Optimal. Leaf size=275 \[ -\frac{b x \sqrt{1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{a+b \csc ^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{d^2 \sqrt{c^2 x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}} \]
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Rubi [A] time = 0.317526, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {271, 191, 5239, 12, 583, 524, 427, 426, 424, 421, 419} \[ -\frac{2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{a+b \csc ^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{d^2 \sqrt{c^2 x^2}}-\frac{b x \sqrt{1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 271
Rule 191
Rule 5239
Rule 12
Rule 583
Rule 524
Rule 427
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac{a+b \csc ^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}+\frac{(b c x) \int \frac{-d-2 e x^2}{d^2 x^2 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{a+b \csc ^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}+\frac{(b c x) \int \frac{-d-2 e x^2}{x^2 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{d^2 \sqrt{c^2 x^2}}\\ &=-\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{d^2 \sqrt{c^2 x^2}}-\frac{a+b \csc ^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}+\frac{(b c x) \int \frac{-2 d e+c^2 d e x^2}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{d^3 \sqrt{c^2 x^2}}\\ &=-\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{d^2 \sqrt{c^2 x^2}}-\frac{a+b \csc ^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}+\frac{\left (b c^3 x\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}} \, dx}{d^2 \sqrt{c^2 x^2}}-\frac{\left (b c \left (c^2 d+2 e\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{d^2 \sqrt{c^2 x^2}}\\ &=-\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{d^2 \sqrt{c^2 x^2}}-\frac{a+b \csc ^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}+\frac{\left (b c^3 x \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}} \, dx}{d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2}}-\frac{\left (b c \left (c^2 d+2 e\right ) 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^2 \sqrt{c^2 x^2} \sqrt{d+e x^2}}\\ &=-\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{d^2 \sqrt{c^2 x^2}}-\frac{a+b \csc ^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}+\frac{\left (b c^3 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2}\right ) \int \frac{\sqrt{1+\frac{e x^2}{d}}}{\sqrt{1-c^2 x^2}} \, dx}{d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (b c \left (c^2 d+2 e\right ) 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^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ &=-\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{d^2 \sqrt{c^2 x^2}}-\frac{a+b \csc ^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}-\frac{b \left (c^2 d+2 e\right ) 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^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.477304, size = 213, normalized size = 0.77 \[ \frac{-a \left (d+2 e x^2\right )-b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (d+e x^2\right )-b \csc ^{-1}(c x) \left (d+2 e x^2\right )}{d^2 x \sqrt{d+e x^2}}+\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left (c^2 d E\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right )|-\frac{e}{c^2 d}\right )-\left (c^2 d+2 e\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right ),-\frac{e}{c^2 d}\right )\right )}{\sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 3.06, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsc} \left (cx\right )}{{x}^{2}} \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^{6} + 2 \, d e x^{4} + d^{2} x^{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}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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