Optimal. Leaf size=590 \[ \frac {a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac {i \text {Li}_2\left (\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \text {Li}_2\left (\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (1-a x)}{\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (a x+1)}{-\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (1-a x)}{-\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (a x+1)}{\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {\tanh ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {x \tanh ^{-1}(a x)}{2 c \left (c+d x^2\right )} \]
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Rubi [A] time = 0.91, antiderivative size = 590, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 13, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {199, 205, 5976, 6725, 517, 444, 36, 31, 4908, 2409, 2394, 2393, 2391} \[ \frac {i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \text {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac {i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (1-a x)}{\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (a x+1)}{-\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (1-a x)}{-\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (a x+1)}{\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {\tanh ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {x \tanh ^{-1}(a x)}{2 c \left (c+d x^2\right )} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 199
Rule 205
Rule 444
Rule 517
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 4908
Rule 5976
Rule 6725
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx &=\frac {x \tanh ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}-a \int \frac {\frac {x}{2 c \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}}{1-a^2 x^2} \, dx\\ &=\frac {x \tanh ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}-a \int \left (-\frac {x}{2 c (-1+a x) (1+a x) \left (c+d x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d} \left (-1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {x \tanh ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}+\frac {a \int \frac {x}{(-1+a x) (1+a x) \left (c+d x^2\right )} \, dx}{2 c}+\frac {a \int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{2 c^{3/2} \sqrt {d}}\\ &=\frac {x \tanh ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}+\frac {a \int \frac {x}{\left (-1+a^2 x^2\right ) \left (c+d x^2\right )} \, dx}{2 c}+\frac {(i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{4 c^{3/2} \sqrt {d}}-\frac {(i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{4 c^{3/2} \sqrt {d}}\\ &=\frac {x \tanh ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\left (-1+a^2 x\right ) (c+d x)} \, dx,x,x^2\right )}{4 c}+\frac {(i a) \int \left (-\frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1-a x)}-\frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1+a x)}\right ) \, dx}{4 c^{3/2} \sqrt {d}}-\frac {(i a) \int \left (-\frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1-a x)}-\frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1+a x)}\right ) \, dx}{4 c^{3/2} \sqrt {d}}\\ &=\frac {x \tanh ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}-\frac {(i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1-a x} \, dx}{8 c^{3/2} \sqrt {d}}-\frac {(i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+a x} \, dx}{8 c^{3/2} \sqrt {d}}+\frac {(i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1-a x} \, dx}{8 c^{3/2} \sqrt {d}}+\frac {(i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+a x} \, dx}{8 c^{3/2} \sqrt {d}}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{-1+a^2 x} \, dx,x,x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {(a d) \operatorname {Subst}\left (\int \frac {1}{c+d x} \, dx,x,x^2\right )}{4 c \left (a^2 c+d\right )}\\ &=\frac {x \tanh ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {\int \frac {\log \left (-\frac {i \sqrt {d} (1-a x)}{\sqrt {c} \left (a-\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1-\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 c^2}-\frac {\int \frac {\log \left (\frac {i \sqrt {d} (1-a x)}{\sqrt {c} \left (a+\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1+\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 c^2}+\frac {\int \frac {\log \left (-\frac {i \sqrt {d} (1+a x)}{\sqrt {c} \left (-a-\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1-\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 c^2}+\frac {\int \frac {\log \left (\frac {i \sqrt {d} (1+a x)}{\sqrt {c} \left (-a+\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1+\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 c^2}\\ &=\frac {x \tanh ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a-\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {a x}{a-\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a+\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {a x}{a+\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}\\ &=\frac {x \tanh ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \tanh ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac {i \text {Li}_2\left (\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \text {Li}_2\left (\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 7.79, size = 746, normalized size = 1.26 \[ \frac {a \left (\frac {i \left (\text {Li}_2\left (\frac {\left (-c a^2+d-2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (c a^2+d\right ) \left (\sqrt {a^2 c d} x-i a c\right )}\right )-\text {Li}_2\left (\frac {\left (-c a^2+d+2 i \sqrt {a^2 c d}\right ) \left (i a c+\sqrt {a^2 c d} x\right )}{\left (c a^2+d\right ) \left (\sqrt {a^2 c d} x-i a c\right )}\right )\right )+2 i \cos ^{-1}\left (\frac {d-a^2 c}{a^2 c+d}\right ) \tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )-4 \tanh ^{-1}(a x) \tan ^{-1}\left (\frac {a c}{x \sqrt {a^2 c d}}\right )+\log \left (\frac {2 i a c (a x-1) \left (\sqrt {a^2 c d}+i d\right )}{\left (a^2 c+d\right ) \left (a c+i x \sqrt {a^2 c d}\right )}\right ) \left (2 \tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )+\cos ^{-1}\left (\frac {d-a^2 c}{a^2 c+d}\right )\right )+\log \left (\frac {2 a c (a x+1) \left (d+i \sqrt {a^2 c d}\right )}{\left (a^2 c+d\right ) \left (a c+i x \sqrt {a^2 c d}\right )}\right ) \left (\cos ^{-1}\left (\frac {d-a^2 c}{a^2 c+d}\right )-2 \tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )-\left (2 \left (\tan ^{-1}\left (\frac {a c}{x \sqrt {a^2 c d}}\right )+\tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )+\cos ^{-1}\left (\frac {d-a^2 c}{a^2 c+d}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{-\tanh ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )+a^2 c-d}}\right )-\left (\cos ^{-1}\left (\frac {d-a^2 c}{a^2 c+d}\right )-2 \left (\tan ^{-1}\left (\frac {a c}{x \sqrt {a^2 c d}}\right )+\tan ^{-1}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{\tanh ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )+a^2 c-d}}\right )}{\sqrt {a^2 c d}}-\frac {2 \log \left (\frac {\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )}{a^2 c-d}+1\right )}{a^2 c+d}+\frac {4 \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )}{\left (a^2 c+d\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )+a^2 c-d}\right )}{8 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (a x\right )}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.35, size = 2346, normalized size = 3.98 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 550, normalized size = 0.93 \[ \frac {1}{2} \, {\left (\frac {x}{c d x^{2} + c^{2}} + \frac {\arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c}\right )} \operatorname {artanh}\left (a x\right ) - \frac {{\left (2 \, a c d \log \left (d x^{2} + c\right ) - 2 \, a c d \log \left (a x + 1\right ) - 2 \, a c d \log \left (a x - 1\right ) + {\left ({\left (a^{2} c + d\right )} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} + 2 \, a d x + d}{a^{2} c + d}\right ) - {\left (a^{2} c + d\right )} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} - 2 \, a d x + d}{a^{2} c + d}\right ) + {\left (i \, a^{2} c + i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c + a d x - {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + {\left (i \, a^{2} c + i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c - a d x + {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + {\left (-i \, a^{2} c - i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c + a d x + {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + {\left (-i \, a^{2} c - i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c - a d x - {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - {\left ({\left (a^{2} c + d\right )} \arctan \left (\frac {{\left (a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, \frac {a d x + d}{a^{2} c + d}\right ) - {\left (a^{2} c + d\right )} \arctan \left (\frac {{\left (a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, -\frac {a d x - d}{a^{2} c + d}\right )\right )} \log \left (d x^{2} + c\right )\right )} \sqrt {c} \sqrt {d}\right )} a}{8 \, {\left (a^{3} c^{3} d + a c^{2} d^{2}\right )}} \]
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 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^2} \,d x \]
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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