\(\int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^3} \, dx\) [50]

Optimal result

Integrand size = 16, antiderivative size = 172 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^3} \, dx=-\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (e+\frac {d}{x}\right )}+\frac {b \csc ^{-1}(c x)}{2 d^2 e}-\frac {a+b \csc ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \left (2 c^2 d^2-e^2\right ) \text {arctanh}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {c^2 d^2-e^2} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{2 d^2 \left (c^2 d^2-e^2\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5335, 1582, 1489, 1665, 858, 222, 739, 212} \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^3} \, dx=-\frac {a+b \csc ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \left (2 c^2 d^2-e^2\right ) \text {arctanh}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d^2-e^2}}\right )}{2 d^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (\frac {d}{x}+e\right )}+\frac {b \csc ^{-1}(c x)}{2 d^2 e} \]

[In]

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

[Out]

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

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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 858

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

Rule 1489

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x
] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1582

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Int[x^(m + mn*q
)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; FreeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (P
osQ[n2] ||  !IntegerQ[p])

Rule 1665

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 5335

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

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.45 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^3} \, dx=\frac {1}{2} \left (-\frac {a}{e (d+e x)^2}-\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}} x}{d \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b \csc ^{-1}(c x)}{e (d+e x)^2}+\frac {b \arcsin \left (\frac {1}{c x}\right )}{d^2 e}+\frac {b \left (2 c^2 d^2-e^2\right ) \log (d+e x)}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}-\frac {b \left (2 c^2 d^2-e^2\right ) \log \left (e+c \left (c d-\sqrt {c^2 d^2-e^2} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(572\) vs. \(2(159)=318\).

Time = 3.24 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.33

method	result	size
parts	\(-\frac {a}{2 \left (e x +d \right )^{2} e}+\frac {b \left (-\frac {c^{3} \operatorname {arccsc}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{3}+\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{2} e x -2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{3}-2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{2} e x -\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e^{3} c x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}+\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c d \,e^{2}+\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) e^{3} c x \right )}{2 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \,d^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +c d \right )}\right )}{c}\)	\(573\)
derivativedivides	\(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{3}+\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{2} e x -2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{3}-2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{2} e x -\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e^{3} c x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}+\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c d \,e^{2}+\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) e^{3} c x \right )}{2 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x \,d^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(583\)
default	\(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{3}+\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{2} e x -2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{3}-2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{2} e x -\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e^{3} c x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}+\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c d \,e^{2}+\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) e^{3} c x \right )}{2 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x \,d^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(583\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (156) = 312\).

Time = 0.56 (sec) , antiderivative size = 1111, normalized size of antiderivative = 6.46 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^3} \, dx=\left [-\frac {a c^{4} d^{6} + b c^{3} d^{5} e - 2 \, a c^{2} d^{4} e^{2} - b c d^{3} e^{3} + a d^{2} e^{4} + {\left (b c^{3} d^{3} e^{3} - b c d e^{5}\right )} x^{2} - {\left (2 \, b c^{2} d^{4} e - b d^{2} e^{3} + {\left (2 \, b c^{2} d^{2} e^{3} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, b c^{2} d^{3} e^{2} - b d e^{4}\right )} x\right )} \sqrt {c^{2} d^{2} - e^{2}} \log \left (\frac {c^{3} d^{2} x + c d e + \sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} + {\left (c^{2} d^{2} + \sqrt {c^{2} d^{2} - e^{2}} c d - e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{e x + d}\right ) + 2 \, {\left (b c^{3} d^{4} e^{2} - b c d^{2} e^{4}\right )} x + {\left (b c^{4} d^{6} - 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4}\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, {\left (b c^{4} d^{6} - 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} - 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e - 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{2} d^{4} e^{2} - b d^{2} e^{4} + {\left (b c^{2} d^{3} e^{3} - b d e^{5}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{2 \, {\left (c^{4} d^{8} e - 2 \, c^{2} d^{6} e^{3} + d^{4} e^{5} + {\left (c^{4} d^{6} e^{3} - 2 \, c^{2} d^{4} e^{5} + d^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{7} e^{2} - 2 \, c^{2} d^{5} e^{4} + d^{3} e^{6}\right )} x\right )}}, -\frac {a c^{4} d^{6} + b c^{3} d^{5} e - 2 \, a c^{2} d^{4} e^{2} - b c d^{3} e^{3} + a d^{2} e^{4} + {\left (b c^{3} d^{3} e^{3} - b c d e^{5}\right )} x^{2} + 2 \, {\left (2 \, b c^{2} d^{4} e - b d^{2} e^{3} + {\left (2 \, b c^{2} d^{2} e^{3} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, b c^{2} d^{3} e^{2} - b d e^{4}\right )} x\right )} \sqrt {-c^{2} d^{2} + e^{2}} \arctan \left (-\frac {\sqrt {-c^{2} d^{2} + e^{2}} \sqrt {c^{2} x^{2} - 1} e - \sqrt {-c^{2} d^{2} + e^{2}} {\left (c e x + c d\right )}}{c^{2} d^{2} - e^{2}}\right ) + 2 \, {\left (b c^{3} d^{4} e^{2} - b c d^{2} e^{4}\right )} x + {\left (b c^{4} d^{6} - 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4}\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, {\left (b c^{4} d^{6} - 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} - 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e - 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{2} d^{4} e^{2} - b d^{2} e^{4} + {\left (b c^{2} d^{3} e^{3} - b d e^{5}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{2 \, {\left (c^{4} d^{8} e - 2 \, c^{2} d^{6} e^{3} + d^{4} e^{5} + {\left (c^{4} d^{6} e^{3} - 2 \, c^{2} d^{4} e^{5} + d^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{7} e^{2} - 2 \, c^{2} d^{5} e^{4} + d^{3} e^{6}\right )} x\right )}}\right ] \]

[In]

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

[Out]

[-1/2*(a*c^4*d^6 + b*c^3*d^5*e - 2*a*c^2*d^4*e^2 - b*c*d^3*e^3 + a*d^2*e^4 + (b*c^3*d^3*e^3 - b*c*d*e^5)*x^2 -
(2*b*c^2*d^4*e - b*d^2*e^3 + (2*b*c^2*d^2*e^3 - b*e^5)*x^2 + 2*(2*b*c^2*d^3*e^2 - b*d*e^4)*x)*sqrt(c^2*d^2 -
e^2)*log((c^3*d^2*x + c*d*e + sqrt(c^2*d^2 - e^2)*(c^2*d*x + e) + (c^2*d^2 + sqrt(c^2*d^2 - e^2)*c*d - e^2)*sq
rt(c^2*x^2 - 1))/(e*x + d)) + 2*(b*c^3*d^4*e^2 - b*c*d^2*e^4)*x + (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4)*ar
ccsc(c*x) + 2*(b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4 + (b*c^4*d^4*e^2 - 2*b*c^2*d^2*e^4 + b*e^6)*x^2 + 2*(b*
c^4*d^5*e - 2*b*c^2*d^3*e^3 + b*d*e^5)*x)*arctan(-c*x + sqrt(c^2*x^2 - 1)) + (b*c^2*d^4*e^2 - b*d^2*e^4 + (b*c
^2*d^3*e^3 - b*d*e^5)*x)*sqrt(c^2*x^2 - 1))/(c^4*d^8*e - 2*c^2*d^6*e^3 + d^4*e^5 + (c^4*d^6*e^3 - 2*c^2*d^4*e^
5 + d^2*e^7)*x^2 + 2*(c^4*d^7*e^2 - 2*c^2*d^5*e^4 + d^3*e^6)*x), -1/2*(a*c^4*d^6 + b*c^3*d^5*e - 2*a*c^2*d^4*e
^2 - b*c*d^3*e^3 + a*d^2*e^4 + (b*c^3*d^3*e^3 - b*c*d*e^5)*x^2 + 2*(2*b*c^2*d^4*e - b*d^2*e^3 + (2*b*c^2*d^2*e
^3 - b*e^5)*x^2 + 2*(2*b*c^2*d^3*e^2 - b*d*e^4)*x)*sqrt(-c^2*d^2 + e^2)*arctan(-(sqrt(-c^2*d^2 + e^2)*sqrt(c^2
*x^2 - 1)*e - sqrt(-c^2*d^2 + e^2)*(c*e*x + c*d))/(c^2*d^2 - e^2)) + 2*(b*c^3*d^4*e^2 - b*c*d^2*e^4)*x + (b*c^
4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4)*arccsc(c*x) + 2*(b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4 + (b*c^4*d^4*e^2
- 2*b*c^2*d^2*e^4 + b*e^6)*x^2 + 2*(b*c^4*d^5*e - 2*b*c^2*d^3*e^3 + b*d*e^5)*x)*arctan(-c*x + sqrt(c^2*x^2 -
1)) + (b*c^2*d^4*e^2 - b*d^2*e^4 + (b*c^2*d^3*e^3 - b*d*e^5)*x)*sqrt(c^2*x^2 - 1))/(c^4*d^8*e - 2*c^2*d^6*e^3
+ d^4*e^5 + (c^4*d^6*e^3 - 2*c^2*d^4*e^5 + d^2*e^7)*x^2 + 2*(c^4*d^7*e^2 - 2*c^2*d^5*e^4 + d^3*e^6)*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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