3.49 \(\int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=102 \[ -\frac {a+b \csc ^{-1}(c x)}{e (d+e x)}+\frac {b \tanh ^{-1}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d^2-e^2}}\right )}{d \sqrt {c^2 d^2-e^2}}+\frac {b \csc ^{-1}(c x)}{d e} \]

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5227, 1568, 1475, 844, 216, 725, 206} \[ -\frac {a+b \csc ^{-1}(c x)}{e (d+e x)}+\frac {b \tanh ^{-1}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d^2-e^2}}\right )}{d \sqrt {c^2 d^2-e^2}}+\frac {b \csc ^{-1}(c x)}{d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*ArcCsc[c*x])/(d*e) - (a + b*ArcCsc[c*x])/(e*(d + e*x)) + (b*ArcTanh[(c^2*d + e/x)/(c*Sqrt[c^2*d^2 - e^2]*Sq
rt[1 - 1/(c^2*x^2)])])/(d*Sqrt[c^2*d^2 - e^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 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)^2} \, dx &=-\frac {a+b \csc ^{-1}(c x)}{e (d+e x)}-\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)} \, dx}{c e}\\ &=-\frac {a+b \csc ^{-1}(c x)}{e (d+e x)}-\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} \left (e+\frac {d}{x}\right ) x^3} \, dx}{c e}\\ &=-\frac {a+b \csc ^{-1}(c x)}{e (d+e x)}+\frac {b \operatorname {Subst}\left (\int \frac {x}{(e+d x) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c e}\\ &=-\frac {a+b \csc ^{-1}(c x)}{e (d+e x)}-\frac {b \operatorname {Subst}\left (\int \frac {1}{(e+d x) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c d e}\\ &=\frac {b \csc ^{-1}(c x)}{d e}-\frac {a+b \csc ^{-1}(c x)}{e (d+e x)}+\frac {b \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 )}{c d}\\ &=\frac {b \csc ^{-1}(c x)}{d e}-\frac {a+b \csc ^{-1}(c x)}{e (d+e x)}+\frac {b \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 )}{d \sqrt {c^2 d^2-e^2}}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 141, normalized size = 1.38 \[ -\frac {a}{e (d+e x)}-\frac {b \log \left (c x \left (c d-\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d^2-e^2}\right )+e\right )}{d \sqrt {c^2 d^2-e^2}}+\frac {b \log (d+e x)}{d \sqrt {c^2 d^2-e^2}}+\frac {b \sin ^{-1}\left (\frac {1}{c x}\right )}{d e}-\frac {b \csc ^{-1}(c x)}{e (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.50, size = 475, normalized size = 4.66 \[ \left [-\frac {a c^{2} d^{3} - a d e^{2} - \sqrt {c^{2} d^{2} - e^{2}} {\left (b e^{2} x + b d e\right )} \log \left (\frac {c^{3} d^{2} x + c d e + \sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} + {\left (c^{2} d^{2} + \sqrt {c^{2} d^{2} - e^{2}} c d - e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{e x + d}\right ) + {\left (b c^{2} d^{3} - b d e^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, {\left (b c^{2} d^{3} - b d e^{2} + {\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}, -\frac {a c^{2} d^{3} - a d e^{2} + 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (b e^{2} x + b d e\right )} \arctan \left (-\frac {\sqrt {-c^{2} d^{2} + e^{2}} \sqrt {c^{2} x^{2} - 1} e - \sqrt {-c^{2} d^{2} + e^{2}} {\left (c e x + c d\right )}}{c^{2} d^{2} - e^{2}}\right ) + {\left (b c^{2} d^{3} - b d e^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, {\left (b c^{2} d^{3} - b d e^{2} + {\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(a*c^2*d^3 - a*d*e^2 - sqrt(c^2*d^2 - e^2)*(b*e^2*x + b*d*e)*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))/(e*x + d)) + (b*c^2*d^3 - b*d*e^2)*
arccsc(c*x) + 2*(b*c^2*d^3 - b*d*e^2 + (b*c^2*d^2*e - b*e^3)*x)*arctan(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^4*e -
d^2*e^3 + (c^2*d^3*e^2 - d*e^4)*x), -(a*c^2*d^3 - a*d*e^2 + 2*sqrt(-c^2*d^2 + e^2)*(b*e^2*x + b*d*e)*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)) + (b*c^2*d^3
- b*d*e^2)*arccsc(c*x) + 2*(b*c^2*d^3 - b*d*e^2 + (b*c^2*d^2*e - b*e^3)*x)*arctan(-c*x + sqrt(c^2*x^2 - 1)))/(
c^2*d^4*e - d^2*e^3 + (c^2*d^3*e^2 - d*e^4)*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)^2,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 = 0.06, size = 214, normalized size = 2.10 \[ -\frac {c a}{\left (c e x +d c \right ) e}-\frac {c b \,\mathrm {arccsc}\left (c x \right )}{\left (c e x +d c \right ) e}+\frac {b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d}-\frac {b \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 c^{2} d x -2 e}{c e x +d c}\right )}{c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*a/(c*e*x+c*d)/e-c*b/(c*e*x+c*d)/e*arccsc(c*x)+1/c*b/e*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x/d*arc
tan(1/(c^2*x^2-1)^(1/2))-1/c*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)*l
n(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 \[ -\frac {b {\left (\frac {2 \, {\left (c^{2} e^{2} x + c^{2} d e\right )} c {\left (\frac {\arctan \left (\frac {{\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{2} e + 2 \, c d}{2 \, \sqrt {-c^{2} d^{2} + e^{2}}}\right ) e}{\sqrt {-c^{2} d^{2} + e^{2}} c d} - \frac {\arctan \left (\frac {1}{2} \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{2}\right )}{c d}\right )}}{c^{2} e} + \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )}}{e^{2} x + d e} - \frac {a}{e^{2} x + d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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