3.2.56 \(\int \frac {x^5 (a+b \csc ^{-1}(c x))}{(d+e x^2)^{5/2}} \, dx\) [156]

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

[Out]

-1/3*d^2*(a+b*arccsc(c*x))/e^3/(e*x^2+d)^(3/2)+b*x*arctanh(e^(1/2)*(c^2*x^2-1)^(1/2)/c/(e*x^2+d)^(1/2))/e^(5/2
)/(c^2*x^2)^(1/2)-8/3*b*c*x*arctan((e*x^2+d)^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))*d^(1/2)/e^3/(c^2*x^2)^(1/2)+2*d*
(a+b*arccsc(c*x))/e^3/(e*x^2+d)^(1/2)+1/3*b*c*d*x*(c^2*x^2-1)^(1/2)/e^2/(c^2*d+e)/(c^2*x^2)^(1/2)/(e*x^2+d)^(1
/2)+(a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/e^3

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Rubi [A]
time = 0.74, antiderivative size = 243, normalized size of antiderivative = 1.00, 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 = {272, 45, 5347, 12, 1628, 163, 65, 223, 212, 95, 210} \begin {gather*} -\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 \text {ArcTan}\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}}+\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}} \end {gather*}

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 12

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

Rule 45

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 65

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 95

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 163

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 210

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

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 272

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 1628

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 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 \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) \text {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) \text {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) \text {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) \text {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) \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 1.23, size = 237, normalized size = 0.98 \begin {gather*} \frac {b c d e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )+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 ) \left (8 d^2+12 d e x^2+3 e^2 x^4\right ) \csc ^{-1}(c x)}{3 e^3 \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (8 c \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )+3 \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{3 e^3 \sqrt {-1+c^2 x^2}} \end {gather*}

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*(8*c*Sqrt[d]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]] + 3*Sqrt[e]*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^
2*x^2])/(c*Sqrt[d + e*x^2])]))/(3*e^3*Sqrt[-1 + c^2*x^2])

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

1/3*(3*x^4*e^(-1)/(x^2*e + d)^(3/2) + 12*d*x^2*e^(-2)/(x^2*e + d)^(3/2) + 8*d^2*e^(-3)/(x^2*e + d)^(3/2))*a +
1/3*(3*x^4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*e^2 + 12*d*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*e +
8*d^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*(x^2*e^4 + d*e^3)*sqrt(x^2*e + d)*integrate(1/3*(3*c^2*x^5*e
^2 + 12*c^2*d*x^3*e + 8*c^2*d^2*x)*e^(-1/2*log(x^2*e + d) + 1/2*log(c*x + 1) + 1/2*log(c*x - 1))/(c^2*x^4*e^4
+ (c^2*d*e^3 - e^4)*x^2 - d*e^3 + (c^2*x^4*e^4 + (c^2*d*e^3 - e^4)*x^2 - d*e^3)*e^(log(c*x + 1) + log(c*x - 1)
)), x))*b/((x^2*e^4 + d*e^3)*sqrt(x^2*e + d))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (205) = 410\).
time = 0.59, size = 1110, normalized size = 4.57 \begin {gather*} \left [\frac {3 \, {\left (b c^{2} d^{3} + b x^{4} e^{3} + {\left (b c^{2} d x^{4} + 2 \, b d x^{2}\right )} e^{2} + {\left (2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e\right )} e^{\frac {1}{2}} \log \left (c^{4} d^{2} + 4 \, {\left (c^{3} d + {\left (2 \, c^{3} x^{2} - c\right )} e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} e^{\frac {1}{2}} + {\left (8 \, c^{4} x^{4} - 8 \, c^{2} x^{2} + 1\right )} e^{2} + 2 \, {\left (4 \, c^{4} d x^{2} - 3 \, c^{2} d\right )} e\right ) + 8 \, {\left (b c^{3} d^{3} + b c x^{4} e^{3} + {\left (b c^{3} d x^{4} + 2 \, b c d x^{2}\right )} e^{2} + {\left (2 \, b c^{3} d^{2} x^{2} + b c d^{2}\right )} e\right )} \sqrt {-d} \log \left (\frac {c^{4} d^{2} x^{4} - 8 \, c^{2} d^{2} x^{2} + x^{4} e^{2} + 4 \, {\left (c^{2} d x^{2} - x^{2} e - 2 \, d\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} \sqrt {-d} + 8 \, d^{2} - 2 \, {\left (3 \, c^{2} d x^{4} - 4 \, d x^{2}\right )} e}{x^{4}}\right ) + 4 \, {\left (8 \, a c^{3} d^{3} + 3 \, a c x^{4} e^{3} + {\left (8 \, b c^{3} d^{3} + 3 \, b c x^{4} e^{3} + 3 \, {\left (b c^{3} d x^{4} + 4 \, b c d x^{2}\right )} e^{2} + 4 \, {\left (3 \, b c^{3} d^{2} x^{2} + 2 \, b c d^{2}\right )} e\right )} \operatorname {arccsc}\left (c x\right ) + 3 \, {\left (a c^{3} d x^{4} + 4 \, a c d x^{2}\right )} e^{2} + 4 \, {\left (3 \, a c^{3} d^{2} x^{2} + 2 \, a c d^{2}\right )} e + {\left (b c d x^{2} e^{2} + b c d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {x^{2} e + d}}{12 \, {\left (c^{3} d^{3} e^{3} + c x^{4} e^{6} + {\left (c^{3} d x^{4} + 2 \, c d x^{2}\right )} e^{5} + {\left (2 \, c^{3} d^{2} x^{2} + c d^{2}\right )} e^{4}\right )}}, \frac {3 \, {\left (b c^{2} d^{3} + b x^{4} e^{3} + {\left (b c^{2} d x^{4} + 2 \, b d x^{2}\right )} e^{2} + {\left (2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e\right )} e^{\frac {1}{2}} \log \left (c^{4} d^{2} + 4 \, {\left (c^{3} d + {\left (2 \, c^{3} x^{2} - c\right )} e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} e^{\frac {1}{2}} + {\left (8 \, c^{4} x^{4} - 8 \, c^{2} x^{2} + 1\right )} e^{2} + 2 \, {\left (4 \, c^{4} d x^{2} - 3 \, c^{2} d\right )} e\right ) - 16 \, {\left (b c^{3} d^{3} + b c x^{4} e^{3} + {\left (b c^{3} d x^{4} + 2 \, b c d x^{2}\right )} e^{2} + {\left (2 \, b c^{3} d^{2} x^{2} + b c d^{2}\right )} e\right )} \sqrt {d} \arctan \left (-\frac {{\left (c^{2} d x^{2} - x^{2} e - 2 \, d\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} \sqrt {d}}{2 \, {\left (c^{2} d^{2} x^{2} - d^{2} + {\left (c^{2} d x^{4} - d x^{2}\right )} e\right )}}\right ) + 4 \, {\left (8 \, a c^{3} d^{3} + 3 \, a c x^{4} e^{3} + {\left (8 \, b c^{3} d^{3} + 3 \, b c x^{4} e^{3} + 3 \, {\left (b c^{3} d x^{4} + 4 \, b c d x^{2}\right )} e^{2} + 4 \, {\left (3 \, b c^{3} d^{2} x^{2} + 2 \, b c d^{2}\right )} e\right )} \operatorname {arccsc}\left (c x\right ) + 3 \, {\left (a c^{3} d x^{4} + 4 \, a c d x^{2}\right )} e^{2} + 4 \, {\left (3 \, a c^{3} d^{2} x^{2} + 2 \, a c d^{2}\right )} e + {\left (b c d x^{2} e^{2} + b c d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {x^{2} e + d}}{12 \, {\left (c^{3} d^{3} e^{3} + c x^{4} e^{6} + {\left (c^{3} d x^{4} + 2 \, c d x^{2}\right )} e^{5} + {\left (2 \, c^{3} d^{2} x^{2} + c d^{2}\right )} e^{4}\right )}}\right ] \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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