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

Optimal. Leaf size=172 \[ -\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (e+\frac {d}{x}\right )}+\frac {b \csc ^{-1}(c x)}{2 d^2 e}-\frac {a+b \csc ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {c^2 d^2-e^2} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{2 d^2 \left (c^2 d^2-e^2\right )^{3/2}} \]

[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
)

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Rubi [A]
time = 0.22, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5335, 1582, 1489, 1665, 858, 222, 739, 212} \begin {gather*} -\frac {a+b \csc ^{-1}(c x)}{2 e (d+e x)^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 \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\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 \csc ^{-1}(c x)}{2 d^2 e} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 \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 ) \tanh ^{-1}\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*}

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Mathematica [A]
time = 0.34, size = 250, normalized size = 1.45 \begin {gather*} \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 \text {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 ) \end {gather*}

Antiderivative was successfully verified.

[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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(970\) vs. \(2(159)=318\).
time = 4.42, size = 971, normalized size = 5.65

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{3}}{2 \left (e c x +c d \right )^{2} e}-\frac {b \,c^{3} \mathrm {arccsc}\left (c x \right )}{2 \left (e c x +c d \right )^{2} e}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )}-\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d \,c^{2} x -2 e}{e c x +c d}\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d \,c^{2} x -2 e}{e c x +c d}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {b c e \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )}-\frac {b c \,e^{2} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )}-\frac {b c e \left (c^{2} x^{2}-1\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )}+\frac {b c e \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d \,c^{2} x -2 e}{e c x +c d}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}+\frac {b c \,e^{2} \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d \,c^{2} x -2 e}{e c x +c d}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(971\)
default	\(\frac {-\frac {a \,c^{3}}{2 \left (e c x +c d \right )^{2} e}-\frac {b \,c^{3} \mathrm {arccsc}\left (c x \right )}{2 \left (e c x +c d \right )^{2} e}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )}-\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d \,c^{2} x -2 e}{e c x +c d}\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d \,c^{2} x -2 e}{e c x +c d}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {b c e \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )}-\frac {b c \,e^{2} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )}-\frac {b c e \left (c^{2} x^{2}-1\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )}+\frac {b c e \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d \,c^{2} x -2 e}{e c x +c d}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}+\frac {b c \,e^{2} \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d \,c^{2} x -2 e}{e c x +c d}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(971\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (154) = 308\).
time = 0.66, size = 1062, normalized size = 6.17 \begin {gather*} \left [-\frac {a c^{4} d^{6} + b c^{3} d^{5} e - b c d x^{2} e^{5} - {\left (4 \, b c^{2} d^{3} x e^{2} + 2 \, b c^{2} d^{4} e - b x^{2} e^{5} - 2 \, b d x e^{4} + {\left (2 \, b c^{2} d^{2} x^{2} - b d^{2}\right )} e^{3}\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}}{x e + d}\right ) + {\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 (2 \, b c^{4} d^{5} x e + b c^{4} d^{6} - 4 \, b c^{2} d^{3} x e^{3} + b x^{2} e^{6} + 2 \, b d x e^{5} - {\left (2 \, b c^{2} d^{2} x^{2} - b d^{2}\right )} e^{4} + {\left (b c^{4} d^{4} x^{2} - 2 \, b c^{2} d^{4}\right )} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c d^{2} x - a d^{2}\right )} e^{4} + {\left (b c^{3} d^{3} x^{2} - b c d^{3}\right )} e^{3} + 2 \, {\left (b c^{3} d^{4} x - a c^{2} d^{4}\right )} e^{2} + {\left (b c^{2} d^{3} x e^{3} + b c^{2} d^{4} e^{2} - b d x e^{5} - b d^{2} e^{4}\right )} \sqrt {c^{2} x^{2} - 1}}{2 \, {\left (2 \, c^{4} d^{7} x e^{2} + c^{4} d^{8} e - 4 \, c^{2} d^{5} x e^{4} + d^{2} x^{2} e^{7} + 2 \, d^{3} x e^{6} - {\left (2 \, c^{2} d^{4} x^{2} - d^{4}\right )} e^{5} + {\left (c^{4} d^{6} x^{2} - 2 \, c^{2} d^{6}\right )} e^{3}\right )}}, -\frac {a c^{4} d^{6} + b c^{3} d^{5} e - b c d x^{2} e^{5} + 2 \, {\left (4 \, b c^{2} d^{3} x e^{2} + 2 \, b c^{2} d^{4} e - b x^{2} e^{5} - 2 \, b d x e^{4} + {\left (2 \, b c^{2} d^{2} x^{2} - b d^{2}\right )} e^{3}\right )} \sqrt {-c^{2} d^{2} + e^{2}} \arctan \left (\frac {\sqrt {-c^{2} d^{2} + e^{2}} {\left (c x e + c d - \sqrt {c^{2} x^{2} - 1} e\right )}}{c^{2} d^{2} - e^{2}}\right ) + {\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 (2 \, b c^{4} d^{5} x e + b c^{4} d^{6} - 4 \, b c^{2} d^{3} x e^{3} + b x^{2} e^{6} + 2 \, b d x e^{5} - {\left (2 \, b c^{2} d^{2} x^{2} - b d^{2}\right )} e^{4} + {\left (b c^{4} d^{4} x^{2} - 2 \, b c^{2} d^{4}\right )} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c d^{2} x - a d^{2}\right )} e^{4} + {\left (b c^{3} d^{3} x^{2} - b c d^{3}\right )} e^{3} + 2 \, {\left (b c^{3} d^{4} x - a c^{2} d^{4}\right )} e^{2} + {\left (b c^{2} d^{3} x e^{3} + b c^{2} d^{4} e^{2} - b d x e^{5} - b d^{2} e^{4}\right )} \sqrt {c^{2} x^{2} - 1}}{2 \, {\left (2 \, c^{4} d^{7} x e^{2} + c^{4} d^{8} e - 4 \, c^{2} d^{5} x e^{4} + d^{2} x^{2} e^{7} + 2 \, d^{3} x e^{6} - {\left (2 \, c^{2} d^{4} x^{2} - d^{4}\right )} e^{5} + {\left (c^{4} d^{6} x^{2} - 2 \, c^{2} d^{6}\right )} e^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 - b*c*d*x^2*e^5 - (4*b*c^2*d^3*x*e^2 + 2*b*c^2*d^4*e - b*x^2*e^5 - 2*b*d*x*e^4
+ (2*b*c^2*d^2*x^2 - b*d^2)*e^3)*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)*sqrt(c^2*x^2 - 1))/(x*e + d)) + (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b
*d^2*e^4)*arccsc(c*x) + 2*(2*b*c^4*d^5*x*e + b*c^4*d^6 - 4*b*c^2*d^3*x*e^3 + b*x^2*e^6 + 2*b*d*x*e^5 - (2*b*c^
2*d^2*x^2 - b*d^2)*e^4 + (b*c^4*d^4*x^2 - 2*b*c^2*d^4)*e^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)) - (2*b*c*d^2*x -
a*d^2)*e^4 + (b*c^3*d^3*x^2 - b*c*d^3)*e^3 + 2*(b*c^3*d^4*x - a*c^2*d^4)*e^2 + (b*c^2*d^3*x*e^3 + b*c^2*d^4*e^
2 - b*d*x*e^5 - b*d^2*e^4)*sqrt(c^2*x^2 - 1))/(2*c^4*d^7*x*e^2 + c^4*d^8*e - 4*c^2*d^5*x*e^4 + d^2*x^2*e^7 + 2
*d^3*x*e^6 - (2*c^2*d^4*x^2 - d^4)*e^5 + (c^4*d^6*x^2 - 2*c^2*d^6)*e^3), -1/2*(a*c^4*d^6 + b*c^3*d^5*e - b*c*d
*x^2*e^5 + 2*(4*b*c^2*d^3*x*e^2 + 2*b*c^2*d^4*e - b*x^2*e^5 - 2*b*d*x*e^4 + (2*b*c^2*d^2*x^2 - b*d^2)*e^3)*sqr
t(-c^2*d^2 + e^2)*arctan(sqrt(-c^2*d^2 + e^2)*(c*x*e + c*d - sqrt(c^2*x^2 - 1)*e)/(c^2*d^2 - e^2)) + (b*c^4*d^
6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4)*arccsc(c*x) + 2*(2*b*c^4*d^5*x*e + b*c^4*d^6 - 4*b*c^2*d^3*x*e^3 + b*x^2*e^6
+ 2*b*d*x*e^5 - (2*b*c^2*d^2*x^2 - b*d^2)*e^4 + (b*c^4*d^4*x^2 - 2*b*c^2*d^4)*e^2)*arctan(-c*x + sqrt(c^2*x^2
- 1)) - (2*b*c*d^2*x - a*d^2)*e^4 + (b*c^3*d^3*x^2 - b*c*d^3)*e^3 + 2*(b*c^3*d^4*x - a*c^2*d^4)*e^2 + (b*c^2*d
^3*x*e^3 + b*c^2*d^4*e^2 - b*d*x*e^5 - b*d^2*e^4)*sqrt(c^2*x^2 - 1))/(2*c^4*d^7*x*e^2 + c^4*d^8*e - 4*c^2*d^5*
x*e^4 + d^2*x^2*e^7 + 2*d^3*x*e^6 - (2*c^2*d^4*x^2 - d^4)*e^5 + (c^4*d^6*x^2 - 2*c^2*d^6)*e^3)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((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: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 {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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