Optimal. Leaf size=126 \[ -\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac{b x \sqrt{1-c^4 x^4}}{2 c^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1}}+\frac{b x \tan ^{-1}\left (\frac{\sqrt{1-c^4 x^4}}{\sqrt{c^2 x^2-1}}\right )}{2 c^3 \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.212074, antiderivative size = 135, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {261, 5247, 12, 1572, 1252, 848, 50, 63, 208} \[ -\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac{b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{2 c^5 x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{2 c^5 x \sqrt{1-\frac{1}{c^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5247
Rule 12
Rule 1572
Rule 1252
Rule 848
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt{1-c^4 x^4}} \, dx &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}+\frac{b \int -\frac{\sqrt{1-c^4 x^4}}{2 c^4 \sqrt{1-\frac{1}{c^2 x^2}} x^2} \, dx}{c}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac{b \int \frac{\sqrt{1-c^4 x^4}}{\sqrt{1-\frac{1}{c^2 x^2}} x^2} \, dx}{2 c^5}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{1-c^4 x^4}}{x \sqrt{1-c^2 x^2}} \, dx}{2 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-c^4 x^2}}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{4 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+c^2 x}}{x} \, dx,x,x^2\right )}{4 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{2 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+c^2 x}} \, dx,x,x^2\right )}{4 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{2 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{2 c^7 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{2 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}+\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{2 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.175771, size = 138, normalized size = 1.1 \[ \frac{\sqrt{1-c^4 x^4} \left (-a c^2 x^2+a-b c x \sqrt{1-\frac{1}{c^2 x^2}}\right )+\left (b-b c^2 x^2\right ) \tan ^{-1}\left (\frac{c x \sqrt{1-\frac{1}{c^2 x^2}}}{\sqrt{1-c^4 x^4}}\right )-b \left (c^2 x^2-1\right ) \sqrt{1-c^4 x^4} \csc ^{-1}(c x)}{2 c^4 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 3.027, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b{\rm arccsc} \left (cx\right ) \right ){\frac{1}{\sqrt{-{c}^{4}{x}^{4}+1}}}}\, dx \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*} \frac{{\left (c^{4} \int \frac{{\left (c^{2} x^{3} + x\right )} e^{\left (-\frac{1}{2} \, \log \left (c^{2} x^{2} + 1\right ) + \frac{1}{2} \, \log \left (c x - 1\right )\right )}}{{\left (c x + 1\right )}{\left (c x - 1\right )} \sqrt{-c x + 1} c^{2} + \sqrt{-c x + 1} c^{2}}\,{d x} - \sqrt{c^{2} x^{2} + 1} \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )\right )} b}{2 \, c^{4}} - \frac{\sqrt{-c^{4} x^{4} + 1} a}{2 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{3}}{\sqrt{-c^{4} x^{4} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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