\(\int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx\) [145]

Optimal result

Integrand size = 23, antiderivative size = 247 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{d x}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {270, 5347, 12, 486, 21, 434, 438, 437, 435, 432, 430} \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{d x}-\frac {b x \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{d \sqrt {c^2 x^2}} \]

[In]

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

[Out]

-((b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(d*Sqrt[c^2*x^2])) - (Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/(d*x) +
(b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(d*Sqrt[c^2*x^2]*Sqrt[-1 + c^
2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(
e/(c^2*d))])/(d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[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 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(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 434

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

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 486

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

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.57 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=-\frac {\sqrt {d+e x^2} \left (a+b c \sqrt {1-\frac {1}{c^2 x^2}} x+b \csc ^{-1}(c x)\right )}{d x}+\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} E\left (\arcsin \left (\sqrt {-\frac {e}{d}} x\right )|-\frac {c^2 d}{e}\right )}{d \sqrt {-\frac {e}{d}} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]

[In]

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

[Out]

-((Sqrt[d + e*x^2]*(a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x + b*ArcCsc[c*x]))/(d*x)) + (b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*
Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[Sqrt[-(e/d)]*x], -((c^2*d)/e)])/(d*Sqrt[-(e/d)]*Sqrt[1 - c^2*x^2]*Sqrt[d
+ e*x^2])

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.43 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=-\frac {{\left (b c d \operatorname {arccsc}\left (c x\right ) + \sqrt {c^{2} x^{2} - 1} b c d + a c d\right )} \sqrt {e x^{2} + d} + {\left (b c^{4} d x E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (b c^{4} d + b e\right )} x F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{c d^{2} x} \]

[In]

integrate((a+b*arccsc(c*x))/x^2/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((a+b*acsc(c*x))/x**2/(e*x**2+d)**(1/2),x)

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((a+b*arccsc(c*x))/x^2/(e*x^2+d)^(1/2),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]

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

[In]

integrate((a+b*arccsc(c*x))/x^2/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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