\(\int \frac {a+b \csc ^{-1}(c x)}{x^2 (d+e x^2)^{3/2}} \, dx\) [155]

Optimal result

Integrand size = 23, antiderivative size = 275 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]

[Out]

(-a-b*arccsc(c*x))/d/x/(e*x^2+d)^(1/2)-2*e*x*(a+b*arccsc(c*x))/d^2/(e*x^2+d)^(1/2)-b*c*(c^2*x^2-1)^(1/2)*(e*x^
2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)+b*c^2*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/
(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)-b*(c^2*d+2*e)*x*EllipticF(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+
1)^(1/2)*(1+e*x^2/d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {277, 197, 5347, 12, 597, 538, 438, 437, 435, 432, 430} \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {b x \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}} \]

[In]

Int[(a + b*ArcCsc[c*x])/(x^2*(d + e*x^2)^(3/2)),x]

[Out]

-((b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(d^2*Sqrt[c^2*x^2])) - (a + b*ArcCsc[c*x])/(d*x*Sqrt[d + e*x^2]) -
(2*e*x*(a + b*ArcCsc[c*x]))/(d^2*Sqrt[d + e*x^2]) + (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSi
n[c*x], -(e/(c^2*d))])/(d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + 2*e)*x*Sqrt[1
- c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqr
t[d + e*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 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 277

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

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 437

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]
, Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 438

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rule 538

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c]))))))

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

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 {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {-d-2 e x^2}{d^2 x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {-d-2 e x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {-2 d e+c^2 d e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^3 \sqrt {c^2 x^2}} \\ & = -\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {\left (b c^3 x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{d^2 \sqrt {c^2 x^2}}-\frac {\left (b c \left (c^2 d+2 e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {\left (b c^3 x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (b c \left (c^2 d+2 e\right ) x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {\left (b c^3 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b c \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.96 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.77 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\frac {-b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )-a \left (d+2 e x^2\right )-b \left (d+2 e x^2\right ) \csc ^{-1}(c x)}{d^2 x \sqrt {d+e x^2}}+\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-\left (c^2 d+2 e\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{\sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]

[In]

Integrate[(a + b*ArcCsc[c*x])/(x^2*(d + e*x^2)^(3/2)),x]

[Out]

(-(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2)) - a*(d + 2*e*x^2) - b*(d + 2*e*x^2)*ArcCsc[c*x])/(d^2*x*Sqrt[d + e
*x^2]) + (I*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2
*d))] - (c^2*d + 2*e)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqr
t[d + e*x^2])

Maple [F]

\[\int \frac {a +b \,\operatorname {arccsc}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]

[In]

int((a+b*arccsc(c*x))/x^2/(e*x^2+d)^(3/2),x)

[Out]

int((a+b*arccsc(c*x))/x^2/(e*x^2+d)^(3/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {{\left (2 \, a c d e x^{2} + a c d^{2} + {\left (2 \, b c d e x^{2} + b c d^{2}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (b c d e x^{2} + b c d^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} + {\left ({\left (b c^{4} d e x^{3} + b c^{4} d^{2} x\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left ({\left (b c^{4} d e + 2 \, b e^{2}\right )} x^{3} + {\left (b c^{4} d^{2} + 2 \, b d e\right )} x\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{c d^{3} e x^{3} + c d^{4} x} \]

[In]

integrate((a+b*arccsc(c*x))/x^2/(e*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

-((2*a*c*d*e*x^2 + a*c*d^2 + (2*b*c*d*e*x^2 + b*c*d^2)*arccsc(c*x) + (b*c*d*e*x^2 + b*c*d^2)*sqrt(c^2*x^2 - 1)
)*sqrt(e*x^2 + d) + ((b*c^4*d*e*x^3 + b*c^4*d^2*x)*elliptic_e(arcsin(c*x), -e/(c^2*d)) - ((b*c^4*d*e + 2*b*e^2
)*x^3 + (b*c^4*d^2 + 2*b*d*e)*x)*elliptic_f(arcsin(c*x), -e/(c^2*d)))*sqrt(-d))/(c*d^3*e*x^3 + c*d^4*x)

Sympy [F]

\[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((a+b*acsc(c*x))/x**2/(e*x**2+d)**(3/2),x)

[Out]

Integral((a + b*acsc(c*x))/(x**2*(d + e*x**2)**(3/2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arccsc(c*x))/x^2/(e*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[In]

integrate((a+b*arccsc(c*x))/x^2/(e*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccsc(c*x) + a)/((e*x^2 + d)^(3/2)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

[In]

int((a + b*asin(1/(c*x)))/(x^2*(d + e*x^2)^(3/2)),x)

[Out]

int((a + b*asin(1/(c*x)))/(x^2*(d + e*x^2)^(3/2)), x)