3.1.49 \(\int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx\) [49]

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

[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)

________________________________________________________________________________________

Rubi [A]
time = 0.11, 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 = {5335, 1582, 1489, 858, 222, 739, 212} \begin {gather*} -\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} \end {gather*}

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 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 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)^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 \text {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 \text {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 \text {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 \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 )}{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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.14, size = 141, normalized size = 1.38 \begin {gather*} -\frac {a}{e (d+e x)}-\frac {b \csc ^{-1}(c x)}{e (d+e x)}+\frac {b \text {ArcSin}\left (\frac {1}{c x}\right )}{d e}+\frac {b \log (d+e x)}{d \sqrt {c^2 d^2-e^2}}-\frac {b \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 \sqrt {c^2 d^2-e^2}} \end {gather*}

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])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(215\) vs. \(2(98)=196\).
time = 2.32, size = 216, normalized size = 2.12

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{2}}{\left (e c x +c d \right ) e}-\frac {b \,c^{2} \mathrm {arccsc}\left (c x \right )}{\left (e c x +c d \right ) e}+\frac {b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d}-\frac {b \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d \,c^{2} x -2 e}{e c x +c d}\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(216\)
default	\(\frac {-\frac {a \,c^{2}}{\left (e c x +c d \right ) e}-\frac {b \,c^{2} \mathrm {arccsc}\left (c x \right )}{\left (e c x +c d \right ) e}+\frac {b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d}-\frac {b \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d \,c^{2} x -2 e}{e c x +c d}\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(216\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (94) = 188\).
time = 0.42, size = 442, normalized size = 4.33 \begin {gather*} \left [-\frac {a c^{2} d^{3} - a d e^{2} - \sqrt {c^{2} d^{2} - e^{2}} {\left (b x e^{2} + 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}}{x e + 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^{2} x e + b c^{2} d^{3} - b x e^{3} - b d e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{3} x e^{2} + c^{2} d^{4} e - d x e^{4} - d^{2} e^{3}}, -\frac {a c^{2} d^{3} - a d e^{2} + 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (b x e^{2} + b d e\right )} \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^{2} d^{3} - b d e^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, {\left (b c^{2} d^{2} x e + b c^{2} d^{3} - b x e^{3} - b d e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{3} x e^{2} + c^{2} d^{4} e - d x e^{4} - d^{2} e^{3}}\right ] \end {gather*}

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*x*e^2 + 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))/(x*e + d)) + (b*c^2*d^3 - b*d*e^2)*
arccsc(c*x) + 2*(b*c^2*d^2*x*e + b*c^2*d^3 - b*x*e^3 - b*d*e^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^3*x*e
^2 + c^2*d^4*e - d*x*e^4 - d^2*e^3), -(a*c^2*d^3 - a*d*e^2 + 2*sqrt(-c^2*d^2 + e^2)*(b*x*e^2 + b*d*e)*arctan(s
qrt(-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^2*d^3 - b*d*e^2)*arccsc(c*x) +
2*(b*c^2*d^2*x*e + b*c^2*d^3 - b*x*e^3 - b*d*e^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^3*x*e^2 + c^2*d^4*
e - d*x*e^4 - d^2*e^3)]

________________________________________________________________________________________

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 )^{2}}\, dx \end {gather*}

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)

________________________________________________________________________________________

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)^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(sageVARx)]s
ym2poly/r2sym(

________________________________________________________________________________________

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 )}^2} \,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)^2,x)

[Out]

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

________________________________________________________________________________________