Optimal. Leaf size=545 \[ \frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (-a-b x+1)}{-\sqrt {-c} a+\sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (-a-b x+1)}{\sqrt {-c} (1-a)+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x+1)}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x+1)}{\sqrt {-c} (a+1)+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (-a-b x+1) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (-a-b x+1) \log \left (\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x+1) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(a+1) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x+1) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c} \]
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Rubi [A] time = 0.91, antiderivative size = 545, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6115, 2409, 2389, 2295, 2394, 2393, 2391} \[ \frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (-a-b x+1)}{a \left (-\sqrt {-c}\right )-b \sqrt {d}+\sqrt {-c}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (-a-b x+1)}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x+1)}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x+1)}{(a+1) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (-a-b x+1) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (-a-b x+1) \log \left (\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x+1) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(a+1) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x+1) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 6115
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx &=-\left (\frac {1}{2} \int \frac {\log (1-a-b x)}{c+\frac {d}{x^2}} \, dx\right )+\frac {1}{2} \int \frac {\log (1+a+b x)}{c+\frac {d}{x^2}} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {\log (1-a-b x)}{c}-\frac {d \log (1-a-b x)}{c \left (d+c x^2\right )}\right ) \, dx\right )+\frac {1}{2} \int \left (\frac {\log (1+a+b x)}{c}-\frac {d \log (1+a+b x)}{c \left (d+c x^2\right )}\right ) \, dx\\ &=-\frac {\int \log (1-a-b x) \, dx}{2 c}+\frac {\int \log (1+a+b x) \, dx}{2 c}+\frac {d \int \frac {\log (1-a-b x)}{d+c x^2} \, dx}{2 c}-\frac {d \int \frac {\log (1+a+b x)}{d+c x^2} \, dx}{2 c}\\ &=\frac {\operatorname {Subst}(\int \log (x) \, dx,x,1-a-b x)}{2 b c}+\frac {\operatorname {Subst}(\int \log (x) \, dx,x,1+a+b x)}{2 b c}+\frac {d \int \left (\frac {\log (1-a-b x)}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\log (1-a-b x)}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{2 c}-\frac {d \int \left (\frac {\log (1+a+b x)}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\log (1+a+b x)}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{2 c}\\ &=\frac {(1-a-b x) \log (1-a-b x)}{2 b c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {\sqrt {d} \int \frac {\log (1-a-b x)}{\sqrt {d}-\sqrt {-c} x} \, dx}{4 c}+\frac {\sqrt {d} \int \frac {\log (1-a-b x)}{\sqrt {d}+\sqrt {-c} x} \, dx}{4 c}-\frac {\sqrt {d} \int \frac {\log (1+a+b x)}{\sqrt {d}-\sqrt {-c} x} \, dx}{4 c}-\frac {\sqrt {d} \int \frac {\log (1+a+b x)}{\sqrt {d}+\sqrt {-c} x} \, dx}{4 c}\\ &=\frac {(1-a-b x) \log (1-a-b x)}{2 b c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {\sqrt {d} \log (1-a-b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{1-a-b x} \, dx}{4 (-c)^{3/2}}+\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{1+a+b x} \, dx}{4 (-c)^{3/2}}-\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{-(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{1-a-b x} \, dx}{4 (-c)^{3/2}}-\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{-(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{1+a+b x} \, dx}{4 (-c)^{3/2}}\\ &=\frac {(1-a-b x) \log (1-a-b x)}{2 b c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {\sqrt {d} \log (1-a-b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c} x}{-(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{x} \, dx,x,1-a-b x\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c} x}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{x} \, dx,x,1-a-b x\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c} x}{-(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,1+a+b x\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c} x}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,1+a+b x\right )}{4 (-c)^{3/2}}\\ &=\frac {(1-a-b x) \log (1-a-b x)}{2 b c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {\sqrt {d} \log (1-a-b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (1-a-b x)}{\sqrt {-c}-a \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (1-a-b x)}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 25.32, size = 1456, normalized size = 2.67 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \operatorname {artanh}\left (b x + a\right )}{c x^{2} + d}, x\right ) \]
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 [C] time = 1.81, size = 20505, normalized size = 37.62 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.55, size = 647, normalized size = 1.19 \[ -{\left (\frac {d \arctan \left (\frac {c x}{\sqrt {c d}}\right )}{\sqrt {c d} c} - \frac {x}{c}\right )} \operatorname {artanh}\left (b x + a\right ) + \frac {2 \, {\left (a + 1\right )} c \log \left (b x + a + 1\right ) - 2 \, {\left (a - 1\right )} c \log \left (b x + a - 1\right ) + {\left (b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, {\left (a + 1\right )} b c x + {\left (a^{2} + 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, {\left (a - 1\right )} b c x + {\left (a^{2} - 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right ) + i \, b {\rm Li}_2\left (\frac {{\left (a + 1\right )} b c x + b^{2} d - {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{{\left (2 i \, a + 2 i\right )} b \sqrt {c} \sqrt {d} + b^{2} d - {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - i \, b {\rm Li}_2\left (-\frac {{\left (a + 1\right )} b c x + b^{2} d + {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{{\left (2 i \, a + 2 i\right )} b \sqrt {c} \sqrt {d} - b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - i \, b {\rm Li}_2\left (\frac {{\left (a - 1\right )} b c x + b^{2} d - {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{{\left (2 i \, a - 2 i\right )} b \sqrt {c} \sqrt {d} + b^{2} d - {\left (a^{2} - 2 \, a + 1\right )} c}\right ) + i \, b {\rm Li}_2\left (-\frac {{\left (a - 1\right )} b c x + b^{2} d + {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{{\left (2 i \, a - 2 i\right )} b \sqrt {c} \sqrt {d} - b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right ) - {\left (b \arctan \left (\frac {{\left (b^{2} x + {\left (a + 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}, \frac {{\left (a + 1\right )} b c x + {\left (a^{2} + 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - b \arctan \left (\frac {{\left (b^{2} x + {\left (a - 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}, \frac {{\left (a - 1\right )} b c x + {\left (a^{2} - 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right )\right )} \log \left (c x^{2} + d\right )\right )} \sqrt {c} \sqrt {d}}{4 \, b c^{2}} \]
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 \[ \int \frac {\mathrm {atanh}\left (a+b\,x\right )}{c+\frac {d}{x^2}} \,d x \]
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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