3.2.13 \(\int \frac {x (a+b \csc ^{-1}(c x))}{(d+e x^2)^3} \, dx\) [113]

Optimal. Leaf size=193 \[ \frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c x \text {ArcTan}\left (\sqrt {-1+c^2 x^2}\right )}{4 d^2 e \sqrt {c^2 x^2}}+\frac {b c \left (3 c^2 d+2 e\right ) x \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2} \sqrt {c^2 x^2}} \]

[Out]

1/4*(-a-b*arccsc(c*x))/e/(e*x^2+d)^2-1/4*b*c*x*arctan((c^2*x^2-1)^(1/2))/d^2/e/(c^2*x^2)^(1/2)+1/8*b*c*(3*c^2*
d+2*e)*x*arctan(e^(1/2)*(c^2*x^2-1)^(1/2)/(c^2*d+e)^(1/2))/d^2/(c^2*d+e)^(3/2)/e^(1/2)/(c^2*x^2)^(1/2)+1/8*b*c
*x*(c^2*x^2-1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)/(c^2*x^2)^(1/2)

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Rubi [A]
time = 0.15, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5345, 457, 105, 162, 65, 211} \begin {gather*} -\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c x \text {ArcTan}\left (\sqrt {c^2 x^2-1}\right )}{4 d^2 e \sqrt {c^2 x^2}}+\frac {b c x \left (3 c^2 d+2 e\right ) \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \sqrt {c^2 x^2} \left (c^2 d+e\right )^{3/2}}+\frac {b c x \sqrt {c^2 x^2-1}}{8 d \sqrt {c^2 x^2} \left (c^2 d+e\right ) \left (d+e x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*x*Sqrt[-1 + c^2*x^2])/(8*d*(c^2*d + e)*Sqrt[c^2*x^2]*(d + e*x^2)) - (a + b*ArcCsc[c*x])/(4*e*(d + e*x^2)^
2) - (b*c*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(4*d^2*e*Sqrt[c^2*x^2]) + (b*c*(3*c^2*d + 2*e)*x*ArcTan[(Sqrt[e]*Sqrt[
-1 + c^2*x^2])/Sqrt[c^2*d + e]])/(8*d^2*Sqrt[e]*(c^2*d + e)^(3/2)*Sqrt[c^2*x^2])

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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 211

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

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} \, dx &=-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {(b c x) \int \frac {1}{x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 e \sqrt {c^2 x^2}}\\ &=-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} (d+e x)^2} \, dx,x,x^2\right )}{8 e \sqrt {c^2 x^2}}\\ &=\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c x) \text {Subst}\left (\int \frac {-c^2 d-e+\frac {1}{2} c^2 e x}{x \sqrt {-1+c^2 x} (d+e x)} \, dx,x,x^2\right )}{8 d e \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{8 d^2 e \sqrt {c^2 x^2}}+\frac {\left (b c \left (3 c^2 d+2 e\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} (d+e x)} \, dx,x,x^2\right )}{16 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {(b x) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{4 c d^2 e \sqrt {c^2 x^2}}+\frac {\left (b \left (3 c^2 d+2 e\right ) x\right ) \text {Subst}\left (\int \frac {1}{d+\frac {e}{c^2}+\frac {e x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{8 c d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c x \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{4 d^2 e \sqrt {c^2 x^2}}+\frac {b c \left (3 c^2 d+2 e\right ) x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2} \sqrt {c^2 x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.65, size = 385, normalized size = 1.99 \begin {gather*} \frac {1}{16} \left (-\frac {4 a}{e \left (d+e x^2\right )^2}+\frac {2 b c \sqrt {1-\frac {1}{c^2 x^2}} x}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {4 b \csc ^{-1}(c x)}{e \left (d+e x^2\right )^2}+\frac {4 b \text {ArcSin}\left (\frac {1}{c x}\right )}{d^2 e}+\frac {b \left (3 c^2 d+2 e\right ) \log \left (\frac {16 d^2 \sqrt {-c^2 d-e} \sqrt {e} \left (i \sqrt {e}+c \left (c \sqrt {d}-i \sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (3 c^2 d+2 e\right ) \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d^2 \left (-c^2 d-e\right )^{3/2} \sqrt {e}}+\frac {b \left (3 c^2 d+2 e\right ) \log \left (-\frac {16 d^2 \sqrt {-c^2 d-e} \sqrt {e} \left (-\sqrt {e}+c \left (-i c \sqrt {d}+\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (3 c^2 d+2 e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{d^2 \left (-c^2 d-e\right )^{3/2} \sqrt {e}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*a)/(e*(d + e*x^2)^2) + (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)/(d*(c^2*d + e)*(d + e*x^2)) - (4*b*ArcCsc[c*x])/(e
*(d + e*x^2)^2) + (4*b*ArcSin[1/(c*x)])/(d^2*e) + (b*(3*c^2*d + 2*e)*Log[(16*d^2*Sqrt[-(c^2*d) - e]*Sqrt[e]*(I
*Sqrt[e] + c*(c*Sqrt[d] - I*Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(3*c^2*d + 2*e)*(Sqrt[d] + I*Sqrt
[e]*x))])/(d^2*(-(c^2*d) - e)^(3/2)*Sqrt[e]) + (b*(3*c^2*d + 2*e)*Log[(-16*d^2*Sqrt[-(c^2*d) - e]*Sqrt[e]*(-Sq
rt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(3*c^2*d + 2*e)*(I*Sqrt[d] + Sqrt
[e]*x))])/(d^2*(-(c^2*d) - e)^(3/2)*Sqrt[e]))/16

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1795\) vs. \(2(168)=336\).
time = 1.46, size = 1796, normalized size = 9.31

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1796\)
default	\(\text {Expression too large to display}\)	\(1796\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(-1/4*a*c^6/e/(c^2*e*x^2+c^2*d)^2-1/4*b*c^6/e/(c^2*e*x^2+c^2*d)^2*arccsc(c*x)-1/4*b*c^5*(c^2*x^2-1)^(1/2
)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x/(c^2*d+e)/(e*c*x+(-c^2*d*e)^(1/2))/(-e*c*x+(-c^2*d*e)^(1/2))*arctan(1/(c^2*x^2
-1)^(1/2))-1/4*b*c^5*(c^2*x^2-1)^(1/2)*e/((c^2*x^2-1)/c^2/x^2)^(1/2)*x/d/(c^2*d+e)/(e*c*x+(-c^2*d*e)^(1/2))/(-
e*c*x+(-c^2*d*e)^(1/2))*arctan(1/(c^2*x^2-1)^(1/2))+3/16*b*c^5*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x
/(-(c^2*d+e)/e)^(1/2)/(c^2*d+e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2))*ln(2*((-(c^2*d+e)/e)^(1/2)*
(c^2*x^2-1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(e*c*x+(-c^2*d*e)^(1/2)))+3/16*b*c^5*(c^2*x^2-1)^(1/2)*e/((c^2*x^2
-1)/c^2/x^2)^(1/2)*x/d/(-(c^2*d+e)/e)^(1/2)/(c^2*d+e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2))*ln(2*
((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(e*c*x+(-c^2*d*e)^(1/2)))+3/16*b*c^5*(c^2*x^
2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x/(-(c^2*d+e)/e)^(1/2)/(c^2*d+e)/(e*c*x+(-c^2*d*e)^(1/2))/(-e*c*x+(-c^2
*d*e)^(1/2))*ln(-2*((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-e*c*x+(-c^2*d*e)^(1/2))
)+3/16*b*c^5*(c^2*x^2-1)^(1/2)*e/((c^2*x^2-1)/c^2/x^2)^(1/2)*x/d/(-(c^2*d+e)/e)^(1/2)/(c^2*d+e)/(e*c*x+(-c^2*d
*e)^(1/2))/(-e*c*x+(-c^2*d*e)^(1/2))*ln(-2*((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(
-e*c*x+(-c^2*d*e)^(1/2)))-1/8*b*c^3*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x/d/(c^2*d+e)/(-e*c*x+(-c^2*d*e)^(
1/2))/(e*c*x+(-c^2*d*e)^(1/2))*e-1/4*b*c^3*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x/d/(c^2*d+e)/(-e*c*x
+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2))*arctan(1/(c^2*x^2-1)^(1/2))*e-1/4*b*c^3*(c^2*x^2-1)^(1/2)/((c^2*x^
2-1)/c^2/x^2)^(1/2)*x/d^2/(c^2*d+e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2))*arctan(1/(c^2*x^2-1)^(1
/2))*e^2+1/8*b*c^3*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x/d/(-(c^2*d+e)/e)^(1/2)/(c^2*d+e)/(-e*c*x+(-
c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2))*ln(2*((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e
)/(e*c*x+(-c^2*d*e)^(1/2)))*e+1/8*b*c^3*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)*x/d^2/(-(c^2*d+e)/e)^(1/
2)/(c^2*d+e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2))*ln(2*((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e
-(-c^2*d*e)^(1/2)*c*x-e)/(e*c*x+(-c^2*d*e)^(1/2)))*e^2+1/8*b*c^3*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)
/x/d/(-(c^2*d+e)/e)^(1/2)/(c^2*d+e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2))*ln(-2*((-(c^2*d+e)/e)^(
1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-e*c*x+(-c^2*d*e)^(1/2)))*e+1/8*b*c^3*(c^2*x^2-1)^(1/2)/((c^
2*x^2-1)/c^2/x^2)^(1/2)*x/d^2/(-(c^2*d+e)/e)^(1/2)/(c^2*d+e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2)
)*ln(-2*((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-e*c*x+(-c^2*d*e)^(1/2)))*e^2)

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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,x, algorithm="maxima")

[Out]

-1/4*(4*(c^2*x^4*e^3 + 2*c^2*d*x^2*e^2 + c^2*d^2*e)*integrate(1/4*x*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))/(c
^2*x^6*e^3 + (2*c^2*d*e^2 - e^3)*x^4 + (c^2*d^2*e - 2*d*e^2)*x^2 - d^2*e + (c^2*x^6*e^3 + (2*c^2*d*e^2 - e^3)*
x^4 + (c^2*d^2*e - 2*d*e^2)*x^2 - d^2*e)*e^(log(c*x + 1) + log(c*x - 1))), x) + arctan2(1, sqrt(c*x + 1)*sqrt(
c*x - 1)))*b/(x^4*e^3 + 2*d*x^2*e^2 + d^2*e) - 1/4*a/(x^4*e^3 + 2*d*x^2*e^2 + d^2*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (167) = 334\).
time = 0.63, size = 888, normalized size = 4.60 \begin {gather*} \left [-\frac {4 \, a c^{4} d^{4} + 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + {\left (3 \, b c^{2} d^{3} + 2 \, b x^{4} e^{3} + {\left (3 \, b c^{2} d x^{4} + 4 \, b d x^{2}\right )} e^{2} + 2 \, {\left (3 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e\right )} \sqrt {-c^{2} d e - e^{2}} \log \left (-\frac {c^{2} d - {\left (c^{2} x^{2} - 2\right )} e + 2 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} d e - e^{2}}}{x^{2} e + d}\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 8 \, {\left (b c^{4} d^{4} + b x^{4} e^{4} + 2 \, {\left (b c^{2} d x^{4} + b d x^{2}\right )} e^{3} + {\left (b c^{4} d^{2} x^{4} + 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2} + 2 \, {\left (b c^{4} d^{3} x^{2} + b c^{2} d^{3}\right )} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, {\left (b c^{2} d^{3} e + b d x^{2} e^{3} + {\left (b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{16 \, {\left (c^{4} d^{6} e + d^{2} x^{4} e^{5} + 2 \, {\left (c^{2} d^{3} x^{4} + d^{3} x^{2}\right )} e^{4} + {\left (c^{4} d^{4} x^{4} + 4 \, c^{2} d^{4} x^{2} + d^{4}\right )} e^{3} + 2 \, {\left (c^{4} d^{5} x^{2} + c^{2} d^{5}\right )} e^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} + 4 \, a c^{2} d^{3} e + 2 \, a d^{2} e^{2} - {\left (3 \, b c^{2} d^{3} + 2 \, b x^{4} e^{3} + {\left (3 \, b c^{2} d x^{4} + 4 \, b d x^{2}\right )} e^{2} + 2 \, {\left (3 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e\right )} \sqrt {c^{2} d e + e^{2}} \arctan \left (\frac {\sqrt {c^{2} x^{2} - 1} \sqrt {c^{2} d e + e^{2}}}{c^{2} d + e}\right ) + 2 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 4 \, {\left (b c^{4} d^{4} + b x^{4} e^{4} + 2 \, {\left (b c^{2} d x^{4} + b d x^{2}\right )} e^{3} + {\left (b c^{4} d^{2} x^{4} + 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2} + 2 \, {\left (b c^{4} d^{3} x^{2} + b c^{2} d^{3}\right )} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} d^{3} e + b d x^{2} e^{3} + {\left (b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{8 \, {\left (c^{4} d^{6} e + d^{2} x^{4} e^{5} + 2 \, {\left (c^{2} d^{3} x^{4} + d^{3} x^{2}\right )} e^{4} + {\left (c^{4} d^{4} x^{4} + 4 \, c^{2} d^{4} x^{2} + d^{4}\right )} e^{3} + 2 \, {\left (c^{4} d^{5} x^{2} + c^{2} d^{5}\right )} e^{2}\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,x, algorithm="fricas")

[Out]

[-1/16*(4*a*c^4*d^4 + 8*a*c^2*d^3*e + 4*a*d^2*e^2 + (3*b*c^2*d^3 + 2*b*x^4*e^3 + (3*b*c^2*d*x^4 + 4*b*d*x^2)*e
^2 + 2*(3*b*c^2*d^2*x^2 + b*d^2)*e)*sqrt(-c^2*d*e - e^2)*log(-(c^2*d - (c^2*x^2 - 2)*e + 2*sqrt(c^2*x^2 - 1)*s
qrt(-c^2*d*e - e^2))/(x^2*e + d)) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*arccsc(c*x) + 8*(b*c^4*d^4 + b*x
^4*e^4 + 2*(b*c^2*d*x^4 + b*d*x^2)*e^3 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + b*d^2)*e^2 + 2*(b*c^4*d^3*x^2 + b*
c^2*d^3)*e)*arctan(-c*x + sqrt(c^2*x^2 - 1)) - 2*(b*c^2*d^3*e + b*d*x^2*e^3 + (b*c^2*d^2*x^2 + b*d^2)*e^2)*sqr
t(c^2*x^2 - 1))/(c^4*d^6*e + d^2*x^4*e^5 + 2*(c^2*d^3*x^4 + d^3*x^2)*e^4 + (c^4*d^4*x^4 + 4*c^2*d^4*x^2 + d^4)
*e^3 + 2*(c^4*d^5*x^2 + c^2*d^5)*e^2), -1/8*(2*a*c^4*d^4 + 4*a*c^2*d^3*e + 2*a*d^2*e^2 - (3*b*c^2*d^3 + 2*b*x^
4*e^3 + (3*b*c^2*d*x^4 + 4*b*d*x^2)*e^2 + 2*(3*b*c^2*d^2*x^2 + b*d^2)*e)*sqrt(c^2*d*e + e^2)*arctan(sqrt(c^2*x
^2 - 1)*sqrt(c^2*d*e + e^2)/(c^2*d + e)) + 2*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*arccsc(c*x) + 4*(b*c^4*d^
4 + b*x^4*e^4 + 2*(b*c^2*d*x^4 + b*d*x^2)*e^3 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + b*d^2)*e^2 + 2*(b*c^4*d^3*x
^2 + b*c^2*d^3)*e)*arctan(-c*x + sqrt(c^2*x^2 - 1)) - (b*c^2*d^3*e + b*d*x^2*e^3 + (b*c^2*d^2*x^2 + b*d^2)*e^2
)*sqrt(c^2*x^2 - 1))/(c^4*d^6*e + d^2*x^4*e^5 + 2*(c^2*d^3*x^4 + d^3*x^2)*e^4 + (c^4*d^4*x^4 + 4*c^2*d^4*x^2 +
d^4)*e^3 + 2*(c^4*d^5*x^2 + c^2*d^5)*e^2)]

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

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(sageVARx)]s
ym2poly/r2sym(

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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} \,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,x)

[Out]

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

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