3.1.83 \(\int x^5 (d+e x^2) (a+b \csc ^{-1}(c x)) \, dx\) [83]

Optimal. Leaf size=196 \[ \frac {b \left (4 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}+\frac {b \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}+\frac {b \left (4 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right ) \]

[Out]

1/6*d*x^6*(a+b*arccsc(c*x))+1/8*e*x^8*(a+b*arccsc(c*x))+1/72*b*(8*c^2*d+9*e)*x*(c^2*x^2-1)^(3/2)/c^7/(c^2*x^2)
^(1/2)+1/120*b*(4*c^2*d+9*e)*x*(c^2*x^2-1)^(5/2)/c^7/(c^2*x^2)^(1/2)+1/56*b*e*x*(c^2*x^2-1)^(7/2)/c^7/(c^2*x^2
)^(1/2)+1/24*b*(4*c^2*d+3*e)*x*(c^2*x^2-1)^(1/2)/c^7/(c^2*x^2)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 5347, 12, 457, 78} \begin {gather*} \frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \left (c^2 x^2-1\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^7 \sqrt {c^2 x^2}}+\frac {b x \left (c^2 x^2-1\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^7 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (4 c^2 d+3 e\right )}{24 c^7 \sqrt {c^2 x^2}}+\frac {b e x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(d + e*x^2)*(a + b*ArcCsc[c*x]),x]

[Out]

(b*(4*c^2*d + 3*e)*x*Sqrt[-1 + c^2*x^2])/(24*c^7*Sqrt[c^2*x^2]) + (b*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(3/2))/(
72*c^7*Sqrt[c^2*x^2]) + (b*(4*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7*Sqrt[c^2*x^2]) + (b*e*x*(-1 + c^2*
x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (d*x^6*(a + b*ArcCsc[c*x]))/6 + (e*x^8*(a + b*ArcCsc[c*x]))/8

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 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 457

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

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^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{24 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{24 \sqrt {c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \text {Subst}\left (\int \frac {x^2 (4 d+3 e x)}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \text {Subst}\left (\int \left (\frac {4 c^2 d+3 e}{c^6 \sqrt {-1+c^2 x}}+\frac {\left (8 c^2 d+9 e\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {\left (4 c^2 d+9 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {3 e \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}}\\ &=\frac {b \left (4 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}+\frac {b \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}+\frac {b \left (4 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 115, normalized size = 0.59 \begin {gather*} \frac {x \left (105 a x^5 \left (4 d+3 e x^2\right )+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} \left (144 e+8 c^2 \left (28 d+9 e x^2\right )+2 c^4 \left (56 d x^2+27 e x^4\right )+c^6 \left (84 d x^4+45 e x^6\right )\right )}{c^7}+105 b x^5 \left (4 d+3 e x^2\right ) \csc ^{-1}(c x)\right )}{2520} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(d + e*x^2)*(a + b*ArcCsc[c*x]),x]

[Out]

(x*(105*a*x^5*(4*d + 3*e*x^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*(144*e + 8*c^2*(28*d + 9*e*x^2) + 2*c^4*(56*d*x^2 + 2
7*e*x^4) + c^6*(84*d*x^4 + 45*e*x^6)))/c^7 + 105*b*x^5*(4*d + 3*e*x^2)*ArcCsc[c*x]))/2520

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Maple [A]
time = 0.42, size = 152, normalized size = 0.78

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arccsc}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\mathrm {arccsc}\left (c x \right ) e \,c^{8} x^{8}}{8}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}}\)	\(152\)
default	\(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arccsc}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\mathrm {arccsc}\left (c x \right ) e \,c^{8} x^{8}}{8}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}}\)	\(152\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(e*x^2+d)*(a+b*arccsc(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c^6*(a/c^2*(1/6*c^8*d*x^6+1/8*e*c^8*x^8)+b/c^2*(1/6*arccsc(c*x)*d*c^8*x^6+1/8*arccsc(c*x)*e*c^8*x^8+1/2520*(
c^2*x^2-1)*(45*c^6*e*x^6+84*c^6*d*x^4+54*c^4*e*x^4+112*c^4*d*x^2+72*c^2*e*x^2+224*c^2*d+144*e)/((c^2*x^2-1)/c^
2/x^2)^(1/2)/c/x))

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Maxima [A]
time = 0.27, size = 185, normalized size = 0.94 \begin {gather*} \frac {1}{8} \, a x^{8} e + \frac {1}{6} \, a d x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arccsc}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} + 21 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 35 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*arccsc(c*x)),x, algorithm="maxima")

[Out]

1/8*a*x^8*e + 1/6*a*d*x^6 + 1/90*(15*x^6*arccsc(c*x) + (3*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 10*c^2*x^3*(-1/(c
^2*x^2) + 1)^(3/2) + 15*x*sqrt(-1/(c^2*x^2) + 1))/c^5)*b*d + 1/280*(35*x^8*arccsc(c*x) + (5*c^6*x^7*(-1/(c^2*x
^2) + 1)^(7/2) + 21*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 35*c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 35*x*sqrt(-1/(c^2
*x^2) + 1))/c^7)*b*e

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Fricas [A]
time = 0.38, size = 130, normalized size = 0.66 \begin {gather*} \frac {315 \, a c^{8} x^{8} e + 420 \, a c^{8} d x^{6} + 105 \, {\left (3 \, b c^{8} x^{8} e + 4 \, b c^{8} d x^{6}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (84 \, b c^{6} d x^{4} + 112 \, b c^{4} d x^{2} + 224 \, b c^{2} d + 9 \, {\left (5 \, b c^{6} x^{6} + 6 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} + 16 \, b\right )} e\right )} \sqrt {c^{2} x^{2} - 1}}{2520 \, c^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*arccsc(c*x)),x, algorithm="fricas")

[Out]

1/2520*(315*a*c^8*x^8*e + 420*a*c^8*d*x^6 + 105*(3*b*c^8*x^8*e + 4*b*c^8*d*x^6)*arccsc(c*x) + (84*b*c^6*d*x^4
+ 112*b*c^4*d*x^2 + 224*b*c^2*d + 9*(5*b*c^6*x^6 + 6*b*c^4*x^4 + 8*b*c^2*x^2 + 16*b)*e)*sqrt(c^2*x^2 - 1))/c^8

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Sympy [A]
time = 5.36, size = 364, normalized size = 1.86 \begin {gather*} \frac {a d x^{6}}{6} + \frac {a e x^{8}}{8} + \frac {b d x^{6} \operatorname {acsc}{\left (c x \right )}}{6} + \frac {b e x^{8} \operatorname {acsc}{\left (c x \right )}}{8} + \frac {b d \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} + \frac {b e \left (\begin {cases} \frac {x^{6} \sqrt {c^{2} x^{2} - 1}}{7 c} + \frac {6 x^{4} \sqrt {c^{2} x^{2} - 1}}{35 c^{3}} + \frac {8 x^{2} \sqrt {c^{2} x^{2} - 1}}{35 c^{5}} + \frac {16 \sqrt {c^{2} x^{2} - 1}}{35 c^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{6} \sqrt {- c^{2} x^{2} + 1}}{7 c} + \frac {6 i x^{4} \sqrt {- c^{2} x^{2} + 1}}{35 c^{3}} + \frac {8 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{5}} + \frac {16 i \sqrt {- c^{2} x^{2} + 1}}{35 c^{7}} & \text {otherwise} \end {cases}\right )}{8 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(e*x**2+d)*(a+b*acsc(c*x)),x)

[Out]

a*d*x**6/6 + a*e*x**8/8 + b*d*x**6*acsc(c*x)/6 + b*e*x**8*acsc(c*x)/8 + b*d*Piecewise((x**4*sqrt(c**2*x**2 - 1
)/(5*c) + 4*x**2*sqrt(c**2*x**2 - 1)/(15*c**3) + 8*sqrt(c**2*x**2 - 1)/(15*c**5), Abs(c**2*x**2) > 1), (I*x**4
*sqrt(-c**2*x**2 + 1)/(5*c) + 4*I*x**2*sqrt(-c**2*x**2 + 1)/(15*c**3) + 8*I*sqrt(-c**2*x**2 + 1)/(15*c**5), Tr
ue))/(6*c) + b*e*Piecewise((x**6*sqrt(c**2*x**2 - 1)/(7*c) + 6*x**4*sqrt(c**2*x**2 - 1)/(35*c**3) + 8*x**2*sqr
t(c**2*x**2 - 1)/(35*c**5) + 16*sqrt(c**2*x**2 - 1)/(35*c**7), Abs(c**2*x**2) > 1), (I*x**6*sqrt(-c**2*x**2 +
1)/(7*c) + 6*I*x**4*sqrt(-c**2*x**2 + 1)/(35*c**3) + 8*I*x**2*sqrt(-c**2*x**2 + 1)/(35*c**5) + 16*I*sqrt(-c**2
*x**2 + 1)/(35*c**7), True))/(8*c)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1244 vs. \(2 (168) = 336\).
time = 0.54, size = 1244, normalized size = 6.35 \begin {gather*} \frac {1}{645120} \, {\left (\frac {315 \, b e x^{8} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{8} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {315 \, a e x^{8} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{8}}{c} + \frac {90 \, b e x^{7} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{7}}{c^{2}} + \frac {1680 \, b d x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {1680 \, a d x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}}{c} + \frac {2520 \, b e x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {2520 \, a e x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}}{c^{3}} + \frac {672 \, b d x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}{c^{2}} + \frac {882 \, b e x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}{c^{4}} + \frac {10080 \, b d x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {10080 \, a d x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c^{3}} + \frac {8820 \, b e x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {8820 \, a e x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c^{5}} + \frac {5600 \, b d x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{4}} + \frac {4410 \, b e x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{6}} + \frac {25200 \, b d x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {25200 \, a d x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{5}} + \frac {17640 \, b e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{7}} + \frac {17640 \, a e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{7}} + \frac {33600 \, b d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{6}} + \frac {22050 \, b e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{8}} + \frac {33600 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{7}} + \frac {33600 \, a d}{c^{7}} + \frac {22050 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{9}} + \frac {22050 \, a e}{c^{9}} - \frac {33600 \, b d}{c^{8} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {22050 \, b e}{c^{10} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {25200 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {25200 \, a d}{c^{9} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {17640 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{11} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {17640 \, a e}{c^{11} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} - \frac {5600 \, b d}{c^{10} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} - \frac {4410 \, b e}{c^{12} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {10080 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{11} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {10080 \, a d}{c^{11} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {8820 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{13} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {8820 \, a e}{c^{13} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} - \frac {672 \, b d}{c^{12} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}} - \frac {882 \, b e}{c^{14} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}} + \frac {1680 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{13} x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}} + \frac {1680 \, a d}{c^{13} x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}} + \frac {2520 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{15} x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}} + \frac {2520 \, a e}{c^{15} x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}} - \frac {90 \, b e}{c^{16} x^{7} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{7}} + \frac {315 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{17} x^{8} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{8}} + \frac {315 \, a e}{c^{17} x^{8} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{8}}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*arccsc(c*x)),x, algorithm="giac")

[Out]

1/645120*(315*b*e*x^8*(sqrt(-1/(c^2*x^2) + 1) + 1)^8*arcsin(1/(c*x))/c + 315*a*e*x^8*(sqrt(-1/(c^2*x^2) + 1) +
1)^8/c + 90*b*e*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7/c^2 + 1680*b*d*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*arcsin(1
/(c*x))/c + 1680*a*d*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6/c + 2520*b*e*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*arcsin
(1/(c*x))/c^3 + 2520*a*e*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6/c^3 + 672*b*d*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c
^2 + 882*b*e*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c^4 + 10080*b*d*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c
*x))/c^3 + 10080*a*d*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c^3 + 8820*b*e*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcs
in(1/(c*x))/c^5 + 8820*a*e*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c^5 + 5600*b*d*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^
3/c^4 + 4410*b*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^6 + 25200*b*d*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(
1/(c*x))/c^5 + 25200*a*d*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^5 + 17640*b*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2
*arcsin(1/(c*x))/c^7 + 17640*a*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^7 + 33600*b*d*x*(sqrt(-1/(c^2*x^2) + 1)
+ 1)/c^6 + 22050*b*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^8 + 33600*b*d*arcsin(1/(c*x))/c^7 + 33600*a*d/c^7 + 2205
0*b*e*arcsin(1/(c*x))/c^9 + 22050*a*e/c^9 - 33600*b*d/(c^8*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) - 22050*b*e/(c^10*x
*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 25200*b*d*arcsin(1/(c*x))/(c^9*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 25200*a*
d/(c^9*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 17640*b*e*arcsin(1/(c*x))/(c^11*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^
2) + 17640*a*e/(c^11*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) - 5600*b*d/(c^10*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3)
- 4410*b*e/(c^12*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 10080*b*d*arcsin(1/(c*x))/(c^11*x^4*(sqrt(-1/(c^2*x^2)
+ 1) + 1)^4) + 10080*a*d/(c^11*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) + 8820*b*e*arcsin(1/(c*x))/(c^13*x^4*(sqrt(
-1/(c^2*x^2) + 1) + 1)^4) + 8820*a*e/(c^13*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) - 672*b*d/(c^12*x^5*(sqrt(-1/(c
^2*x^2) + 1) + 1)^5) - 882*b*e/(c^14*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1680*b*d*arcsin(1/(c*x))/(c^13*x^6*
(sqrt(-1/(c^2*x^2) + 1) + 1)^6) + 1680*a*d/(c^13*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6) + 2520*b*e*arcsin(1/(c*x)
)/(c^15*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6) + 2520*a*e/(c^15*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6) - 90*b*e/(c^1
6*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7) + 315*b*e*arcsin(1/(c*x))/(c^17*x^8*(sqrt(-1/(c^2*x^2) + 1) + 1)^8) + 31
5*a*e/(c^17*x^8*(sqrt(-1/(c^2*x^2) + 1) + 1)^8))*c

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^5\,\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^5*(d + e*x^2)*(a + b*asin(1/(c*x))),x)

[Out]

int(x^5*(d + e*x^2)*(a + b*asin(1/(c*x))), x)

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