Optimal. Leaf size=88 \[ \frac {a \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+\frac {a \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+b \tan ^{-1}\left (\sqrt [4]{-x^2-1}\right )-b \tanh ^{-1}\left (\sqrt [4]{-x^2-1}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1010, 398, 444, 63, 298, 203, 206} \[ \frac {a \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+\frac {a \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+b \tan ^{-1}\left (\sqrt [4]{-x^2-1}\right )-b \tanh ^{-1}\left (\sqrt [4]{-x^2-1}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 298
Rule 398
Rule 444
Rule 1010
Rubi steps
\begin {align*} \int \frac {a+b x}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx &=a \int \frac {1}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx+b \int \frac {x}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx\\ &=\frac {a \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+\frac {a \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-x} (2+x)} \, dx,x,x^2\right )\\ &=\frac {a \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+\frac {a \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}-(2 b) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\sqrt [4]{-1-x^2}\right )\\ &=\frac {a \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+\frac {a \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}-b \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{-1-x^2}\right )+b \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{-1-x^2}\right )\\ &=\frac {a \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+b \tan ^{-1}\left (\sqrt [4]{-1-x^2}\right )+\frac {a \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}-b \tanh ^{-1}\left (\sqrt [4]{-1-x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.26, size = 162, normalized size = 1.84 \[ \frac {x \left (b x \sqrt [4]{x^2+1} F_1\left (1;\frac {1}{4},1;2;-x^2,-\frac {x^2}{2}\right )-\frac {24 a F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-x^2,-\frac {x^2}{2}\right )}{\left (x^2+2\right ) \left (x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};-x^2,-\frac {x^2}{2}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};-x^2,-\frac {x^2}{2}\right )\right )-6 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-x^2,-\frac {x^2}{2}\right )\right )}\right )}{4 \sqrt [4]{-x^2-1}} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b x + a}{{\left (x^{2} + 2\right )} {\left (-x^{2} - 1\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {b x +a}{\left (-x^{2}-1\right )^{\frac {1}{4}} \left (x^{2}+2\right )}\, dx \]
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 \[ \int \frac {b x + a}{{\left (x^{2} + 2\right )} {\left (-x^{2} - 1\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,x}{{\left (-x^2-1\right )}^{1/4}\,\left (x^2+2\right )} \,d x \]
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 \[ \int \frac {a + b x}{\sqrt [4]{- x^{2} - 1} \left (x^{2} + 2\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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