Optimal. Leaf size=561 \[ -\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {b c \log (x)}{d} \]
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Rubi [A] time = 0.53, antiderivative size = 561, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {4918, 4852, 266, 36, 29, 31, 4910, 205, 4908, 2409, 2394, 2393, 2391} \[ \frac {i b \sqrt {e} \text {PolyLog}\left (2,\frac {\sqrt {e} (-c x+i)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \text {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{\sqrt {e}+i c \sqrt {-d}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \text {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{\sqrt {e}+i c \sqrt {-d}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \text {PolyLog}\left (2,\frac {\sqrt {e} (c x+i)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {b c \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 205
Rule 266
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 4852
Rule 4908
Rule 4910
Rule 4918
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx &=\frac {\int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx}{d}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{d+e x^2} \, dx}{d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d}-\frac {(a e) \int \frac {1}{d+e x^2} \, dx}{d}-\frac {(b e) \int \frac {\tan ^{-1}(c x)}{d+e x^2} \, dx}{d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d}-\frac {(i b e) \int \frac {\log (1-i c x)}{d+e x^2} \, dx}{2 d}+\frac {(i b e) \int \frac {\log (1+i c x)}{d+e x^2} \, dx}{2 d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}-\frac {\left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d}-\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}+\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}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+\frac {b c \log (x)}{d}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d}-\frac {(i b e) \int \frac {\log (1-i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{3/2}}-\frac {(i b e) \int \frac {\log (1-i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{3/2}}+\frac {(i b e) \int \frac {\log (1+i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{3/2}}+\frac {(i b e) \int \frac {\log (1+i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{3/2}}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+\frac {b c \log (x)}{d}-\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)^{3/2}}+\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)^{3/2}}-\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)^{3/2}}+\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)^{3/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d}-\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)^{3/2}}-\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)^{3/2}}+\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)^{3/2}}+\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)^{3/2}}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+\frac {b c \log (x)}{d}-\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)^{3/2}}+\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)^{3/2}}-\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)^{3/2}}+\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)^{3/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d}+\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)^{3/2}}-\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)^{3/2}}-\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)^{3/2}}+\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)^{3/2}}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+\frac {b c \log (x)}{d}-\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)^{3/2}}+\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)^{3/2}}-\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)^{3/2}}+\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)^{3/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\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)^{3/2}}-\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)^{3/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 468, normalized size = 0.83 \[ \frac {-\frac {\sqrt {e} \left (4 a \sqrt {-d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+i b \sqrt {d} \left (\text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )+\log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )\right )-i b \sqrt {d} \left (\text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )+\log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )\right )-i b \sqrt {d} \left (\text {Li}_2\left (\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )+\log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )\right )+i b \sqrt {d} \left (\text {Li}_2\left (\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )+\log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )\right )\right )}{4 \sqrt {-d^2}}-\frac {a+b \tan ^{-1}(c x)}{x}-\frac {1}{2} b c \log \left (c^2 x^2+1\right )+b c \log (x)}{d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (c x\right ) + a}{e x^{4} + d x^{2}}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.84, size = 2439, normalized size = 4.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -a {\left (\frac {e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} d} + \frac {1}{d x}\right )} + 2 \, b \int \frac {\arctan \left (c x\right )}{2 \, {\left (e x^{4} + d x^{2}\right )}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^2\,\left (e\,x^2+d\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {atan}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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