3.1.17 \(\int x (a+b \sec ^{-1}(c x))^2 \, dx\) [17]

Optimal. Leaf size=56 \[ -\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5330, 4494, 4269, 3556} \begin {gather*} -\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSec[c*x])^2,x]

[Out]

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

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4494

Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[
(c + d*x)^m*(Sec[a + b*x]^n/(b*n)), x] - Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; Fre
eQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5330

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
ec[x]^(m + 1)*Tan[x], x], x, ArcSec[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n,
0] || LtQ[m, -1])

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 90, normalized size = 1.61 \begin {gather*} \frac {a c x \left (-2 b \sqrt {1-\frac {1}{c^2 x^2}}+a c x\right )+2 b c x \left (-b \sqrt {1-\frac {1}{c^2 x^2}}+a c x\right ) \sec ^{-1}(c x)+b^2 c^2 x^2 \sec ^{-1}(c x)^2+2 b^2 \log (c x)}{2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcSec[c*x])^2,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(127\) vs. \(2(52)=104\).
time = 0.26, size = 128, normalized size = 2.29

method	result	size
derivativedivides	\(\frac {\frac {c^{2} x^{2} a^{2}}{2}+\frac {b^{2} \mathrm {arcsec}\left (c x \right )^{2} c^{2} x^{2}}{2}-b^{2} \mathrm {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-b^{2} \ln \left (\frac {1}{c x}\right )+2 a b \left (\frac {c^{2} x^{2} \mathrm {arcsec}\left (c x \right )}{2}-\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\)	\(128\)
default	\(\frac {\frac {c^{2} x^{2} a^{2}}{2}+\frac {b^{2} \mathrm {arcsec}\left (c x \right )^{2} c^{2} x^{2}}{2}-b^{2} \mathrm {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-b^{2} \ln \left (\frac {1}{c x}\right )+2 a b \left (\frac {c^{2} x^{2} \mathrm {arcsec}\left (c x \right )}{2}-\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 87, normalized size = 1.55 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} \operatorname {arcsec}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (x^{2} \operatorname {arcsec}\left (c x\right ) - \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} a b - {\left (\frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arcsec}\left (c x\right )}{c} - \frac {\log \left (x\right )}{c^{2}}\right )} b^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsec(c*x))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (52) = 104\).
time = 2.15, size = 111, normalized size = 1.98 \begin {gather*} \frac {b^{2} c^{2} x^{2} \operatorname {arcsec}\left (c x\right )^{2} + a^{2} c^{2} x^{2} + 4 \, a b c^{2} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, b^{2} \log \left (x\right ) + 2 \, {\left (a b c^{2} x^{2} - a b c^{2}\right )} \operatorname {arcsec}\left (c x\right ) - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (b^{2} \operatorname {arcsec}\left (c x\right ) + a b\right )}}{2 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsec(c*x))^2,x, algorithm="fricas")

[Out]

1/2*(b^2*c^2*x^2*arcsec(c*x)^2 + a^2*c^2*x^2 + 4*a*b*c^2*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 2*b^2*log(x) + 2*(
a*b*c^2*x^2 - a*b*c^2)*arcsec(c*x) - 2*sqrt(c^2*x^2 - 1)*(b^2*arcsec(c*x) + a*b))/c^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a + b*asec(c*x))**2, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2181 vs. \(2 (52) = 104\).
time = 0.56, size = 2181, normalized size = 38.95 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsec(c*x))^2,x, algorithm="giac")

[Out]

1/2*(b^2*arccos(1/(c*x))^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) + 2*a*b*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^2*(1/(c^2*x^2) - 1)*arccos(1/(c*x))^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)^2) - 2*b^2*log(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) + 2*b^2*log(2/(c*x) + 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) - 2*b^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) - 2*b^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) - 4*b^2*sqrt(-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)) + a^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) - 4*a*b*(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) + b^2*(1/(c^2*x^2) - 1)^2*ar
ccos(1/(c*x))^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) - 4*b^2*(1/(c^2*x^2) - 1)*log(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)^2) + 4*b^2*(1/(c^2*x^2) - 1)*log(2/(c*x) + 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)^2) - 4*b^2*(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^2*(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*a*b*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)) + 4*b^2*(-1/(c^2*x^2) + 1)^(3/2)*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)^3) - 2*a^2*(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*b*(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^2*(1/(c^2*x^2) - 1)^2*log(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) + 2*b^2*(1/(c^2*x^2) - 1)^2*log(2/(c*x)
+ 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) - 2*b^2*(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^2*(1/(c^2*x^2) - 1)^2*log(a
bs(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) + 4*a*b*(-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^2*(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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*acos(1/(c*x)))^2,x)

[Out]

int(x*(a + b*acos(1/(c*x)))^2, x)

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