Integrand size = 14, antiderivative size = 63 \[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\frac {c^2 \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}-\frac {c^2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{2 b} \]
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Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5331, 4491, 12, 3384, 3380, 3383} \[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\frac {c^2 \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{2 b}-\frac {c^2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{2 b} \]
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Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 5331
Rubi steps \begin{align*} \text {integral}& = -\left (c^2 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right ) \\ & = -\left (c^2 \text {Subst}\left (\int \frac {\sin (2 x)}{2 (a+b x)} \, dx,x,\csc ^{-1}(c x)\right )\right ) \\ & = -\left (\frac {1}{2} c^2 \text {Subst}\left (\int \frac {\sin (2 x)}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right ) \\ & = -\left (\frac {1}{2} \left (c^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right )+\frac {1}{2} \left (c^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \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} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=-\frac {c^2 \left (-\operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right ) \sin \left (\frac {2 a}{b}\right )+\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )\right )}{2 b} \]
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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 | \(c^{2} \left (-\frac {\operatorname {Si}\left (\frac {2 a}{b}+2 \,\operatorname {arccsc}\left (c x \right )\right ) \cos \left (\frac {2 a}{b}\right )}{2 b}+\frac {\operatorname {Ci}\left (\frac {2 a}{b}+2 \,\operatorname {arccsc}\left (c x \right )\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}\right )\) | \(58\) |
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\[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}} \,d x } \]
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\[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int \frac {1}{x^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}\, dx \]
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\[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}} \,d x } \]
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none
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.51 \[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\frac {1}{2} \, {\left (\frac {2 \, c \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (\frac {1}{c x}\right )\right ) \sin \left (\frac {a}{b}\right )}{b} - \frac {2 \, c \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b} + \frac {c \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b}\right )} c \]
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Timed out. \[ \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx=\int \frac {1}{x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )} \,d x \]
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