*x
*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 120*a*d/(c^5*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 60*b*e*arcsin(1/(c*x))/(c^7*x*
(sqrt(-1/(c^2*x^2) + 1) + 1)) + 60*a*e/(c^7*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) - 40*b*d/(c^6*x^2*(sqrt(-1/(c^2*x^
2) + 1) + 1)^2) - 24*b*e/(c^8*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 40*b*d*arcsin(1/(c*x))/(c^7*x^3*(sqrt(-1/(
c^2*x^2) + 1) + 1)^3) + 40*a*d/(c^7*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 30*b*e*arcsin(1/(c*x))/(c^9*x^3*(sqr
t(-1/(c^2*x^2) + 1) + 1)^3) + 30*a*e/(c^9*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) - 3*b*e/(c^10*x^4*(sqrt(-1/(c^2*
x^2) + 1) + 1)^4) + 6*b*e*arcsin(1/(c*x))/(c^11*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 6*a*e/(c^11*x^5*(sqrt(-1
/(c^2*x^2) + 1) + 1)^5))*c

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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