Optimal. Leaf size=296 \[ \frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c e x^2 \sqrt {c^2 x^2-1}}{3 d^2 \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1}} \]
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Rubi [A] time = 0.24, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {192, 191, 5229, 12, 527, 524, 427, 426, 424, 421, 419} \[ \frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c e x^2 \sqrt {c^2 x^2-1}}{3 d^2 \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 192
Rule 419
Rule 421
Rule 424
Rule 426
Rule 427
Rule 524
Rule 527
Rule 5229
Rubi steps
\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {3 d+2 e x^2}{3 d^2 \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {3 d+2 e x^2}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {c^2 x^2}}\\ &=-\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {d \left (3 c^2 d+2 e\right )+c^2 d e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=-\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {(2 b c x) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^2 \sqrt {c^2 x^2}}+\frac {\left (b c^3 x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=-\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 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}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (2 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}{3 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}\\ &=-\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 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}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (2 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}{3 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ &=-\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \csc ^{-1}(c x)\right )}{3 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 )}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {2 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 )}{3 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 = 0.56, size = 249, normalized size = 0.84 \[ \frac {x \left (a \left (c^2 d+e\right ) \left (3 d+2 e x^2\right )-b c e x \sqrt {1-\frac {1}{c^2 x^2}} \left (d+e x^2\right )+b \left (c^2 d+e\right ) \csc ^{-1}(c x) \left (3 d+2 e x^2\right )\right )}{3 d^2 \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {i b c x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {e x^2}{d}+1} \left (2 \left (c^2 d+e\right ) F\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+c^2 d E\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )\right )}{3 \sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.32, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsc}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
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 \[ \frac {1}{3} \, a {\left (\frac {2 \, x}{\sqrt {e x^{2} + d} d^{2}} + \frac {x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} d}\right )} + b \int \frac {\arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e x^{2} + d}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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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