Optimal. Leaf size=47 \[ -\frac{c \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{b}-\frac{c \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{b} \]
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Rubi [A] time = 0.106107, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5223, 3303, 3299, 3302} \[ -\frac{c \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{b}-\frac{c \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 5223
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b \csc ^{-1}(c x)\right )} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\left (\left (c \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\right )-\left (c \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac{c \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{b}-\frac{c \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0811525, size = 43, normalized size = 0.91 \[ -\frac{c \left (\cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )+\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\csc ^{-1}(c x)\right )\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.214, size = 48, normalized size = 1. \begin{align*} c \left ( -{\frac{1}{b}{\it Si} \left ({\frac{a}{b}}+{\rm arccsc} \left (cx\right ) \right ) \sin \left ({\frac{a}{b}} \right ) }-{\frac{1}{b}{\it Ci} \left ({\frac{a}{b}}+{\rm arccsc} \left (cx\right ) \right ) \cos \left ({\frac{a}{b}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b x^{2} \operatorname{arccsc}\left (c x\right ) + a x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (a + b \operatorname{acsc}{\left (c x \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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