Optimal. Leaf size=285 \[ \frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-\frac {b \log \left (\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {b \log \left (\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {2 b \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}-\frac {\sqrt {2} b \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}+\frac {\sqrt {2} b \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt [4]{c} \sqrt {d}}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}} \]
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Rubi [A] time = 0.24, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {6097, 16, 329, 301, 211, 1165, 628, 1162, 617, 204, 212, 208, 205} \[ \frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-\frac {b \log \left (\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {b \log \left (\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {2 b \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}-\frac {\sqrt {2} b \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}+\frac {\sqrt {2} b \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt [4]{c} \sqrt {d}}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 204
Rule 205
Rule 208
Rule 211
Rule 212
Rule 301
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6097
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^2\right )}{\sqrt {d x}} \, dx &=\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-\frac {(4 b c) \int \frac {x \sqrt {d x}}{1-c^2 x^4} \, dx}{d}\\ &=\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-\frac {(4 b c) \int \frac {(d x)^{3/2}}{1-c^2 x^4} \, dx}{d^2}\\ &=\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-\frac {(8 b c) \operatorname {Subst}\left (\int \frac {x^4}{1-\frac {c^2 x^8}{d^4}} \, dx,x,\sqrt {d x}\right )}{d^3}\\ &=\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-(4 b d) \operatorname {Subst}\left (\int \frac {1}{d^2-c x^4} \, dx,x,\sqrt {d x}\right )+(4 b d) \operatorname {Subst}\left (\int \frac {1}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )\\ &=\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-(2 b) \operatorname {Subst}\left (\int \frac {1}{d-\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )-(2 b) \operatorname {Subst}\left (\int \frac {1}{d+\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )+(2 b) \operatorname {Subst}\left (\int \frac {d-\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )+(2 b) \operatorname {Subst}\left (\int \frac {d+\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )\\ &=-\frac {2 b \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}+\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d x}\right )}{\sqrt {c}}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d x}\right )}{\sqrt {c}}-\frac {b \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt [4]{c}}+2 x}{-\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {d x}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {b \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt [4]{c}}-2 x}{-\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {d x}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}\\ &=-\frac {2 b \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}+\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}-\frac {b \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {b \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\left (\sqrt {2} b\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}-\frac {\left (\sqrt {2} b\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}\\ &=-\frac {2 b \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}-\frac {\sqrt {2} b \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}+\frac {\sqrt {2} b \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}+\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}-\frac {b \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {b \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 227, normalized size = 0.80 \[ \frac {\sqrt {x} \left (4 a \sqrt [4]{c} \sqrt {x}+4 b \sqrt [4]{c} \sqrt {x} \tanh ^{-1}\left (c x^2\right )+2 b \log \left (1-\sqrt [4]{c} \sqrt {x}\right )-2 b \log \left (\sqrt [4]{c} \sqrt {x}+1\right )-\sqrt {2} b \log \left (\sqrt {c} x-\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right )+\sqrt {2} b \log \left (\sqrt {c} x+\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right )-2 \sqrt {2} b \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )+2 \sqrt {2} b \tan ^{-1}\left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right )-4 b \tan ^{-1}\left (\sqrt [4]{c} \sqrt {x}\right )\right )}{2 \sqrt [4]{c} \sqrt {d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.96, size = 34, normalized size = 0.12 \[ \frac {\sqrt {d x} {\left (b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 493, normalized size = 1.73 \[ \frac {{\left (c d^{2} {\left (\frac {2 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} + \frac {2 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} - \frac {2 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} - \frac {2 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} + \frac {\sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{c^{2} d^{2}} - \frac {\sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{c^{2} d^{2}} - \frac {\sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{c^{2} d^{2}} + \frac {\sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{c^{2} d^{2}}\right )} + 2 \, \sqrt {d x} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )\right )} b + 4 \, \sqrt {d x} a}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 273, normalized size = 0.96 \[ \frac {2 a \sqrt {d x}}{d}+\frac {2 b \sqrt {d x}\, \arctanh \left (c \,x^{2}\right )}{d}-\frac {b \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{d}-\frac {2 b \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{d}+\frac {b \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )}{2 d}+\frac {b \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )}{d}+\frac {b \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 296, normalized size = 1.04 \[ \frac {{\left (4 \, \sqrt {d x} \operatorname {artanh}\left (c x^{2}\right ) + \frac {c {\left (\frac {\frac {2 \, \sqrt {2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} + 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} - 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} + \frac {\sqrt {2} d^{\frac {5}{2}} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {1}{4}}} - \frac {\sqrt {2} d^{\frac {5}{2}} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {1}{4}}}}{c} - \frac {2 \, {\left (\frac {2 \, d^{3} \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} - \frac {d^{3} \log \left (\frac {\sqrt {d x} \sqrt {c} - \sqrt {\sqrt {c} d}}{\sqrt {d x} \sqrt {c} + \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}}\right )}}{c}\right )}}{d^{2}}\right )} b + 4 \, \sqrt {d x} a}{2 \, d} \]
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 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x^2\right )}{\sqrt {d\,x}} \,d x \]
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 \[ \int \frac {a + b \operatorname {atanh}{\left (c x^{2} \right )}}{\sqrt {d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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