3.2.49 \(\int \frac {x (a+b \csc ^{-1}(c x))}{(d+e x^2)^{3/2}} \, dx\) [149]

Optimal. Leaf size=79 \[ -\frac {a+b \csc ^{-1}(c x)}{e \sqrt {d+e x^2}}+\frac {b c x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{\sqrt {d} e \sqrt {c^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5345, 457, 95, 210} \begin {gather*} \frac {b c x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{\sqrt {d} e \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{e \sqrt {d+e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.09, size = 96, normalized size = 1.22 \begin {gather*} -\frac {a+b \csc ^{-1}(c x)}{e \sqrt {d+e x^2}}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )}{\sqrt {d} e \sqrt {-1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (67) = 134\).
time = 0.40, size = 304, normalized size = 3.85 \begin {gather*} \left [-\frac {{\left (b x^{2} e + b d\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 (b d \operatorname {arccsc}\left (c x\right ) + a d\right )} \sqrt {x^{2} e + d}}{4 \, {\left (d x^{2} e^{2} + d^{2} e\right )}}, \frac {{\left (b x^{2} e + b d\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 ) - 2 \, {\left (b d \operatorname {arccsc}\left (c x\right ) + a d\right )} \sqrt {x^{2} e + d}}{2 \, {\left (d x^{2} e^{2} + d^{2} e\right )}}\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)^(3/2),x, algorithm="fricas")

[Out]

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

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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 )}{\left (d + e x^{2}\right )^{\frac {3}{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)**(3/2),x)

[Out]

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

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

[Out]

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

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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 )}{{\left (e\,x^2+d\right )}^{3/2}} \,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)^(3/2),x)

[Out]

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

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