Optimal. Leaf size=1382 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.58037, antiderivative size = 1382, normalized size of antiderivative = 1., number of steps used = 50, number of rules used = 17, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.81, Rules used = {4980, 4852, 266, 36, 29, 31, 199, 205, 4912, 6725, 444, 4908, 2409, 2394, 2393, 2391, 4910} \[ -\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 d^{5/2}}-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (e x^2+d\right )}-\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2}}+\frac{b c \log (x)}{d^2}+\frac{i b \sqrt{e} \log (i c x+1) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} c+i \sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{e} x+\sqrt{-d}\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (i c x+1) \log \left (\frac{c \left (\sqrt{e} x+\sqrt{-d}\right )}{\sqrt{-d} c+i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b c \sqrt{e} \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left (-\frac{\sqrt{e} \left (\sqrt{-c^2} x+1\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right )}{8 \sqrt{-c^2} d^{5/2}}-\frac{i b c \sqrt{e} \log \left (\frac{\sqrt{e} \left (\sqrt{-c^2} x+1\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{b c e \log \left (c^2 x^2+1\right )}{4 d^2 \left (c^2 d-e\right )}-\frac{b c \log \left (c^2 x^2+1\right )}{2 d^2}-\frac{b c e \log \left (e x^2+d\right )}{4 d^2 \left (c^2 d-e\right )}-\frac{i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b c \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{5/2}}-\frac{i b c \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4980
Rule 4852
Rule 266
Rule 36
Rule 29
Rule 31
Rule 199
Rule 205
Rule 4912
Rule 6725
Rule 444
Rule 4908
Rule 2409
Rule 2394
Rule 2393
Rule 2391
Rule 4910
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx &=\int \left (\frac{a+b \tan ^{-1}(c x)}{d^2 x^2}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{d \left (d+e x^2\right )^2}-\frac{e \left (a+b \tan ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx}{d^2}-\frac{e \int \frac{a+b \tan ^{-1}(c x)}{d+e x^2} \, dx}{d^2}-\frac{e \int \frac{a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{d}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{(b c) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx}{d^2}-\frac{(a e) \int \frac{1}{d+e x^2} \, dx}{d^2}-\frac{(b e) \int \frac{\tan ^{-1}(c x)}{d+e x^2} \, dx}{d^2}+\frac{(b c e) \int \frac{\frac{x}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}}{1+c^2 x^2} \, dx}{d}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^2}-\frac{(i b e) \int \frac{\log (1-i c x)}{d+e x^2} \, dx}{2 d^2}+\frac{(i b e) \int \frac{\log (1+i c x)}{d+e x^2} \, dx}{2 d^2}+\frac{(b c e) \int \left (\frac{x}{2 d \left (1+c^2 x^2\right ) \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e} \left (1+c^2 x^2\right )}\right ) \, dx}{d}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 d^2}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )}{2 d^2}+\frac{\left (b c \sqrt{e}\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{2 d^{5/2}}-\frac{(i b e) \int \left (\frac{\sqrt{-d} \log (1-i c x)}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \log (1-i c x)}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 d^2}+\frac{(i b e) \int \left (\frac{\sqrt{-d} \log (1+i c x)}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \log (1+i c x)}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 d^2}+\frac{(b c e) \int \frac{x}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 d^2}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{b c \log (x)}{d^2}-\frac{b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{\left (i b c \sqrt{e}\right ) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{4 d^{5/2}}-\frac{\left (i b c \sqrt{e}\right ) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{4 d^{5/2}}+\frac{(i b e) \int \frac{\log (1-i c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 (-d)^{5/2}}+\frac{(i b e) \int \frac{\log (1-i c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 (-d)^{5/2}}-\frac{(i b e) \int \frac{\log (1+i c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 (-d)^{5/2}}-\frac{(i b e) \int \frac{\log (1+i c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 (-d)^{5/2}}+\frac{(b c e) \operatorname{Subst}\left (\int \frac{1}{\left (1+c^2 x\right ) (d+e x)} \, dx,x,x^2\right )}{4 d^2}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{b c \log (x)}{d^2}+\frac{i b \sqrt{e} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{\left (b c \sqrt{e}\right ) \int \frac{\log \left (-\frac{i c \left (\sqrt{-d}-\sqrt{e} x\right )}{-i c \sqrt{-d}+\sqrt{e}}\right )}{1-i c x} \, dx}{4 (-d)^{5/2}}+\frac{\left (b c \sqrt{e}\right ) \int \frac{\log \left (\frac{i c \left (\sqrt{-d}-\sqrt{e} x\right )}{i c \sqrt{-d}+\sqrt{e}}\right )}{1+i c x} \, dx}{4 (-d)^{5/2}}-\frac{\left (b c \sqrt{e}\right ) \int \frac{\log \left (-\frac{i c \left (\sqrt{-d}+\sqrt{e} x\right )}{-i c \sqrt{-d}-\sqrt{e}}\right )}{1-i c x} \, dx}{4 (-d)^{5/2}}-\frac{\left (b c \sqrt{e}\right ) \int \frac{\log \left (\frac{i c \left (\sqrt{-d}+\sqrt{e} x\right )}{i c \sqrt{-d}-\sqrt{e}}\right )}{1+i c x} \, dx}{4 (-d)^{5/2}}+\frac{\left (i b c \sqrt{e}\right ) \int \left (\frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1-\sqrt{-c^2} x\right )}+\frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1+\sqrt{-c^2} x\right )}\right ) \, dx}{4 d^{5/2}}-\frac{\left (i b c \sqrt{e}\right ) \int \left (\frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1-\sqrt{-c^2} x\right )}+\frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 \left (1+\sqrt{-c^2} x\right )}\right ) \, dx}{4 d^{5/2}}+\frac{\left (b c^3 e\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac{\left (b c e^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x} \, dx,x,x^2\right )}{4 d^2 \left (c^2 d-e\right )}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{b c \log (x)}{d^2}+\frac{i b \sqrt{e} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{b c e \log \left (1+c^2 x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac{b c e \log \left (d+e x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac{\left (i b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{-i c \sqrt{-d}-\sqrt{e}}\right )}{x} \, dx,x,1-i c x\right )}{4 (-d)^{5/2}}+\frac{\left (i b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{i c \sqrt{-d}-\sqrt{e}}\right )}{x} \, dx,x,1+i c x\right )}{4 (-d)^{5/2}}+\frac{\left (i b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{-i c \sqrt{-d}+\sqrt{e}}\right )}{x} \, dx,x,1-i c x\right )}{4 (-d)^{5/2}}-\frac{\left (i b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{i c \sqrt{-d}+\sqrt{e}}\right )}{x} \, dx,x,1+i c x\right )}{4 (-d)^{5/2}}+\frac{\left (i b c \sqrt{e}\right ) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1-\sqrt{-c^2} x} \, dx}{8 d^{5/2}}+\frac{\left (i b c \sqrt{e}\right ) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+\sqrt{-c^2} x} \, dx}{8 d^{5/2}}-\frac{\left (i b c \sqrt{e}\right ) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1-\sqrt{-c^2} x} \, dx}{8 d^{5/2}}-\frac{\left (i b c \sqrt{e}\right ) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+\sqrt{-c^2} x} \, dx}{8 d^{5/2}}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{b c \log (x)}{d^2}+\frac{i b \sqrt{e} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b c \sqrt{e} \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left (-\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}-\frac{i b c \sqrt{e} \log \left (\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}-\frac{b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{b c e \log \left (1+c^2 x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac{b c e \log \left (d+e x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (i-c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (1-i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (1+i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (i+c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{(b c e) \int \frac{\log \left (-\frac{i \sqrt{e} \left (1-\sqrt{-c^2} x\right )}{\sqrt{d} \left (\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1-\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d^3}+\frac{(b c e) \int \frac{\log \left (\frac{i \sqrt{e} \left (1-\sqrt{-c^2} x\right )}{\sqrt{d} \left (\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1+\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d^3}-\frac{(b c e) \int \frac{\log \left (-\frac{i \sqrt{e} \left (1+\sqrt{-c^2} x\right )}{\sqrt{d} \left (-\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1-\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d^3}-\frac{(b c e) \int \frac{\log \left (\frac{i \sqrt{e} \left (1+\sqrt{-c^2} x\right )}{\sqrt{d} \left (-\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1+\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d^3}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{b c \log (x)}{d^2}+\frac{i b \sqrt{e} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b c \sqrt{e} \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left (-\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}-\frac{i b c \sqrt{e} \log \left (\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}-\frac{b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{b c e \log \left (1+c^2 x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac{b c e \log \left (d+e x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (i-c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (1-i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (1+i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (i+c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{\left (i b c \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c^2} x}{-\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{\left (i b c \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c^2} x}{\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{\left (i b c \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c^2} x}{-\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}-\frac{\left (i b c \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c^2} x}{\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}+\frac{b c \log (x)}{d^2}+\frac{i b \sqrt{e} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b c \sqrt{e} \log \left (\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left (-\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left (-\frac{\sqrt{e} \left (1-\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}-\frac{i b c \sqrt{e} \log \left (\frac{\sqrt{e} \left (1+\sqrt{-c^2} x\right )}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right ) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} d^{5/2}}-\frac{b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{b c e \log \left (1+c^2 x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac{b c e \log \left (d+e x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (i-c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (1-i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (1+i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \text{Li}_2\left (\frac{\sqrt{e} (i+c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{i b c \sqrt{e} \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{5/2}}-\frac{i b c \sqrt{e} \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}+i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}+i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} d^{5/2}}\\ \end{align*}
Mathematica [A] time = 12.9503, size = 982, normalized size = 0.71 \[ b \left (-\frac{e \sin \left (2 \tan ^{-1}(c x)\right ) \tan ^{-1}(c x)}{2 c^4 d^2 \left (d c^2+d \cos \left (2 \tan ^{-1}(c x)\right ) c^2+e-e \cos \left (2 \tan ^{-1}(c x)\right )\right )}-\frac{\tan ^{-1}(c x)}{c^5 d^2 x}+\frac{\log \left (\frac{c x}{\sqrt{c^2 x^2+1}}\right )}{c^4 d^2}-\frac{e \log \left (\frac{\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}{d c^2+e}+1\right )}{4 c^4 d^2 \left (c^2 d-e\right )}-\frac{3 e \left (4 \tan ^{-1}(c x) \tanh ^{-1}\left (\frac{c d}{\sqrt{-c^2 d e} x}\right )+2 \cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right ) \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )-\left (\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )-2 i \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right ) \log \left (1-\frac{\left (d c^2+e-2 i \sqrt{-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt{-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 d c^2+2 \sqrt{-c^2 d e} x c\right )}\right )+\left (-\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )-2 i \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right ) \log \left (1-\frac{\left (d c^2+e+2 i \sqrt{-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt{-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 d c^2+2 \sqrt{-c^2 d e} x c\right )}\right )+\left (\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )-2 i \left (\tanh ^{-1}\left (\frac{c d}{\sqrt{-c^2 d e} x}\right )+\tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}}\right )+\left (\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )+2 i \left (\tanh ^{-1}\left (\frac{c d}{\sqrt{-c^2 d e} x}\right )+\tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}}\right )+i \left (\text{PolyLog}\left (2,\frac{\left (d c^2+e-2 i \sqrt{-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt{-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 d c^2+2 \sqrt{-c^2 d e} x c\right )}\right )-\text{PolyLog}\left (2,\frac{\left (d c^2+e+2 i \sqrt{-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt{-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 d c^2+2 \sqrt{-c^2 d e} x c\right )}\right )\right )\right )}{8 c^4 d^2 \sqrt{-c^2 d e}}\right ) c^5-\frac{3 a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2}}-\frac{a}{d^2 x}-\frac{a e x}{2 d^2 \left (e x^2+d\right )} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.638, size = 3851, normalized size = 2.8 \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{b \arctan \left (c x\right ) + a}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, 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{b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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