\(\int \frac {1}{x^4 (a+b \csc ^{-1}(c x))} \, dx\) [38]

Optimal result

Integrand size = 14, antiderivative size = 117 \[ \int \frac {1}{x^4 \left (a+b \csc ^{-1}(c x)\right )} \, dx=-\frac {c^3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac {c^3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b}-\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac {c^3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5331, 4491, 3384, 3380, 3383} \[ \int \frac {1}{x^4 \left (a+b \csc ^{-1}(c x)\right )} \, dx=-\frac {c^3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac {c^3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b}-\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )}{4 b}+\frac {c^3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \csc ^{-1}(c x)\right )}{4 b} \]

[In]

Int[1/(x^4*(a + b*ArcCsc[c*x])),x]

[Out]

-1/4*(c^3*Cos[a/b]*CosIntegral[a/b + ArcCsc[c*x]])/b + (c^3*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcCsc[c*x]])
/(4*b) - (c^3*Sin[a/b]*SinIntegral[a/b + ArcCsc[c*x]])/(4*b) + (c^3*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcCs
c[c*x]])/(4*b)

Rule 3380

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 3383

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]

Rule 3384

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 4491

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 5331

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G
tQ[n, 0] || LtQ[m, -1])

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^4 \left (a+b \csc ^{-1}(c x)\right )} \, dx=-\frac {c^3 \left (\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )-\cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\csc ^{-1}(c x)\right )\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\csc ^{-1}(c x)\right )-\sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\csc ^{-1}(c x)\right )\right )\right )}{4 b} \]

[In]

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

[Out]

-1/4*(c^3*(Cos[a/b]*CosIntegral[a/b + ArcCsc[c*x]] - Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcCsc[c*x])] + Sin[a/b
]*SinIntegral[a/b + ArcCsc[c*x]] - Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcCsc[c*x])]))/b

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(c^{3} \left (-\frac {\operatorname {Si}\left (\frac {a}{b}+\operatorname {arccsc}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )}{4 b}-\frac {\operatorname {Ci}\left (\frac {a}{b}+\operatorname {arccsc}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{4 b}+\frac {\operatorname {Si}\left (\frac {3 a}{b}+3 \,\operatorname {arccsc}\left (c x \right )\right ) \sin \left (\frac {3 a}{b}\right )}{4 b}+\frac {\operatorname {Ci}\left (\frac {3 a}{b}+3 \,\operatorname {arccsc}\left (c x \right )\right ) \cos \left (\frac {3 a}{b}\right )}{4 b}\right )\)	\(102\)
default	\(c^{3} \left (-\frac {\operatorname {Si}\left (\frac {a}{b}+\operatorname {arccsc}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )}{4 b}-\frac {\operatorname {Ci}\left (\frac {a}{b}+\operatorname {arccsc}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{4 b}+\frac {\operatorname {Si}\left (\frac {3 a}{b}+3 \,\operatorname {arccsc}\left (c x \right )\right ) \sin \left (\frac {3 a}{b}\right )}{4 b}+\frac {\operatorname {Ci}\left (\frac {3 a}{b}+3 \,\operatorname {arccsc}\left (c x \right )\right ) \cos \left (\frac {3 a}{b}\right )}{4 b}\right )\)	\(102\)

[In]

int(1/x^4/(a+b*arccsc(c*x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {1}{x^4 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{4}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{x^4 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int \frac {1}{x^{4} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{x^4 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{4}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.71 \[ \int \frac {1}{x^4 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\frac {1}{4} \, {\left (\frac {4 \, c^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b} + \frac {4 \, c^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b} - \frac {3 \, c^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b} - \frac {c^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (\frac {1}{c x}\right )\right )}{b} - \frac {c^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b} - \frac {c^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (\frac {1}{c x}\right )\right )}{b}\right )} c \]

[In]

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

[Out]

1/4*(4*c^2*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(1/(c*x)))/b + 4*c^2*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b
+ 3*arcsin(1/(c*x)))/b - 3*c^2*cos(a/b)*cos_integral(3*a/b + 3*arcsin(1/(c*x)))/b - c^2*cos(a/b)*cos_integral
(a/b + arcsin(1/(c*x)))/b - c^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(1/(c*x)))/b - c^2*sin(a/b)*sin_integral
(a/b + arcsin(1/(c*x)))/b)*c

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int \frac {1}{x^4\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )} \,d x \]

[In]

int(1/(x^4*(a + b*asin(1/(c*x)))),x)

[Out]

int(1/(x^4*(a + b*asin(1/(c*x)))), x)