Integrand size = 14, antiderivative size = 63 \[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\frac {c^2 \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}-\frac {c^2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{2 b} \]
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=-\frac {c^2 \left (-\operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right ) \sin \left (\frac {2 a}{b}\right )+\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )\right )}{2 b} \]
-1/2*(c^2*(-(CosIntegral[(2*a)/b + 2*ArcCsc[c*x]]*Sin[(2*a)/b]) + Cos[(2*a )/b]*SinIntegral[(2*a)/b + 2*ArcCsc[c*x]]))/b
Time = 0.52 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5746, 4906, 27, 3042, 3784, 3042, 3780, 3783}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx\) |
\(\Big \downarrow \) 5746 |
\(\displaystyle -c^2 \int \frac {\sqrt {1-\frac {1}{c^2 x^2}}}{c x \left (a+b \csc ^{-1}(c x)\right )}d\csc ^{-1}(c x)\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -c^2 \int \frac {\sin \left (2 \csc ^{-1}(c x)\right )}{2 \left (a+b \csc ^{-1}(c x)\right )}d\csc ^{-1}(c x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{2} c^2 \int \frac {\sin \left (2 \csc ^{-1}(c x)\right )}{a+b \csc ^{-1}(c x)}d\csc ^{-1}(c x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} c^2 \int \frac {\sin \left (2 \csc ^{-1}(c x)\right )}{a+b \csc ^{-1}(c x)}d\csc ^{-1}(c x)\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle -\frac {1}{2} c^2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{a+b \csc ^{-1}(c x)}d\csc ^{-1}(c x)-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{a+b \csc ^{-1}(c x)}d\csc ^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} c^2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{a+b \csc ^{-1}(c x)}d\csc ^{-1}(c x)-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 a}{b}+2 \csc ^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \csc ^{-1}(c x)}d\csc ^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle -\frac {1}{2} c^2 \left (\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{b}-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 a}{b}+2 \csc ^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \csc ^{-1}(c x)}d\csc ^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle -\frac {1}{2} c^2 \left (\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{b}-\frac {\sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{b}\right )\) |
-1/2*(c^2*(-((CosIntegral[(2*a)/b + 2*ArcCsc[c*x]]*Sin[(2*a)/b])/b) + (Cos [(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcCsc[c*x]])/b))
3.1.37.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG tQ[p, 0]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- (c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcC sc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n , 0] || LtQ[m, -1])
Time = 0.50 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {\operatorname {Si}\left (\frac {2 a}{b}+2 \,\operatorname {arccsc}\left (c x \right )\right ) \cos \left (\frac {2 a}{b}\right )}{2 b}+\frac {\operatorname {Ci}\left (\frac {2 a}{b}+2 \,\operatorname {arccsc}\left (c x \right )\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}\right )\) | \(58\) |
default | \(c^{2} \left (-\frac {\operatorname {Si}\left (\frac {2 a}{b}+2 \,\operatorname {arccsc}\left (c x \right )\right ) \cos \left (\frac {2 a}{b}\right )}{2 b}+\frac {\operatorname {Ci}\left (\frac {2 a}{b}+2 \,\operatorname {arccsc}\left (c x \right )\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}\right )\) | \(58\) |
\[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}} \,d x } \]
\[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int \frac {1}{x^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}\, dx \]
\[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.51 \[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\frac {1}{2} \, {\left (\frac {2 \, c \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (\frac {1}{c x}\right )\right ) \sin \left (\frac {a}{b}\right )}{b} - \frac {2 \, c \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b} + \frac {c \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b}\right )} c \]
1/2*(2*c*cos(a/b)*cos_integral(2*a/b + 2*arcsin(1/(c*x)))*sin(a/b)/b - 2*c *cos(a/b)^2*sin_integral(2*a/b + 2*arcsin(1/(c*x)))/b + c*sin_integral(2*a /b + 2*arcsin(1/(c*x)))/b)*c
Timed out. \[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int \frac {1}{x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )} \,d x \]