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

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

[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 = 0.29, 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 = {5227, 1568, 1475, 1651, 844, 216, 725, 206} \[ -\frac {a+b \csc ^{-1}(c x)}{2 e (d+e x)^2}-\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (\frac {d}{x}+e\right )}+\frac {b \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d^2-e^2}}\right )}{2 d^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b \csc ^{-1}(c x)}{2 d^2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*e*Sqrt[1 - 1/(c^2*x^2)])/(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 206

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

Rule 216

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 725

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 844

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 1475

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 1568

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 1651

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 5227

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 \operatorname {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) \operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.51, size = 250, normalized size = 1.45 \[ \frac {1}{2} \left (-\frac {a}{e (d+e x)^2}-\frac {b c e x \sqrt {1-\frac {1}{c^2 x^2}}}{d \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b \left (2 c^2 d^2-e^2\right ) \log \left (c x \left (c d-\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d^2-e^2}\right )+e\right )}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}+\frac {b \left (2 c^2 d^2-e^2\right ) \log (d+e x)}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}+\frac {b \sin ^{-1}\left (\frac {1}{c x}\right )}{d^2 e}-\frac {b \csc ^{-1}(c x)}{e (d+e x)^2}\right ) \]

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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fricas [B]  time = 0.95, size = 1111, normalized size = 6.46 \[ \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 ] \]

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 - 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)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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(x)]sym2poly
/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.07, size = 1007, normalized size = 5.85 \[ -\frac {c^{2} a}{2 \left (c e x +d c \right )^{2} e}-\frac {c^{2} b \,\mathrm {arccsc}\left (c x \right )}{2 \left (c e x +d c \right )^{2} e}+\frac {c^{2} b \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 (c e x +d c \right )}+\frac {c^{2} b \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 (c e x +d c \right )}-\frac {c^{2} b \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 c^{2} d x -2 e}{c e x +d c}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}-\frac {c^{2} b \sqrt {c^{2} x^{2}-1}\, d \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 c^{2} d x -2 e}{c e x +d c}\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}-\frac {c^{2} b e x}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}+\frac {b e}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}-\frac {b \,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 (c e x +d c \right )}-\frac {b 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 (c e x +d c \right )}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 c^{2} d x -2 e}{c e x +d c}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}+\frac {b e \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 c^{2} d x -2 e}{c e x +d c}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left ({\left (c^{2} e^{3} x^{2} + 2 \, c^{2} d e^{2} x + c^{2} d^{2} e\right )} \int \frac {x e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{3} - 2 \, d e^{2} x + {\left (c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{3} - 2 \, d e^{2} x - d^{2} e + {\left (c^{2} d^{2} e - e^{3}\right )} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} - d^{2} e + {\left (c^{2} d^{2} e - e^{3}\right )} x^{2}}\,{d x} + \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} b}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {a}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]

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*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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^3} \,d x \]

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

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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