Optimal. Leaf size=63 \[ \frac {c^2 \sin \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{2 b}-\frac {c^2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{2 b} \]
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Rubi [A] time = 0.13, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5223, 4406, 12, 3303, 3299, 3302} \[ \frac {c^2 \sin \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{2 b}-\frac {c^2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 3302
Rule 3303
Rule 4406
Rule 5223
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b \csc ^{-1}(c x)\right )} \, dx &=-\left (c^2 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\left (c^2 \operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 \text {Ci}\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*}
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Mathematica [A] time = 0.08, size = 56, normalized size = 0.89 \[ -\frac {c^2 \left (\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )-\sin \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 a}{b}+2 \csc ^{-1}(c x)\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b x^{3} \operatorname {arccsc}\left (c x\right ) + a x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 95, normalized size = 1.51 \[ \frac {1}{2} \, {\left (\frac {2 \, c \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (\frac {1}{c x}\right )\right ) \sin \left (\frac {a}{b}\right )}{b} - \frac {2 \, c \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b} + \frac {c \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (\frac {1}{c x}\right )\right )}{b}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 58, normalized size = 0.92 \[ c^{2} \left (-\frac {\Si \left (\frac {2 a}{b}+2 \,\mathrm {arccsc}\left (c x \right )\right ) \cos \left (\frac {2 a}{b}\right )}{2 b}+\frac {\Ci \left (\frac {2 a}{b}+2 \,\mathrm {arccsc}\left (c x \right )\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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