Optimal. Leaf size=79 \[ \frac {b c x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{\sqrt {d} e \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{e \sqrt {d+e x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5237, 446, 93, 204} \[ \frac {b c x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{\sqrt {d} e \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{e \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 204
Rule 446
Rule 5237
Rubi steps
\begin {align*} \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac {a+b \csc ^{-1}(c x)}{e \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {1}{x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{e \sqrt {c^2 x^2}}\\ &=-\frac {a+b \csc ^{-1}(c x)}{e \sqrt {d+e x^2}}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e \sqrt {c^2 x^2}}\\ &=-\frac {a+b \csc ^{-1}(c x)}{e \sqrt {d+e x^2}}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}}\right )}{e \sqrt {c^2 x^2}}\\ &=-\frac {a+b \csc ^{-1}(c x)}{e \sqrt {d+e x^2}}+\frac {b c x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{\sqrt {d} e \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 96, normalized size = 1.22 \[ -\frac {a+b \csc ^{-1}(c x)}{e \sqrt {d+e x^2}}-\frac {b c x \sqrt {1-\frac {1}{c^2 x^2}} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2-1}}{\sqrt {d+e x^2}}\right )}{\sqrt {d} e \sqrt {c^2 x^2-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 283, normalized size = 3.58 \[ \left [-\frac {{\left (b e x^{2} + b d\right )} \sqrt {-d} \log \left (\frac {{\left (c^{4} d^{2} - 6 \, c^{2} d e + e^{2}\right )} x^{4} - 8 \, {\left (c^{2} d^{2} - d e\right )} x^{2} + 4 \, \sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {-d} + 8 \, d^{2}}{x^{4}}\right ) + 4 \, \sqrt {e x^{2} + d} {\left (b d \operatorname {arccsc}\left (c x\right ) + a d\right )}}{4 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}, \frac {{\left (b e x^{2} + b d\right )} \sqrt {d} \arctan \left (-\frac {\sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {d}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right ) - 2 \, \sqrt {e x^{2} + d} {\left (b d \operatorname {arccsc}\left (c x\right ) + a d\right )}}{2 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.79, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
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 \[ -\frac {{\left (\sqrt {e x^{2} + d} \int \frac {1}{\sqrt {e x^{2} + d} \sqrt {c x + 1} \sqrt {c x - 1} x}\,{d x} + \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} b}{\sqrt {e x^{2} + d} e} - \frac {a}{\sqrt {e x^{2} + d} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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