Optimal. Leaf size=735 \[ -\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{b \left (\sqrt{d}-i \sqrt{c} x\right )}{b \sqrt{d}+(1+i a) \sqrt{c}}\right )}{4 c^{3/2}}+\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{b \left (\sqrt{d}-i \sqrt{c} x\right )}{b \sqrt{d}+i (a+i) \sqrt{c}}\right )}{4 c^{3/2}}+\frac{\sqrt{d} \text{PolyLog}\left (2,-\frac{b \left (\sqrt{d}+i \sqrt{c} x\right )}{-b \sqrt{d}+(1+i a) \sqrt{c}}\right )}{4 c^{3/2}}-\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{b \left (\sqrt{d}+i \sqrt{c} x\right )}{b \sqrt{d}+(1-i a) \sqrt{c}}\right )}{4 c^{3/2}}-\frac{\sqrt{d} \log \left (1-\frac{i \sqrt{c} x}{\sqrt{d}}\right ) \log \left (\frac{\sqrt{c} (-a-b x+i)}{(-a+i) \sqrt{c}+i b \sqrt{d}}\right )}{4 c^{3/2}}+\frac{\sqrt{d} \log \left (1-\frac{i \sqrt{c} x}{\sqrt{d}}\right ) \log \left (\frac{\sqrt{c} (a+b x+i)}{(a+i) \sqrt{c}-i b \sqrt{d}}\right )}{4 c^{3/2}}+\frac{\sqrt{d} \log \left (1+\frac{i \sqrt{c} x}{\sqrt{d}}\right ) \log \left (\frac{\sqrt{c} (-a-b x+i)}{(-a+i) \sqrt{c}-i b \sqrt{d}}\right )}{4 c^{3/2}}-\frac{\sqrt{d} \log \left (1+\frac{i \sqrt{c} x}{\sqrt{d}}\right ) \log \left (\frac{\sqrt{c} (a+b x+i)}{(a+i) \sqrt{c}+i b \sqrt{d}}\right )}{4 c^{3/2}}-\frac{i \sqrt{d} \log \left (-\frac{-a-b x+i}{a+b x}\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{2 c^{3/2}}+\frac{i \sqrt{d} \log \left (\frac{a+b x+i}{a+b x}\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{2 c^{3/2}}+\frac{\log (-a-b x+i)}{2 b c}+\frac{i (a+b x) \log \left (-\frac{-a-b x+i}{a+b x}\right )}{2 b c}+\frac{\log (a+b x+i)}{2 b c}-\frac{i (a+b x) \log \left (\frac{a+b x+i}{a+b x}\right )}{2 b c} \]
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Rubi [A] time = 1.51786, antiderivative size = 818, normalized size of antiderivative = 1.11, number of steps used = 57, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688, Rules used = {5052, 2513, 2409, 2389, 2295, 2394, 2393, 2391, 193, 321, 205} \[ \frac{i x \left (\log \left (-\frac{-a-b x+i}{a+b x}\right )+\log (a+b x)-\log (a+b x-i)\right )}{2 c}-\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log \left (-\frac{-a-b x+i}{a+b x}\right )+\log (a+b x)-\log (a+b x-i)\right )}{2 c^{3/2}}-\frac{i (-a-b x+i) \log (a+b x-i)}{2 b c}-\frac{i (a+b x+i) \log (a+b x+i)}{2 b c}-\frac{i x \left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac{a+b x+i}{a+b x}\right )\right )}{2 c}+\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac{a+b x+i}{a+b x}\right )\right )}{2 c^{3/2}}-\frac{i \sqrt{d} \log (a+b x-i) \log \left (-\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(i-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (a+b x+i) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{\sqrt{-c} (a+i)+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \log (a+b x+i) \log \left (-\frac{b \left (\sqrt{-c} x+\sqrt{d}\right )}{(a+i) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (a+b x-i) \log \left (\frac{b \left (\sqrt{-c} x+\sqrt{d}\right )}{\sqrt{-c} (i-a)+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (-a-b x+i)}{(i-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (-a-b x+i)}{\sqrt{-c} (i-a)+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x+i)}{(a+i) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x+i)}{\sqrt{-c} (a+i)+b \sqrt{d}}\right )}{4 (-c)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Rule 5052
Rule 2513
Rule 2409
Rule 2389
Rule 2295
Rule 2394
Rule 2393
Rule 2391
Rule 193
Rule 321
Rule 205
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(a+b x)}{c+\frac{d}{x^2}} \, dx &=\frac{1}{2} i \int \frac{\log \left (\frac{-i+a+b x}{a+b x}\right )}{c+\frac{d}{x^2}} \, dx-\frac{1}{2} i \int \frac{\log \left (\frac{i+a+b x}{a+b x}\right )}{c+\frac{d}{x^2}} \, dx\\ &=\frac{1}{2} i \int \frac{\log (-i+a+b x)}{c+\frac{d}{x^2}} \, dx-\frac{1}{2} i \int \frac{\log (i+a+b x)}{c+\frac{d}{x^2}} \, dx-\frac{1}{2} \left (i \left (-\log (a+b x)+\log (-i+a+b x)-\log \left (\frac{-i+a+b x}{a+b x}\right )\right )\right ) \int \frac{1}{c+\frac{d}{x^2}} \, dx+\frac{1}{2} \left (i \left (-\log (a+b x)+\log (i+a+b x)-\log \left (\frac{i+a+b x}{a+b x}\right )\right )\right ) \int \frac{1}{c+\frac{d}{x^2}} \, dx\\ &=\frac{1}{2} i \int \left (\frac{\log (-i+a+b x)}{c}-\frac{d \log (-i+a+b x)}{c \left (d+c x^2\right )}\right ) \, dx-\frac{1}{2} i \int \left (\frac{\log (i+a+b x)}{c}-\frac{d \log (i+a+b x)}{c \left (d+c x^2\right )}\right ) \, dx-\frac{1}{2} \left (i \left (-\log (a+b x)+\log (-i+a+b x)-\log \left (\frac{-i+a+b x}{a+b x}\right )\right )\right ) \int \frac{x^2}{d+c x^2} \, dx+\frac{1}{2} \left (i \left (-\log (a+b x)+\log (i+a+b x)-\log \left (\frac{i+a+b x}{a+b x}\right )\right )\right ) \int \frac{x^2}{d+c x^2} \, dx\\ &=\frac{i x \left (\log \left (-\frac{i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c}-\frac{i x \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac{i+a+b x}{a+b x}\right )\right )}{2 c}+\frac{i \int \log (-i+a+b x) \, dx}{2 c}-\frac{i \int \log (i+a+b x) \, dx}{2 c}-\frac{(i d) \int \frac{\log (-i+a+b x)}{d+c x^2} \, dx}{2 c}+\frac{(i d) \int \frac{\log (i+a+b x)}{d+c x^2} \, dx}{2 c}+\frac{\left (i d \left (-\log (a+b x)+\log (-i+a+b x)-\log \left (\frac{-i+a+b x}{a+b x}\right )\right )\right ) \int \frac{1}{d+c x^2} \, dx}{2 c}-\frac{\left (i d \left (-\log (a+b x)+\log (i+a+b x)-\log \left (\frac{i+a+b x}{a+b x}\right )\right )\right ) \int \frac{1}{d+c x^2} \, dx}{2 c}\\ &=\frac{i x \left (\log \left (-\frac{i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c}-\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log \left (-\frac{i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c^{3/2}}-\frac{i x \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac{i+a+b x}{a+b x}\right )\right )}{2 c}+\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac{i+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac{i \operatorname{Subst}(\int \log (x) \, dx,x,-i+a+b x)}{2 b c}-\frac{i \operatorname{Subst}(\int \log (x) \, dx,x,i+a+b x)}{2 b c}-\frac{(i d) \int \left (\frac{\log (-i+a+b x)}{2 \sqrt{d} \left (\sqrt{d}-\sqrt{-c} x\right )}+\frac{\log (-i+a+b x)}{2 \sqrt{d} \left (\sqrt{d}+\sqrt{-c} x\right )}\right ) \, dx}{2 c}+\frac{(i d) \int \left (\frac{\log (i+a+b x)}{2 \sqrt{d} \left (\sqrt{d}-\sqrt{-c} x\right )}+\frac{\log (i+a+b x)}{2 \sqrt{d} \left (\sqrt{d}+\sqrt{-c} x\right )}\right ) \, dx}{2 c}\\ &=\frac{i x \left (\log \left (-\frac{i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c}-\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log \left (-\frac{i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c^{3/2}}-\frac{i (i-a-b x) \log (-i+a+b x)}{2 b c}-\frac{i (i+a+b x) \log (i+a+b x)}{2 b c}-\frac{i x \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac{i+a+b x}{a+b x}\right )\right )}{2 c}+\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac{i+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}-\frac{\left (i \sqrt{d}\right ) \int \frac{\log (-i+a+b x)}{\sqrt{d}-\sqrt{-c} x} \, dx}{4 c}-\frac{\left (i \sqrt{d}\right ) \int \frac{\log (-i+a+b x)}{\sqrt{d}+\sqrt{-c} x} \, dx}{4 c}+\frac{\left (i \sqrt{d}\right ) \int \frac{\log (i+a+b x)}{\sqrt{d}-\sqrt{-c} x} \, dx}{4 c}+\frac{\left (i \sqrt{d}\right ) \int \frac{\log (i+a+b x)}{\sqrt{d}+\sqrt{-c} x} \, dx}{4 c}\\ &=\frac{i x \left (\log \left (-\frac{i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c}-\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log \left (-\frac{i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c^{3/2}}-\frac{i (i-a-b x) \log (-i+a+b x)}{2 b c}-\frac{i (i+a+b x) \log (i+a+b x)}{2 b c}-\frac{i x \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac{i+a+b x}{a+b x}\right )\right )}{2 c}+\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac{i+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}-\frac{i \sqrt{d} \log (-i+a+b x) \log \left (-\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(i-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (i+a+b x) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(i+a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \log (i+a+b x) \log \left (-\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(i+a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (-i+a+b x) \log \left (\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(i-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{\left (i b \sqrt{d}\right ) \int \frac{\log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(-i+a) \sqrt{-c}+b \sqrt{d}}\right )}{-i+a+b x} \, dx}{4 (-c)^{3/2}}-\frac{\left (i b \sqrt{d}\right ) \int \frac{\log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(i+a) \sqrt{-c}+b \sqrt{d}}\right )}{i+a+b x} \, dx}{4 (-c)^{3/2}}-\frac{\left (i b \sqrt{d}\right ) \int \frac{\log \left (\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{-(-i+a) \sqrt{-c}+b \sqrt{d}}\right )}{-i+a+b x} \, dx}{4 (-c)^{3/2}}+\frac{\left (i b \sqrt{d}\right ) \int \frac{\log \left (\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{-(i+a) \sqrt{-c}+b \sqrt{d}}\right )}{i+a+b x} \, dx}{4 (-c)^{3/2}}\\ &=\frac{i x \left (\log \left (-\frac{i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c}-\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log \left (-\frac{i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c^{3/2}}-\frac{i (i-a-b x) \log (-i+a+b x)}{2 b c}-\frac{i (i+a+b x) \log (i+a+b x)}{2 b c}-\frac{i x \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac{i+a+b x}{a+b x}\right )\right )}{2 c}+\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac{i+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}-\frac{i \sqrt{d} \log (-i+a+b x) \log \left (-\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(i-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (i+a+b x) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(i+a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \log (i+a+b x) \log \left (-\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(i+a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (-i+a+b x) \log \left (\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(i-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{\left (i \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c} x}{-(-i+a) \sqrt{-c}+b \sqrt{d}}\right )}{x} \, dx,x,-i+a+b x\right )}{4 (-c)^{3/2}}+\frac{\left (i \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c} x}{(-i+a) \sqrt{-c}+b \sqrt{d}}\right )}{x} \, dx,x,-i+a+b x\right )}{4 (-c)^{3/2}}+\frac{\left (i \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c} x}{-(i+a) \sqrt{-c}+b \sqrt{d}}\right )}{x} \, dx,x,i+a+b x\right )}{4 (-c)^{3/2}}-\frac{\left (i \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c} x}{(i+a) \sqrt{-c}+b \sqrt{d}}\right )}{x} \, dx,x,i+a+b x\right )}{4 (-c)^{3/2}}\\ &=\frac{i x \left (\log \left (-\frac{i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c}-\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log \left (-\frac{i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c^{3/2}}-\frac{i (i-a-b x) \log (-i+a+b x)}{2 b c}-\frac{i (i+a+b x) \log (i+a+b x)}{2 b c}-\frac{i x \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac{i+a+b x}{a+b x}\right )\right )}{2 c}+\frac{i \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ) \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac{i+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}-\frac{i \sqrt{d} \log (-i+a+b x) \log \left (-\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(i-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (i+a+b x) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{(i+a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \log (i+a+b x) \log \left (-\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(i+a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \log (-i+a+b x) \log \left (\frac{b \left (\sqrt{d}+\sqrt{-c} x\right )}{(i-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \text{Li}_2\left (\frac{\sqrt{-c} (i-a-b x)}{(i-a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \text{Li}_2\left (\frac{\sqrt{-c} (i-a-b x)}{(i-a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}-\frac{i \sqrt{d} \text{Li}_2\left (\frac{\sqrt{-c} (i+a+b x)}{(i+a) \sqrt{-c}-b \sqrt{d}}\right )}{4 (-c)^{3/2}}+\frac{i \sqrt{d} \text{Li}_2\left (\frac{\sqrt{-c} (i+a+b x)}{(i+a) \sqrt{-c}+b \sqrt{d}}\right )}{4 (-c)^{3/2}}\\ \end{align*}
Mathematica [B] time = 33.8933, size = 5117, normalized size = 6.96 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 1.987, size = 52954, normalized size = 72.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2} \operatorname{arccot}\left (b x + a\right )}{c x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (b x + a\right )}{c + \frac{d}{x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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