Optimal. Leaf size=362 \[ \frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}-\frac {b c \sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {2 b x \sqrt {1-c^2 x^2} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}} \]
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Rubi [A] time = 0.57, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {271, 264, 5239, 12, 580, 583, 524, 427, 426, 424, 421, 419} \[ \frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}-\frac {b c \sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {2 b x \sqrt {1-c^2 x^2} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 271
Rule 419
Rule 421
Rule 424
Rule 426
Rule 427
Rule 524
Rule 580
Rule 583
Rule 5239
Rubi steps
\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx &=-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-d+2 e x^2\right )}{3 d^2 x^4 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-d+2 e x^2\right )}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{3 d^2 \sqrt {c^2 x^2}}\\ &=-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}-\frac {(b c x) \int \frac {d \left (2 c^2 d-5 e\right )+\left (c^2 d-6 e\right ) e x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {c^2 x^2}}\\ &=-\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}-\frac {(b c x) \int \frac {d \left (c^2 d-6 e\right ) e-c^2 d \left (2 c^2 d-5 e\right ) e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^3 \sqrt {c^2 x^2}}\\ &=-\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {\left (b c^3 \left (2 c^2 d-5 e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{9 d^2 \sqrt {c^2 x^2}}-\frac {\left (2 b c \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {c^2 x^2}}\\ &=-\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {\left (b c^3 \left (2 c^2 d-5 e\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (2 b c \left (c^2 d-3 e\right ) \left (c^2 d+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}{9 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}\\ &=-\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {\left (b c^3 \left (2 c^2 d-5 e\right ) 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}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (2 b c \left (c^2 d-3 e\right ) \left (c^2 d+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}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ &=-\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{3 d^2 x}+\frac {b c^2 \left (2 c^2 d-5 e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {2 b \left (c^2 d-3 e\right ) \left (c^2 d+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 )}{9 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 = 0.66, size = 249, normalized size = 0.69 \[ -\frac {\sqrt {d+e x^2} \left (3 a \left (d-2 e x^2\right )+b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (2 c^2 d x^2+d-5 e x^2\right )+3 b \csc ^{-1}(c x) \left (d-2 e x^2\right )\right )}{9 d^2 x^3}+\frac {i b c x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {e x^2}{d}+1} \left (c^2 d \left (2 c^2 d-5 e\right ) E\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+2 \left (c^4 \left (-d^2\right )+2 c^2 d e+3 e^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )\right )}{9 \sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{e x^{6} + d x^{4}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 8.42, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsc}\left (c x \right )}{x^{4} \sqrt {e \,x^{2}+d}}\, dx \]
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 \[ \frac {1}{3} \, a {\left (\frac {2 \, \sqrt {e x^{2} + d} e}{d^{2} x} - \frac {\sqrt {e x^{2} + d}}{d x^{3}}\right )} + \frac {{\left (2 \, e^{2} x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + \sqrt {e x^{2} + d} d^{2} x^{3} \int \frac {{\left (2 \, c^{2} e^{2} x^{4} + c^{2} d e x^{2} - c^{2} d^{2}\right )} e^{\left (-\frac {1}{2} \, \log \left (e x^{2} + d\right ) + \frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} d^{2} x^{4} - d^{2} x^{2} + {\left (c^{2} d^{2} x^{4} - d^{2} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )}}\,{d x} + d e x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - d^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} b}{3 \, \sqrt {e x^{2} + d} d^{2} x^{3}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^4\,\sqrt {e\,x^2+d}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x^{4} \sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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