Optimal. Leaf size=43 \[ \text {Int}\left (\frac {1}{\left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2},x\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx &=\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.84, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{a^{2} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname {artanh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{2} - a^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \operatorname {artanh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {artanh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.92, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-c^{2} x^{2}+1\right ) \left (a +b \arctanh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {4 \, c x}{\sqrt {c x + 1} \sqrt {-c x + 1} b^{2} c \log \left (\sqrt {c x + 1} + \sqrt {-c x + 1}\right ) - \sqrt {c x + 1} \sqrt {-c x + 1} b^{2} c \log \left (\sqrt {c x + 1} - \sqrt {-c x + 1}\right ) + 2 \, \sqrt {c x + 1} \sqrt {-c x + 1} a b c} - \int -\frac {4}{{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \log \left (\sqrt {c x + 1} + \sqrt {-c x + 1}\right ) - {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \log \left (\sqrt {c x + 1} - \sqrt {-c x + 1}\right ) + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {1}{{\left (a+b\,\mathrm {atanh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2\,\left (c^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a^{2} c^{2} x^{2} - a^{2} + 2 a b c^{2} x^{2} \operatorname {atanh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} - 2 a b \operatorname {atanh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} + b^{2} c^{2} x^{2} \operatorname {atanh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} - b^{2} \operatorname {atanh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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