Optimal. Leaf size=100 \[ -\frac {\sqrt {d+e x^2} (a+b \text {ArcTan}(c x))}{d x}-\frac {b c \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {b \sqrt {c^2 d-e} \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{d} \]
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Rubi [A]
time = 0.14, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {270, 5096, 457,
85, 65, 214} \begin {gather*} -\frac {\sqrt {d+e x^2} (a+b \text {ArcTan}(c x))}{d x}+\frac {b \sqrt {c^2 d-e} \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{d}-\frac {b c \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 85
Rule 214
Rule 270
Rule 457
Rule 5096
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx &=-\frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{d x}-(b c) \int \frac {\sqrt {d+e x^2}}{x \left (-d-c^2 d x^2\right )} \, dx\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{d x}-\frac {1}{2} (b c) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x \left (-d-c^2 d x\right )} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{d x}+\frac {1}{2} (b c) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )+\frac {1}{2} \left (b c \left (c^2 d-e\right )\right ) \text {Subst}\left (\int \frac {1}{\left (-d-c^2 d x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{d x}+\frac {(b c) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{e}+\frac {\left (b c \left (c^2 d-e\right )\right ) \text {Subst}\left (\int \frac {1}{-d+\frac {c^2 d^2}{e}-\frac {c^2 d x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{e}\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{d x}-\frac {b c \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {b \sqrt {c^2 d-e} \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.30, size = 247, normalized size = 2.47 \begin {gather*} \frac {-2 a \sqrt {d+e x^2}-2 b \sqrt {d+e x^2} \text {ArcTan}(c x)+2 b c \sqrt {d} x \log (x)-2 b c \sqrt {d} x \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )+b \sqrt {c^2 d-e} x \log \left (-\frac {4 c d \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{3/2} (i+c x)}\right )+b \sqrt {c^2 d-e} x \log \left (-\frac {4 c d \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{3/2} (-i+c x)}\right )}{2 d x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {a +b \arctan \left (c x \right )}{x^{2} \sqrt {e \,x^{2}+d}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.75, size = 688, normalized size = 6.88 \begin {gather*} \left [\frac {2 \, b c \sqrt {d} x \log \left (-\frac {x^{2} e - 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) + \sqrt {c^{2} d - e} b x \log \left (\frac {8 \, c^{4} d^{2} + 4 \, {\left (2 \, c^{3} d + {\left (c^{3} x^{2} - c\right )} e\right )} \sqrt {c^{2} d - e} \sqrt {x^{2} e + d} + {\left (c^{4} x^{4} - 6 \, c^{2} x^{2} + 1\right )} e^{2} + 8 \, {\left (c^{4} d x^{2} - c^{2} d\right )} e}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) - 4 \, \sqrt {x^{2} e + d} {\left (b \arctan \left (c x\right ) + a\right )}}{4 \, d x}, \frac {b c \sqrt {d} x \log \left (-\frac {x^{2} e - 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) + \sqrt {-c^{2} d + e} b x \arctan \left (-\frac {{\left (2 \, c^{2} d + {\left (c^{2} x^{2} - 1\right )} e\right )} \sqrt {-c^{2} d + e} \sqrt {x^{2} e + d}}{2 \, {\left (c^{3} d^{2} - c x^{2} e^{2} + {\left (c^{3} d x^{2} - c d\right )} e\right )}}\right ) - 2 \, \sqrt {x^{2} e + d} {\left (b \arctan \left (c x\right ) + a\right )}}{2 \, d x}, \frac {4 \, b c \sqrt {-d} x \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) + \sqrt {c^{2} d - e} b x \log \left (\frac {8 \, c^{4} d^{2} + 4 \, {\left (2 \, c^{3} d + {\left (c^{3} x^{2} - c\right )} e\right )} \sqrt {c^{2} d - e} \sqrt {x^{2} e + d} + {\left (c^{4} x^{4} - 6 \, c^{2} x^{2} + 1\right )} e^{2} + 8 \, {\left (c^{4} d x^{2} - c^{2} d\right )} e}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) - 4 \, \sqrt {x^{2} e + d} {\left (b \arctan \left (c x\right ) + a\right )}}{4 \, d x}, \frac {2 \, b c \sqrt {-d} x \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) + \sqrt {-c^{2} d + e} b x \arctan \left (-\frac {{\left (2 \, c^{2} d + {\left (c^{2} x^{2} - 1\right )} e\right )} \sqrt {-c^{2} d + e} \sqrt {x^{2} e + d}}{2 \, {\left (c^{3} d^{2} - c x^{2} e^{2} + {\left (c^{3} d x^{2} - c d\right )} e\right )}}\right ) - 2 \, \sqrt {x^{2} e + d} {\left (b \arctan \left (c x\right ) + a\right )}}{2 \, d x}\right ] \end {gather*}
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 \begin {gather*} \int \frac {a + b \operatorname {atan}{\left (c x \right )}}{x^{2} \sqrt {d + e x^{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^2\,\sqrt {e\,x^2+d}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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