\(\int (d+e x) (a+b \sec ^{-1}(c x)) \, dx\) [58]

Optimal result

Integrand size = 14, antiderivative size = 84 \[ \int (d+e x) \left (a+b \sec ^{-1}(c x)\right ) \, dx=-\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {b d^2 \csc ^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e}-\frac {b d \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5334, 1582, 1410, 1821, 858, 222, 272, 65, 214} \[ \int (d+e x) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {(d+e x)^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e}-\frac {b d \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c}-\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}}}{2 c}+\frac {b d^2 \csc ^{-1}(c x)}{2 e} \]

[In]

Int[(d + e*x)*(a + b*ArcSec[c*x]),x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 1410

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> -Subst[Int[(d + e/x^n)^q*((a +
c/x^(2*n))^p/x^2), x], x, 1/x] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]

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 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

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 {(d+e x)^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e}-\frac {b \int \frac {(d+e x)^2}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{2 c e} \\ & = \frac {(d+e x)^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e}-\frac {b \int \frac {\left (e+\frac {d}{x}\right )^2}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{2 c e} \\ & = \frac {(d+e x)^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e}+\frac {b \text {Subst}\left (\int \frac {(e+d x)^2}{x^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e} \\ & = -\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {(d+e x)^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e}-\frac {b \text {Subst}\left (\int \frac {-2 d e-d^2 x}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e} \\ & = -\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {(d+e x)^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e}+\frac {(b d) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e} \\ & = -\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {b d^2 \csc ^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e}+\frac {(b d) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 c} \\ & = -\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {b d^2 \csc ^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e}-(b c d) \text {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right ) \\ & = -\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {b d^2 \csc ^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \sec ^{-1}(c x)\right )}{2 e}-\frac {b d \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.36 \[ \int (d+e x) \left (a+b \sec ^{-1}(c x)\right ) \, dx=a d x+\frac {1}{2} a e x^2-\frac {b e x \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{2 c}+b d x \sec ^{-1}(c x)+\frac {1}{2} b e x^2 \sec ^{-1}(c x)-\frac {b d \sqrt {1-\frac {1}{c^2 x^2}} x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {-1+c^2 x^2}} \]

[In]

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

[Out]

a*d*x + (a*e*x^2)/2 - (b*e*x*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + b*d*x*ArcSec[c*x] + (b*e*x^2*ArcSec[c*x])
/2 - (b*d*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2]

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.31

method	result	size
parts	\(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \left (\frac {c \,\operatorname {arcsec}\left (c x \right ) x^{2} e}{2}+\operatorname {arcsec}\left (c x \right ) x c d -\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 d c \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e \sqrt {c^{2} x^{2}-1}\right )}{2 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )}{c}\)	\(110\)
derivativedivides	\(\frac {\frac {a \left (d x \,c^{2}+\frac {1}{2} e \,c^{2} x^{2}\right )}{c}+\frac {b \left (\operatorname {arcsec}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arcsec}\left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 d c \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e \sqrt {c^{2} x^{2}-1}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c}}{c}\)	\(127\)
default	\(\frac {\frac {a \left (d x \,c^{2}+\frac {1}{2} e \,c^{2} x^{2}\right )}{c}+\frac {b \left (\operatorname {arcsec}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arcsec}\left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 d c \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e \sqrt {c^{2} x^{2}-1}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c}}{c}\)	\(127\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.55 \[ \int (d+e x) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {a c^{2} e x^{2} + 2 \, a c^{2} d x + 2 \, b c d \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - \sqrt {c^{2} x^{2} - 1} b e + {\left (b c^{2} e x^{2} + 2 \, b c^{2} d x - 2 \, b c^{2} d - b c^{2} e\right )} \operatorname {arcsec}\left (c x\right ) + 2 \, {\left (2 \, b c^{2} d + b c^{2} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, c^{2}} \]

[In]

integrate((e*x+d)*(a+b*arcsec(c*x)),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 2.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.24 \[ \int (d+e x) \left (a+b \sec ^{-1}(c x)\right ) \, dx=a d x + \frac {a e x^{2}}{2} + b d x \operatorname {asec}{\left (c x \right )} + \frac {b e x^{2} \operatorname {asec}{\left (c x \right )}}{2} - \frac {b d \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} - \frac {b e \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} \]

[In]

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

[Out]

a*d*x + a*e*x**2/2 + b*d*x*asec(c*x) + b*e*x**2*asec(c*x)/2 - b*d*Piecewise((acosh(c*x), Abs(c**2*x**2) > 1),
(-I*asin(c*x), True))/c - b*e*Piecewise((sqrt(c**2*x**2 - 1)/c, Abs(c**2*x**2) > 1), (I*sqrt(-c**2*x**2 + 1)/c
, True))/(2*c)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.11 \[ \int (d+e x) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{2} \, a e x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arcsec}\left (c x\right ) - \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b e + a d x + \frac {{\left (2 \, c x \operatorname {arcsec}\left (c x\right ) - \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) + \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d}{2 \, c} \]

[In]

integrate((e*x+d)*(a+b*arcsec(c*x)),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1547 vs. \(2 (74) = 148\).

Time = 0.53 (sec) , antiderivative size = 1547, normalized size of antiderivative = 18.42 \[ \int (d+e x) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)*(a+b*arcsec(c*x)),x, algorithm="giac")

[Out]

1/2*(2*b*c*d*arccos(1/(c*x))/(c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x)
+ 1)^4) - 2*b*c*d*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)
^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4) + 2*b*c*d*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(c^3 +
2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4) + 2*a*c*d/(c^3 + 2*c^3*(1/(
c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4) + b*e*arccos(1/(c*x))/(c^3 + 2*c^3*(1
/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4) - 4*b*c*d*(1/(c^2*x^2) - 1)*log(abs
(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1
)^2/(1/(c*x) + 1)^4)*(1/(c*x) + 1)^2) + 4*b*c*d*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1
))/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4)*(1/(c*x) + 1)^2)
+ a*e/(c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4) - 2*b*e*(1/(c
^2*x^2) - 1)*arccos(1/(c*x))/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x
) + 1)^4)*(1/(c*x) + 1)^2) - 2*b*c*d*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c
*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4)*(1/(c*x) + 1)^4) - 2*b*c*d*(1/(c^2*x^2) - 1)^2*log(abs(s
qrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^
2/(1/(c*x) + 1)^4)*(1/(c*x) + 1)^4) + 2*b*c*d*(1/(c^2*x^2) - 1)^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1
))/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4)*(1/(c*x) + 1)^4)
- 2*b*e*sqrt(-1/(c^2*x^2) + 1)/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(
c*x) + 1)^4)*(1/(c*x) + 1)) - 2*a*e*(1/(c^2*x^2) - 1)/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1
/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4)*(1/(c*x) + 1)^2) - 2*a*c*d*(1/(c^2*x^2) - 1)^2/((c^3 + 2*c^3*(1/(c^2*x^2) -
1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4)*(1/(c*x) + 1)^4) + b*e*(1/(c^2*x^2) - 1)^2*arcc
os(1/(c*x))/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4)*(1/(c*x
) + 1)^4) + 2*b*e*(-1/(c^2*x^2) + 1)^(3/2)/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + c^3*(1/(c^2*x^2)
- 1)^2/(1/(c*x) + 1)^4)*(1/(c*x) + 1)^3) + a*e*(1/(c^2*x^2) - 1)^2/((c^3 + 2*c^3*(1/(c^2*x^2) - 1)/(1/(c*x) +
1)^2 + c^3*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4)*(1/(c*x) + 1)^4))*c

Mupad [B] (verification not implemented)

Time = 0.98 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92 \[ \int (d+e x) \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {a\,x\,\left (2\,d+e\,x\right )}{2}-\frac {b\,d\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {1}{c^2\,x^2}}}\right )}{c}+b\,d\,x\,\mathrm {acos}\left (\frac {1}{c\,x}\right )-\frac {b\,e\,x\,\left (\sqrt {1-\frac {1}{c^2\,x^2}}-c\,x\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{2\,c} \]

[In]

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

[Out]

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