\(\int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^2} \, dx\) [61]

Optimal result

Integrand size = 16, antiderivative size = 104 \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^2} \, dx=-\frac {b \csc ^{-1}(c x)}{d e}-\frac {a+b \sec ^{-1}(c x)}{e (d+e x)}-\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {c^2 d^2-e^2} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{d \sqrt {c^2 d^2-e^2}} \]

[Out]

-b*arccsc(c*x)/d/e+(-a-b*arcsec(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] (verified)

Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5334, 1582, 1489, 858, 222, 739, 212} \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^2} \, dx=-\frac {a+b \sec ^{-1}(c x)}{e (d+e x)}-\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d^2-e^2}}\right )}{d \sqrt {c^2 d^2-e^2}}-\frac {b \csc ^{-1}(c x)}{d e} \]

[In]

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

[Out]

-((b*ArcCsc[c*x])/(d*e)) - (a + b*ArcSec[c*x])/(e*(d + e*x)) - (b*ArcTanh[(c^2*d + e/x)/(c*Sqrt[c^2*d^2 - e^2]
*Sqrt[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 5334

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b
*ArcSec[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 \sec ^{-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 \sec ^{-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 \sec ^{-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 \sec ^{-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 \sec ^{-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 \sec ^{-1}(c x)}{e (d+e x)}-\frac {b \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 )}{d \sqrt {c^2 d^2-e^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.37 \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^2} \, dx=-\frac {a}{e (d+e x)}-\frac {b \sec ^{-1}(c x)}{e (d+e x)}-\frac {b \arcsin \left (\frac {1}{c x}\right )}{d e}-\frac {b \log (d+e x)}{d \sqrt {c^2 d^2-e^2}}+\frac {b \log \left (e+c \left (c d-\sqrt {c^2 d^2-e^2} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{d \sqrt {c^2 d^2-e^2}} \]

[In]

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

[Out]

-(a/(e*(d + e*x))) - (b*ArcSec[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 [A] (verified)

Time = 2.19 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.86

method	result	size
parts	\(-\frac {a}{\left (e x +d \right ) e}+\frac {b \left (-\frac {c^{2} \operatorname {arcsec}\left (c x \right )}{\left (c e x +c d \right ) e}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right )\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(193\)
derivativedivides	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}+b \,c^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{\left (c e x +c d \right ) e}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right )\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x d \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(203\)
default	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}+b \,c^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{\left (c e x +c d \right ) e}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right )\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x d \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(203\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (98) = 196\).

Time = 0.31 (sec) , antiderivative size = 477, normalized size of antiderivative = 4.59 \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^2} \, dx=\left [-\frac {a c^{2} d^{3} - a d e^{2} - \sqrt {c^{2} d^{2} - e^{2}} {\left (b e^{2} x + b d e\right )} \log \left (\frac {c^{3} d^{2} x + c d e - \sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} + {\left (c^{2} d^{2} - \sqrt {c^{2} d^{2} - e^{2}} c d - e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{e x + d}\right ) + {\left (b c^{2} d^{3} - b d e^{2}\right )} \operatorname {arcsec}\left (c x\right ) - 2 \, {\left (b c^{2} d^{3} - b d e^{2} + {\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}, -\frac {a c^{2} d^{3} - a d e^{2} - 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (b e^{2} x + b d e\right )} \arctan \left (-\frac {\sqrt {-c^{2} d^{2} + e^{2}} \sqrt {c^{2} x^{2} - 1} e - \sqrt {-c^{2} d^{2} + e^{2}} {\left (c e x + c d\right )}}{c^{2} d^{2} - e^{2}}\right ) + {\left (b c^{2} d^{3} - b d e^{2}\right )} \operatorname {arcsec}\left (c x\right ) - 2 \, {\left (b c^{2} d^{3} - b d e^{2} + {\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}\right ] \]

[In]

integrate((a+b*arcsec(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)*
arcsec(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)*arcsec(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)]

Sympy [F]

\[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^2} \, dx=\int \frac {a + b \operatorname {asec}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^2} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((a+b*arcsec(c*x))/(e*x+d)^2,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. \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^2} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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