Optimal. Leaf size=80 \[ \frac {a \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {a \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}-b \tan ^{-1}\left (\sqrt [4]{x^2-1}\right )+b \tanh ^{-1}\left (\sqrt [4]{x^2-1}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 80, 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}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx &=a \int \frac {1}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx+b \int \frac {x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, 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}{(2-x) \sqrt [4]{-1+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 = 157, normalized size = 1.96 \[ \frac {x \left (b x \sqrt [4]{1-x^2} \left (x^2-2\right ) F_1\left (1;\frac {1}{4},1;2;x^2,\frac {x^2}{2}\right )-\frac {24 a F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};x^2,\frac {x^2}{2}\right )}{x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};x^2,\frac {x^2}{2}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};x^2,\frac {x^2}{2}\right )\right )+6 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};x^2,\frac {x^2}{2}\right )}\right )}{4 \left (x^2-2\right ) \sqrt [4]{x^2-1}} \]
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 \[ \int -\frac {b x + a}{{\left (x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {b x +a}{\left (-x^{2}+2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}}\, dx \]
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 \[ -\int \frac {b x + a}{{\left (x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 2\right )}}\,{d x} \]
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 \[ - \int \frac {a}{x^{2} \sqrt [4]{x^{2} - 1} - 2 \sqrt [4]{x^{2} - 1}}\, dx - \int \frac {b x}{x^{2} \sqrt [4]{x^{2} - 1} - 2 \sqrt [4]{x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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