Optimal. Leaf size=481 \[ -\frac {\text {Li}_2\left (-\frac {\sqrt {d} (-a-b x+1)}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {\sqrt {d} (-a-b x+1)}{\sqrt {d} (1-a)+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\text {Li}_2\left (-\frac {\sqrt {d} (a+b x+1)}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {\sqrt {d} (a+b x+1)}{\sqrt {d} (a+1)+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (-a-b x+1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (a+b x+1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{(a+1) \sqrt {d}+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (-a-b x+1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{(1-a) \sqrt {d}+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (a+b x+1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}} \]
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Rubi [A] time = 0.61, antiderivative size = 481, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6115, 2409, 2394, 2393, 2391} \[ -\frac {\text {PolyLog}\left (2,-\frac {\sqrt {d} (-a-b x+1)}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\text {PolyLog}\left (2,\frac {\sqrt {d} (-a-b x+1)}{(1-a) \sqrt {d}+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\text {PolyLog}\left (2,-\frac {\sqrt {d} (a+b x+1)}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\text {PolyLog}\left (2,\frac {\sqrt {d} (a+b x+1)}{(a+1) \sqrt {d}+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (-a-b x+1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (a+b x+1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{(a+1) \sqrt {d}+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (-a-b x+1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{(1-a) \sqrt {d}+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (a+b x+1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 6115
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a+b x)}{c+d x^2} \, dx &=-\left (\frac {1}{2} \int \frac {\log (1-a-b x)}{c+d x^2} \, dx\right )+\frac {1}{2} \int \frac {\log (1+a+b x)}{c+d x^2} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {\sqrt {-c} \log (1-a-b x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \log (1-a-b x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx\right )+\frac {1}{2} \int \left (\frac {\sqrt {-c} \log (1+a+b x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \log (1+a+b x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx\\ &=\frac {\int \frac {\log (1-a-b x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{4 \sqrt {-c}}+\frac {\int \frac {\log (1-a-b x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{4 \sqrt {-c}}-\frac {\int \frac {\log (1+a+b x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{4 \sqrt {-c}}-\frac {\int \frac {\log (1+a+b x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{4 \sqrt {-c}}\\ &=-\frac {\log (1-a-b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (1+a+b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(1+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (1-a-b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}+(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (1+a+b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(1+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {b \int \frac {\log \left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{-b \sqrt {-c}+(1-a) \sqrt {d}}\right )}{1-a-b x} \, dx}{4 \sqrt {-c} \sqrt {d}}-\frac {b \int \frac {\log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(1+a) \sqrt {d}}\right )}{1+a+b x} \, dx}{4 \sqrt {-c} \sqrt {d}}+\frac {b \int \frac {\log \left (-\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{-b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{1-a-b x} \, dx}{4 \sqrt {-c} \sqrt {d}}+\frac {b \int \frac {\log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(1+a) \sqrt {d}}\right )}{1+a+b x} \, dx}{4 \sqrt {-c} \sqrt {d}}\\ &=-\frac {\log (1-a-b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (1+a+b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(1+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (1-a-b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}+(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (1+a+b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(1+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {d} x}{-b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{x} \, dx,x,1-a-b x\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {d} x}{-b \sqrt {-c}+(1-a) \sqrt {d}}\right )}{x} \, dx,x,1-a-b x\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {d} x}{b \sqrt {-c}-(1+a) \sqrt {d}}\right )}{x} \, dx,x,1+a+b x\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {d} x}{b \sqrt {-c}+(1+a) \sqrt {d}}\right )}{x} \, dx,x,1+a+b x\right )}{4 \sqrt {-c} \sqrt {d}}\\ &=-\frac {\log (1-a-b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (1+a+b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(1+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (1-a-b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}+(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (1+a+b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(1+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\text {Li}_2\left (-\frac {\sqrt {d} (1-a-b x)}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {\sqrt {d} (1-a-b x)}{b \sqrt {-c}+(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\text {Li}_2\left (-\frac {\sqrt {d} (1+a+b x)}{b \sqrt {-c}-(1+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {\sqrt {d} (1+a+b x)}{b \sqrt {-c}+(1+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 365, normalized size = 0.76 \[ \frac {\text {Li}_2\left (-\frac {\sqrt {d} (a+b x-1)}{b \sqrt {-c}-(a-1) \sqrt {d}}\right )-\text {Li}_2\left (\frac {\sqrt {d} (a+b x-1)}{\sqrt {d} (a-1)+b \sqrt {-c}}\right )-\text {Li}_2\left (-\frac {\sqrt {d} (a+b x+1)}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )+\text {Li}_2\left (\frac {\sqrt {d} (a+b x+1)}{\sqrt {d} (a+1)+b \sqrt {-c}}\right )-\log (-a-b x+1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{(a-1) \sqrt {d}+b \sqrt {-c}}\right )+\log (a+b x+1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{(a+1) \sqrt {d}+b \sqrt {-c}}\right )+\log (-a-b x+1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(a-1) \sqrt {d}}\right )-\log (a+b x+1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (b x + a\right )}{d x^{2} + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 0.74, size = 1300, normalized size = 2.70 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.59, size = 589, normalized size = 1.22 \[ \frac {\arctan \left (\frac {d x}{\sqrt {c d}}\right ) \operatorname {artanh}\left (b x + a\right )}{\sqrt {c d}} + \frac {{\left (\arctan \left (\frac {{\left (b^{2} x + {\left (a + 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c + {\left (a^{2} + 2 \, a + 1\right )} d}, \frac {{\left (a + 1\right )} b d x + {\left (a^{2} + 2 \, a + 1\right )} d}{b^{2} c + {\left (a^{2} + 2 \, a + 1\right )} d}\right ) - \arctan \left (\frac {{\left (b^{2} x + {\left (a - 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c + {\left (a^{2} - 2 \, a + 1\right )} d}, \frac {{\left (a - 1\right )} b d x + {\left (a^{2} - 2 \, a + 1\right )} d}{b^{2} c + {\left (a^{2} - 2 \, a + 1\right )} d}\right )\right )} \log \left (d x^{2} + c\right ) - \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {b^{2} d x^{2} + 2 \, {\left (a + 1\right )} b d x + {\left (a^{2} + 2 \, a + 1\right )} d}{b^{2} c + {\left (a^{2} + 2 \, a + 1\right )} d}\right ) + \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {b^{2} d x^{2} + 2 \, {\left (a - 1\right )} b d x + {\left (a^{2} - 2 \, a + 1\right )} d}{b^{2} c + {\left (a^{2} - 2 \, a + 1\right )} d}\right ) - i \, {\rm Li}_2\left (\frac {{\left (a + 1\right )} b d x + b^{2} c - {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c + {\left (2 i \, a + 2 i\right )} b \sqrt {c} \sqrt {d} - {\left (a^{2} + 2 \, a + 1\right )} d}\right ) + i \, {\rm Li}_2\left (\frac {{\left (a + 1\right )} b d x + b^{2} c + {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c - {\left (2 i \, a + 2 i\right )} b \sqrt {c} \sqrt {d} - {\left (a^{2} + 2 \, a + 1\right )} d}\right ) + i \, {\rm Li}_2\left (\frac {{\left (a - 1\right )} b d x + b^{2} c - {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c + {\left (2 i \, a - 2 i\right )} b \sqrt {c} \sqrt {d} - {\left (a^{2} - 2 \, a + 1\right )} d}\right ) - i \, {\rm Li}_2\left (\frac {{\left (a - 1\right )} b d x + b^{2} c + {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} c - {\left (2 i \, a - 2 i\right )} b \sqrt {c} \sqrt {d} - {\left (a^{2} - 2 \, a + 1\right )} d}\right )}{4 \, \sqrt {c 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 \[ \int \frac {\mathrm {atanh}\left (a+b\,x\right )}{d\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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