Optimal. Leaf size=130 \[ -\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{(1-i a) x}+\frac {2 i b \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{\sqrt {i-a} (i+a)^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5203, 96, 95,
214} \begin {gather*} \frac {2 i b \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{\sqrt {-a+i} (a+i)^{3/2}}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{(1-i a) x} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 214
Rule 5203
Rubi steps
\begin {align*} \int \frac {e^{i \tan ^{-1}(a+b x)}}{x^2} \, dx &=\int \frac {\sqrt {1+i a+i b x}}{x^2 \sqrt {1-i a-i b x}} \, dx\\ &=-\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{(1-i a) x}-\frac {b \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{i+a}\\ &=-\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{(1-i a) x}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{-1-i a-(-1+i a) x^2} \, dx,x,\frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}}\right )}{i+a}\\ &=-\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{(1-i a) x}+\frac {2 i b \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{\sqrt {i-a} (i+a)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 120, normalized size = 0.92 \begin {gather*} -i \left (\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{i x+a x}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )}{\sqrt {-1-i a} (-1+i a)^{3/2}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.11, size = 152, normalized size = 1.17
method | result | size |
risch | \(-\frac {i \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (i+a \right ) x}+\frac {b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (i+a \right ) \sqrt {a^{2}+1}}\) | \(93\) |
default | \(-\frac {i b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\sqrt {a^{2}+1}}+\left (i a +1\right ) \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 239 vs. \(2 (86) = 172\).
time = 0.29, size = 239, normalized size = 1.84 \begin {gather*} \frac {a {\left (i \, a + 1\right )} b \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {i \, b \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{\sqrt {a^{2} + 1}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-i \, a - 1\right )}}{{\left (a^{2} + 1\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 224 vs. \(2 (86) = 172\).
time = 1.53, size = 224, normalized size = 1.72 \begin {gather*} -\frac {{\left (a + i\right )} \sqrt {\frac {b^{2}}{a^{4} + 2 i \, a^{3} + 2 i \, a - 1}} x \log \left (-\frac {b^{2} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b + {\left (a^{3} + i \, a^{2} + a + i\right )} \sqrt {\frac {b^{2}}{a^{4} + 2 i \, a^{3} + 2 i \, a - 1}}}{b}\right ) - {\left (a + i\right )} \sqrt {\frac {b^{2}}{a^{4} + 2 i \, a^{3} + 2 i \, a - 1}} x \log \left (-\frac {b^{2} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b - {\left (a^{3} + i \, a^{2} + a + i\right )} \sqrt {\frac {b^{2}}{a^{4} + 2 i \, a^{3} + 2 i \, a - 1}}}{b}\right ) + i \, b x + i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (a + i\right )} x} \end {gather*}
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 \begin {gather*} i \left (\int \left (- \frac {i}{x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a}{x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {b}{x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 145, normalized size = 1.12 \begin {gather*} \frac {b \log \left (\frac {{\left | 2 \, x {\left | b \right |} - 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | 2 \, x {\left | b \right |} - 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{\sqrt {a^{2} + 1} {\left (a + i\right )}} - \frac {2 \, {\left ({\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a b + a^{2} {\left | b \right |} + {\left | b \right |}\right )}}{{\left ({\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} - a^{2} - 1\right )} {\left (i \, a - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.68, size = 218, normalized size = 1.68 \begin {gather*} \frac {a\,b\,\mathrm {atanh}\left (\frac {a^2+b\,x\,a+1}{\sqrt {a^2+1}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}\right )}{{\left (a^2+1\right )}^{3/2}}-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{x\,\left (a^2+1\right )}-\frac {b\,\ln \left (a\,b+\frac {a^2+1}{x}+\frac {\sqrt {a^2+1}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{x}\right )\,1{}\mathrm {i}}{\sqrt {a^2+1}}+\frac {a^2\,b\,\mathrm {atanh}\left (\frac {a^2+b\,x\,a+1}{\sqrt {a^2+1}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}\right )\,1{}\mathrm {i}}{{\left (a^2+1\right )}^{3/2}}-\frac {a\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}\,1{}\mathrm {i}}{x\,\left (a^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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