3.44 \(\int \frac{1}{x^4 (a+b \sec ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=178 \[ \frac{c^3 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac{3 c^3 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}+\frac{c^3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac{3 c^3 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}-\frac{c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )}-\frac{c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )} \]

[Out]

-(c^3*Sqrt[1 - 1/(c^2*x^2)])/(4*b*(a + b*ArcSec[c*x])) + (c^3*Cos[a/b]*CosIntegral[a/b + ArcSec[c*x]])/(4*b^2)
+ (3*c^3*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcSec[c*x]])/(4*b^2) - (c^3*Sin[3*ArcSec[c*x]])/(4*b*(a + b*Ar
cSec[c*x])) + (c^3*Sin[a/b]*SinIntegral[a/b + ArcSec[c*x]])/(4*b^2) + (3*c^3*Sin[(3*a)/b]*SinIntegral[(3*a)/b
+ 3*ArcSec[c*x]])/(4*b^2)

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Rubi [A]  time = 0.266526, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5222, 4406, 3297, 3303, 3299, 3302} \[ \frac{c^3 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac{3 c^3 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}+\frac{c^3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac{3 c^3 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}-\frac{c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )}-\frac{c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*ArcSec[c*x])^2),x]

[Out]

-(c^3*Sqrt[1 - 1/(c^2*x^2)])/(4*b*(a + b*ArcSec[c*x])) + (c^3*Cos[a/b]*CosIntegral[a/b + ArcSec[c*x]])/(4*b^2)
+ (3*c^3*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcSec[c*x]])/(4*b^2) - (c^3*Sin[3*ArcSec[c*x]])/(4*b*(a + b*Ar
cSec[c*x])) + (c^3*Sin[a/b]*SinIntegral[a/b + ArcSec[c*x]])/(4*b^2) + (3*c^3*Sin[(3*a)/b]*SinIntegral[(3*a)/b
+ 3*ArcSec[c*x]])/(4*b^2)

Rule 5222

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])

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx &=c^3 \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )\\ &=c^3 \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 (a+b x)^2}+\frac{\sin (3 x)}{4 (a+b x)^2}\right ) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{1}{4} c^3 \operatorname{Subst}\left (\int \frac{\sin (x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )+\frac{1}{4} c^3 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac{c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )}-\frac{c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )}+\frac{c^3 \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}\\ &=-\frac{c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )}-\frac{c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )}+\frac{\left (c^3 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}+\frac{\left (3 c^3 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}+\frac{\left (c^3 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}+\frac{\left (3 c^3 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}\\ &=-\frac{c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )}+\frac{c^3 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac{3 c^3 \cos \left (\frac{3 a}{b}\right ) \text{Ci}\left (\frac{3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}-\frac{c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )}+\frac{c^3 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac{3 c^3 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}\\ \end{align*}

Mathematica [A]  time = 0.445924, size = 223, normalized size = 1.25 \[ \frac{c^3 x^2 \cos \left (\frac{a}{b}\right ) \left (a+b \sec ^{-1}(c x)\right ) \text{CosIntegral}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )+3 c^3 x^2 \cos \left (\frac{3 a}{b}\right ) \left (a+b \sec ^{-1}(c x)\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sec ^{-1}(c x)\right )\right )+a c^3 x^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )+b c^3 x^2 \sin \left (\frac{a}{b}\right ) \sec ^{-1}(c x) \text{Si}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )+3 a c^3 x^2 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sec ^{-1}(c x)\right )\right )+3 b c^3 x^2 \sin \left (\frac{3 a}{b}\right ) \sec ^{-1}(c x) \text{Si}\left (3 \left (\frac{a}{b}+\sec ^{-1}(c x)\right )\right )-4 b c \sqrt{1-\frac{1}{c^2 x^2}}}{4 b^2 x^2 \left (a+b \sec ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*ArcSec[c*x])^2),x]

[Out]

(-4*b*c*Sqrt[1 - 1/(c^2*x^2)] + c^3*x^2*(a + b*ArcSec[c*x])*Cos[a/b]*CosIntegral[a/b + ArcSec[c*x]] + 3*c^3*x^
2*(a + b*ArcSec[c*x])*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSec[c*x])] + a*c^3*x^2*Sin[a/b]*SinIntegral[a/b + A
rcSec[c*x]] + b*c^3*x^2*ArcSec[c*x]*Sin[a/b]*SinIntegral[a/b + ArcSec[c*x]] + 3*a*c^3*x^2*Sin[(3*a)/b]*SinInte
gral[3*(a/b + ArcSec[c*x])] + 3*b*c^3*x^2*ArcSec[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSec[c*x])])/(4*b^2*
x^2*(a + b*ArcSec[c*x]))

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Maple [A]  time = 0.247, size = 153, normalized size = 0.9 \begin{align*}{c}^{3} \left ( -{\frac{\sin \left ( 3\,{\rm arcsec} \left (cx\right ) \right ) }{ \left ( 4\,a+4\,b{\rm arcsec} \left (cx\right ) \right ) b}}+{\frac{3}{4\,{b}^{2}} \left ({\it Si} \left ( 3\,{\frac{a}{b}}+3\,{\rm arcsec} \left (cx\right ) \right ) \sin \left ( 3\,{\frac{a}{b}} \right ) +{\it Ci} \left ( 3\,{\frac{a}{b}}+3\,{\rm arcsec} \left (cx\right ) \right ) \cos \left ( 3\,{\frac{a}{b}} \right ) \right ) }-{\frac{1}{ \left ( 4\,a+4\,b{\rm arcsec} \left (cx\right ) \right ) b}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{1}{4\,{b}^{2}} \left ({\it Si} \left ({\frac{a}{b}}+{\rm arcsec} \left (cx\right ) \right ) \sin \left ({\frac{a}{b}} \right ) +{\it Ci} \left ({\frac{a}{b}}+{\rm arcsec} \left (cx\right ) \right ) \cos \left ({\frac{a}{b}} \right ) \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(a+b*arcsec(c*x))^2,x)

[Out]

c^3*(-1/4*sin(3*arcsec(c*x))/(a+b*arcsec(c*x))/b+3/4*(Si(3*a/b+3*arcsec(c*x))*sin(3*a/b)+Ci(3*a/b+3*arcsec(c*x
))*cos(3*a/b))/b^2-1/4*((c^2*x^2-1)/c^2/x^2)^(1/2)/(a+b*arcsec(c*x))/b+1/4*(Si(a/b+arcsec(c*x))*sin(a/b)+Ci(a/
b+arcsec(c*x))*cos(a/b))/b^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*arcsec(c*x))^2,x, algorithm="maxima")

[Out]

-(4*sqrt(c*x + 1)*sqrt(c*x - 1)*(b*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + a) + (4*b^3*x^3*arctan(sqrt(c*x + 1)*
sqrt(c*x - 1))^2 + b^3*x^3*log(c^2*x^2)^2 + 8*b^3*x^3*log(c)*log(x) + 4*b^3*x^3*log(x)^2 + 8*a*b^2*x^3*arctan(
sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*(b^3*log(c)^2 + a^2*b)*x^3 - 4*(b^3*x^3*log(c) + b^3*x^3*log(x))*log(c^2*x^2)
)*integrate(4*(2*a*c^2*x^2 + (2*b*c^2*x^2 - 3*b)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - 3*a)*sqrt(c*x + 1)*sqrt
(c*x - 1)/(4*(b^3*c^2*log(c)^2 + a^2*b*c^2)*x^6 - 4*(b^3*log(c)^2 + a^2*b)*x^4 + 4*(b^3*c^2*x^6 - b^3*x^4)*arc
tan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + (b^3*c^2*x^6 - b^3*x^4)*log(c^2*x^2)^2 + 4*(b^3*c^2*x^6 - b^3*x^4)*log(x)
^2 + 8*(a*b^2*c^2*x^6 - a*b^2*x^4)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - 4*(b^3*c^2*x^6*log(c) - b^3*x^4*log(c
) + (b^3*c^2*x^6 - b^3*x^4)*log(x))*log(c^2*x^2) + 8*(b^3*c^2*x^6*log(c) - b^3*x^4*log(c))*log(x)), x))/(4*b^3
*x^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + b^3*x^3*log(c^2*x^2)^2 + 8*b^3*x^3*log(c)*log(x) + 4*b^3*x^3*log(
x)^2 + 8*a*b^2*x^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*(b^3*log(c)^2 + a^2*b)*x^3 - 4*(b^3*x^3*log(c) + b^
3*x^3*log(x))*log(c^2*x^2))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{2} x^{4} \operatorname{arcsec}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname{arcsec}\left (c x\right ) + a^{2} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*arcsec(c*x))^2,x, algorithm="fricas")

[Out]

integral(1/(b^2*x^4*arcsec(c*x)^2 + 2*a*b*x^4*arcsec(c*x) + a^2*x^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \left (a + b \operatorname{asec}{\left (c x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(a+b*asec(c*x))**2,x)

[Out]

Integral(1/(x**4*(a + b*asec(c*x))**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}^{2} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*arcsec(c*x))^2,x, algorithm="giac")

[Out]

integrate(1/((b*arcsec(c*x) + a)^2*x^4), x)