Optimal. Leaf size=275 \[ -\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {b c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {b x \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}} \]
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Rubi [A] time = 0.32, antiderivative size = 275, normalized size of antiderivative = 1.00, 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 12
Rule 191
Rule 271
Rule 419
Rule 421
Rule 424
Rule 426
Rule 427
Rule 524
Rule 583
Rule 5239
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*}
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Mathematica [C] time = 0.49, size = 213, normalized size = 0.77 \[ \frac {-a \left (d+2 e x^2\right )-b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (d+e x^2\right )-b \csc ^{-1}(c x) \left (d+2 e x^2\right )}{d^2 x \sqrt {d+e x^2}}+\frac {i b c x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {e x^2}{d}+1} \left (c^2 d E\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-\left (c^2 d+2 e\right ) F\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )\right )}{\sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.03, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsc}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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