Optimal. Leaf size=673 \[ \frac {\text {Li}_2\left (-\frac {\left (d a^2+b^2 c\right ) (-a-b x+1)}{\left (c b^2-\sqrt {-c} \sqrt {d} b-(1-a) a d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\text {Li}_2\left (-\frac {\left (d a^2+b^2 c\right ) (-a-b x+1)}{\left (c b^2+\sqrt {-c} \sqrt {d} b-(1-a) a d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {\left (d a^2+b^2 c\right ) (a+b x+1)}{\left (c b^2-\sqrt {-c} \sqrt {d} b+a (a+1) d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\text {Li}_2\left (\frac {\left (d a^2+b^2 c\right ) (a+b x+1)}{\left (c b^2+\sqrt {-c} \sqrt {d} b+a (a+1) d\right ) (a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log \left (-\frac {-a-b x+1}{a+b x}\right ) \log \left (\frac {(-a-b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (-(1-a) a d+b^2 c-b \sqrt {-c} \sqrt {d}\right )}+1\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log \left (-\frac {-a-b x+1}{a+b x}\right ) \log \left (\frac {(-a-b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (-(1-a) a d+b^2 c+b \sqrt {-c} \sqrt {d}\right )}+1\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log \left (\frac {a+b x+1}{a+b x}\right ) \log \left (1-\frac {(a+b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (a (a+1) d+b^2 c-b \sqrt {-c} \sqrt {d}\right )}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log \left (\frac {a+b x+1}{a+b x}\right ) \log \left (1-\frac {(a+b x+1) \left (a^2 d+b^2 c\right )}{(a+b x) \left (a (a+1) d+b^2 c+b \sqrt {-c} \sqrt {d}\right )}\right )}{4 \sqrt {-c} \sqrt {d}} \]
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Rubi [A] time = 1.10, antiderivative size = 597, normalized size of antiderivative = 0.89, number of steps used = 37, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6116, 2513, 2409, 2394, 2393, 2391, 205} \[ -\frac {\text {PolyLog}\left (2,-\frac {\sqrt {d} (-a-b x+1)}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\text {PolyLog}\left (2,\frac {\sqrt {d} (-a-b x+1)}{(1-a) \sqrt {d}+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\text {PolyLog}\left (2,-\frac {\sqrt {d} (a+b x+1)}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\text {PolyLog}\left (2,\frac {\sqrt {d} (a+b x+1)}{(a+1) \sqrt {d}+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (a+b x-1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (a+b x+1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{(a+1) \sqrt {d}+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\log (a+b x-1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{(1-a) \sqrt {d}+b \sqrt {-c}}\right )}{4 \sqrt {-c} \sqrt {d}}-\frac {\log (a+b x+1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\left (\log (a+b x-1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}} \]
Warning: Unable to verify antiderivative.
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Rule 205
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 2513
Rule 6116
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a+b x)}{c+d x^2} \, dx &=-\left (\frac {1}{2} \int \frac {\log \left (\frac {-1+a+b x}{a+b x}\right )}{c+d x^2} \, dx\right )+\frac {1}{2} \int \frac {\log \left (\frac {1+a+b x}{a+b x}\right )}{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-\frac {1}{2} \left (-\log (-1+a+b x)+\log \left (\frac {-1+a+b x}{a+b x}\right )+\log (a+b x)\right ) \int \frac {1}{c+d x^2} \, dx+\frac {1}{2} \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right ) \int \frac {1}{c+d x^2} \, dx\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 \sqrt {c} \sqrt {d}}-\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 {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 {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 \sqrt {c} \sqrt {d}}+\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 {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 \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 {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 {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 \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 {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 \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 = 0.62, size = 529, normalized size = 0.79 \[ \frac {\text {Li}_2\left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d} (a-1)+b \sqrt {-c}}\right )-\text {Li}_2\left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d} (a+1)+b \sqrt {-c}}\right )-\text {Li}_2\left (\frac {b \left (\sqrt {d} x+\sqrt {-c}\right )}{b \sqrt {-c}-(a-1) \sqrt {d}}\right )+\text {Li}_2\left (\frac {b \left (\sqrt {d} x+\sqrt {-c}\right )}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )+\log \left (\sqrt {-c}-\sqrt {d} x\right ) \log \left (\frac {\sqrt {d} (a+b x-1)}{(a-1) \sqrt {d}+b \sqrt {-c}}\right )-\log \left (\frac {a+b x-1}{a+b x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (\sqrt {-c}-\sqrt {d} x\right ) \log \left (\frac {\sqrt {d} (a+b x+1)}{(a+1) \sqrt {d}+b \sqrt {-c}}\right )+\log \left (\frac {a+b x+1}{a+b x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (\sqrt {-c}+\sqrt {d} x\right ) \log \left (-\frac {\sqrt {d} (a+b x-1)}{b \sqrt {-c}-(a-1) \sqrt {d}}\right )+\log \left (\frac {a+b x-1}{a+b x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\log \left (\sqrt {-c}+\sqrt {d} x\right ) \log \left (-\frac {\sqrt {d} (a+b x+1)}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )-\log \left (\frac {a+b x+1}{a+b x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )}{4 \sqrt {-c} \sqrt {d}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (b x + a\right )}{d x^{2} + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (b x + a\right )}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.75, size = 1230, normalized size = 1.83 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.59, size = 589, normalized size = 0.88 \[ \frac {\operatorname {arcoth}\left (b x + a\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\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 \[ \int \frac {\mathrm {acoth}\left (a+b\,x\right )}{d\,x^2+c} \,d x \]
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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