3.156 \(\int \frac{x^5 (a+b \csc ^{-1}(c x))}{(d+e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=243 \[ -\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac{2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{b c d x \sqrt{c^2 x^2-1}}{3 e^2 \sqrt{c^2 x^2} \left (c^2 d+e\right ) \sqrt{d+e x^2}}-\frac{8 b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{3 e^3 \sqrt{c^2 x^2}}+\frac{b x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{e^{5/2} \sqrt{c^2 x^2}} \]

[Out]

(b*c*d*x*Sqrt[-1 + c^2*x^2])/(3*e^2*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2]) - (d^2*(a + b*ArcCsc[c*x]))/(3*
e^3*(d + e*x^2)^(3/2)) + (2*d*(a + b*ArcCsc[c*x]))/(e^3*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]
))/e^3 - (8*b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*e^3*Sqrt[c^2*x^2]) + (b*x*A
rcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(e^(5/2)*Sqrt[c^2*x^2])

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Rubi [A]  time = 1.0566, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {266, 43, 5239, 12, 1614, 157, 63, 217, 206, 93, 204} \[ -\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac{2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{b c d x \sqrt{c^2 x^2-1}}{3 e^2 \sqrt{c^2 x^2} \left (c^2 d+e\right ) \sqrt{d+e x^2}}-\frac{8 b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{3 e^3 \sqrt{c^2 x^2}}+\frac{b x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{e^{5/2} \sqrt{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*d*x*Sqrt[-1 + c^2*x^2])/(3*e^2*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2]) - (d^2*(a + b*ArcCsc[c*x]))/(3*
e^3*(d + e*x^2)^(3/2)) + (2*d*(a + b*ArcCsc[c*x]))/(e^3*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]
))/e^3 - (8*b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*e^3*Sqrt[c^2*x^2]) + (b*x*A
rcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(e^(5/2)*Sqrt[c^2*x^2])

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 5239

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]))

Rule 12

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

Rule 1614

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(b*R*(a + b*x)^(m + 1)
*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

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 206

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac{2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{(b c x) \int \frac{8 d^2+12 d e x^2+3 e^2 x^4}{3 e^3 x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac{2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{(b c x) \int \frac{8 d^2+12 d e x^2+3 e^2 x^4}{x \sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e^3 \sqrt{c^2 x^2}}\\ &=-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac{2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{(b c x) \operatorname{Subst}\left (\int \frac{8 d^2+12 d e x+3 e^2 x^2}{x \sqrt{-1+c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^3 \sqrt{c^2 x^2}}\\ &=\frac{b c d x \sqrt{-1+c^2 x^2}}{3 e^2 \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{d+e x^2}}-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac{2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{-4 d^2 \left (c^2 d+e\right )-\frac{3}{2} d e \left (c^2 d+e\right ) x}{x \sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{3 d e^3 \left (c^2 d+e\right ) \sqrt{c^2 x^2}}\\ &=\frac{b c d x \sqrt{-1+c^2 x^2}}{3 e^2 \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{d+e x^2}}-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac{2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{(4 b c d x) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{3 e^3 \sqrt{c^2 x^2}}+\frac{(b c x) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{2 e^2 \sqrt{c^2 x^2}}\\ &=\frac{b c d x \sqrt{-1+c^2 x^2}}{3 e^2 \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{d+e x^2}}-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac{2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{(8 b c d x) \operatorname{Subst}\left (\int \frac{1}{-d-x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}}\right )}{3 e^3 \sqrt{c^2 x^2}}+\frac{(b x) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+\frac{e}{c^2}+\frac{e x^2}{c^2}}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{c e^2 \sqrt{c^2 x^2}}\\ &=\frac{b c d x \sqrt{-1+c^2 x^2}}{3 e^2 \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{d+e x^2}}-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac{2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{8 b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{3 e^3 \sqrt{c^2 x^2}}+\frac{(b x) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{c^2}} \, dx,x,\frac{\sqrt{-1+c^2 x^2}}{\sqrt{d+e x^2}}\right )}{c e^2 \sqrt{c^2 x^2}}\\ &=\frac{b c d x \sqrt{-1+c^2 x^2}}{3 e^2 \left (c^2 d+e\right ) \sqrt{c^2 x^2} \sqrt{d+e x^2}}-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac{2 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{8 b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{3 e^3 \sqrt{c^2 x^2}}+\frac{b x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-1+c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{e^{5/2} \sqrt{c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.552179, size = 311, normalized size = 1.28 \[ \frac{a \left (c^2 d+e\right ) \left (8 d^2+12 d e x^2+3 e^2 x^4\right )+b \left (c^2 d+e\right ) \csc ^{-1}(c x) \left (8 d^2+12 d e x^2+3 e^2 x^4\right )+b c d e x \sqrt{1-\frac{1}{c^2 x^2}} \left (d+e x^2\right )}{3 e^3 \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left (8 c^3 \sqrt{d} \sqrt{d+e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right )+3 \sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right )\right )}{3 c^2 e^3 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*d*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2) + a*(c^2*d + e)*(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4) + b*(c^2*d + e)*
(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4)*ArcCsc[c*x])/(3*e^3*(c^2*d + e)*(d + e*x^2)^(3/2)) + (b*Sqrt[1 - 1/(c^2*x^2)]
*x*(3*Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x
^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 8*c^3*Sqrt[d]*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d +
e*x^2]]))/(3*c^2*e^3*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])

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Maple [F]  time = 3.697, size = 0, normalized size = 0. \begin{align*} \int{{x}^{5} \left ( a+b{\rm arccsc} \left (cx\right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.85175, size = 4487, normalized size = 18.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*(b*c^2*d^3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2*e + b*d*e^2)*x^2)*sqrt(e)*log(8*c^4*e
^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e - c^2*e^2)*x^2 + 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*sqr
t(e*x^2 + d)*sqrt(e) + e^2) + 8*(b*c^3*d^3 + b*c*d^2*e + (b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(b*c^3*d^2*e + b*c*d*
e^2)*x^2)*sqrt(-d)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 + 4*sqrt(c^2*x^2 - 1)*((c^2*d
- e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(-d) + 8*d^2)/x^4) + 4*(8*a*c^3*d^3 + 8*a*c*d^2*e + 3*(a*c^3*d*e^2 + a*c*e
^3)*x^4 + 12*(a*c^3*d^2*e + a*c*d*e^2)*x^2 + (8*b*c^3*d^3 + 8*b*c*d^2*e + 3*(b*c^3*d*e^2 + b*c*e^3)*x^4 + 12*(
b*c^3*d^2*e + b*c*d*e^2)*x^2)*arccsc(c*x) + (b*c*d*e^2*x^2 + b*c*d^2*e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c
^3*d^3*e^3 + c*d^2*e^4 + (c^3*d*e^5 + c*e^6)*x^4 + 2*(c^3*d^2*e^4 + c*d*e^5)*x^2), -1/12*(16*(b*c^3*d^3 + b*c*
d^2*e + (b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(b*c^3*d^2*e + b*c*d*e^2)*x^2)*sqrt(d)*arctan(-1/2*sqrt(c^2*x^2 - 1)*(
(c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(d)/(c^2*d*e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) - 3*(b*c^2*d^3 + (b*
c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2*e + b*d*e^2)*x^2)*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*
d*e + 8*(c^4*d*e - c^2*e^2)*x^2 + 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(e) + e^
2) - 4*(8*a*c^3*d^3 + 8*a*c*d^2*e + 3*(a*c^3*d*e^2 + a*c*e^3)*x^4 + 12*(a*c^3*d^2*e + a*c*d*e^2)*x^2 + (8*b*c^
3*d^3 + 8*b*c*d^2*e + 3*(b*c^3*d*e^2 + b*c*e^3)*x^4 + 12*(b*c^3*d^2*e + b*c*d*e^2)*x^2)*arccsc(c*x) + (b*c*d*e
^2*x^2 + b*c*d^2*e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^3*d^3*e^3 + c*d^2*e^4 + (c^3*d*e^5 + c*e^6)*x^4 + 2
*(c^3*d^2*e^4 + c*d*e^5)*x^2), -1/6*(3*(b*c^2*d^3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2*e + b*d
*e^2)*x^2)*sqrt(-e)*arctan(1/2*(2*c^2*e*x^2 + c^2*d - e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(-e)/(c^3*e^2*x
^4 - c*d*e + (c^3*d*e - c*e^2)*x^2)) - 4*(b*c^3*d^3 + b*c*d^2*e + (b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(b*c^3*d^2*e
+ b*c*d*e^2)*x^2)*sqrt(-d)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 + 4*sqrt(c^2*x^2 - 1)
*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(-d) + 8*d^2)/x^4) - 2*(8*a*c^3*d^3 + 8*a*c*d^2*e + 3*(a*c^3*d*e^
2 + a*c*e^3)*x^4 + 12*(a*c^3*d^2*e + a*c*d*e^2)*x^2 + (8*b*c^3*d^3 + 8*b*c*d^2*e + 3*(b*c^3*d*e^2 + b*c*e^3)*x
^4 + 12*(b*c^3*d^2*e + b*c*d*e^2)*x^2)*arccsc(c*x) + (b*c*d*e^2*x^2 + b*c*d^2*e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2
+ d))/(c^3*d^3*e^3 + c*d^2*e^4 + (c^3*d*e^5 + c*e^6)*x^4 + 2*(c^3*d^2*e^4 + c*d*e^5)*x^2), -1/6*(8*(b*c^3*d^3
+ b*c*d^2*e + (b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(b*c^3*d^2*e + b*c*d*e^2)*x^2)*sqrt(d)*arctan(-1/2*sqrt(c^2*x^2
- 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(d)/(c^2*d*e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) + 3*(b*c^2*d^
3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2*e + b*d*e^2)*x^2)*sqrt(-e)*arctan(1/2*(2*c^2*e*x^2 + c^
2*d - e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(-e)/(c^3*e^2*x^4 - c*d*e + (c^3*d*e - c*e^2)*x^2)) - 2*(8*a*c^
3*d^3 + 8*a*c*d^2*e + 3*(a*c^3*d*e^2 + a*c*e^3)*x^4 + 12*(a*c^3*d^2*e + a*c*d*e^2)*x^2 + (8*b*c^3*d^3 + 8*b*c*
d^2*e + 3*(b*c^3*d*e^2 + b*c*e^3)*x^4 + 12*(b*c^3*d^2*e + b*c*d*e^2)*x^2)*arccsc(c*x) + (b*c*d*e^2*x^2 + b*c*d
^2*e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^3*d^3*e^3 + c*d^2*e^4 + (c^3*d*e^5 + c*e^6)*x^4 + 2*(c^3*d^2*e^4
+ c*d*e^5)*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccsc(c*x) + a)*x^5/(e*x^2 + d)^(5/2), x)