Optimal. Leaf size=108 \[ \frac{b x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt{d+e x^2}} \]
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Rubi [A] time = 0.0911184, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {191, 5229, 12, 421, 419} \[ \frac{x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{b x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 5229
Rule 12
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac{x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{(b c x) \int \frac{1}{d \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=\frac{x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{(b c x) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{d \sqrt{c^2 x^2}}\\ &=\frac{x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{\left (b c x \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}} \, dx}{d \sqrt{c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{\left (b c x \sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}}} \, dx}{d \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{b x \sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{d \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [A] time = 0.192367, size = 112, normalized size = 1.04 \[ \frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{d \left (c^3 x^2-c\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.883, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\rm arccsc} \left (cx\right )) \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{\sqrt{e x^{2} + d}{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{b \operatorname{arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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