3.2.40 \(\int \frac {x (a+b \csc ^{-1}(c x))}{\sqrt {d+e x^2}} \, dx\) [140]

Optimal. Leaf size=132 \[ \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {b c \sqrt {d} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e \sqrt {c^2 x^2}}+\frac {b x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {c^2 x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.10, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5345, 457, 132, 65, 223, 212, 12, 95, 210} \begin {gather*} \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {b c \sqrt {d} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{e \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 )}{\sqrt {e} \sqrt {c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e - (b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])
/(e*Sqrt[c^2*x^2]) + (b*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(Sqrt[e]*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 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 132

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[b*d^(m
+ n)*f^p, Int[(a + b*x)^(m - 1)/(c + d*x)^m, x], x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandTo
Sum[(a + b*x)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] ||  !(GtQ[n, 0] || SumSimplerQ[n,
-1]))

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 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 5345

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1
)*((a + b*ArcCsc[c*x])/(2*e*(p + 1))), x] + Dist[b*c*(x/(2*e*(p + 1)*Sqrt[c^2*x^2])), Int[(d + e*x^2)^(p + 1)/
(x*Sqrt[c^2*x^2 - 1]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 136, normalized size = 1.03 \begin {gather*} \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (c \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )+\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{e \sqrt {-1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(c*Sqrt[d]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*
x^2])/Sqrt[d + e*x^2]] + Sqrt[e]*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])]))/(e*Sqrt[-1 + c^2*
x^2])

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (109) = 218\).
time = 0.44, size = 466, normalized size = 3.53 \begin {gather*} \left [\frac {{\left (b c \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 ) + b 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 ) + 4 \, {\left (b c \operatorname {arccsc}\left (c x\right ) + a c\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-1\right )}}{4 \, c}, -\frac {{\left (2 \, b c \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 ) - b 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 ) - 4 \, {\left (b c \operatorname {arccsc}\left (c x\right ) + a c\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-1\right )}}{4 \, c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(b*c*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) + b*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) + 4*(b*c*arccsc(c*x) + a*c)*sqrt(x^2*e + d))*e^(-1)/c, -1/4*(2*b*c*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)) - b*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) - 4*(b*c*arccsc(c*x) + a*c)*sqrt(x^2*e + d))*e^(-1)/c]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________