Optimal. Leaf size=201 \[ -\frac {(1+2 i a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 (i-a) (i+a)^2 x}-\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 \left (1+a^2\right ) x^2}+\frac {(1+2 i a) b^2 \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{3/2} (i+a)^{5/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5203, 98, 96,
95, 214} \begin {gather*} -\frac {\sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{2 \left (a^2+1\right ) x^2}+\frac {(1+2 i a) b^2 \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 )}{(-a+i)^{3/2} (a+i)^{5/2}}-\frac {(1+2 i a) b \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 (-a+i) (a+i)^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 98
Rule 214
Rule 5203
Rubi steps
\begin {align*} \int \frac {e^{i \tan ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac {\sqrt {1+i a+i b x}}{x^3 \sqrt {1-i a-i b x}} \, dx\\ &=-\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 \left (1+a^2\right ) x^2}+\frac {((i-2 a) b) \int \frac {\sqrt {1+i a+i b x}}{x^2 \sqrt {1-i a-i b x}} \, dx}{2 \left (1+a^2\right )}\\ &=-\frac {(i-2 a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 (1-i a) \left (1+a^2\right ) x}-\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 \left (1+a^2\right ) x^2}-\frac {\left ((i-2 a) b^2\right ) \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 (i+a) \left (1+a^2\right )}\\ &=-\frac {(i-2 a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 (1-i a) \left (1+a^2\right ) x}-\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 \left (1+a^2\right ) x^2}-\frac {\left ((i-2 a) b^2\right ) \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) \left (1+a^2\right )}\\ &=-\frac {(i-2 a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 (1-i a) \left (1+a^2\right ) x}-\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 \left (1+a^2\right ) x^2}+\frac {(1+2 i a) b^2 \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 )}{(i-a)^{3/2} (i+a)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 154, normalized size = 0.77 \begin {gather*} \frac {-\frac {i \left (1+a^2+2 i b x-a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{x^2}+\frac {2 (-i+2 a) b^2 \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} \sqrt {-1+i a}}}{2 (-i+a) (i+a)^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 287, normalized size = 1.43
method | result | size |
risch | \(-\frac {i \left (-a \,b^{3} x^{3}+2 i b^{3} x^{3}-a^{2} b^{2} x^{2}+4 i a \,b^{2} x^{2}+a^{3} b x +2 i x \,a^{2} b +a^{4}+b^{2} x^{2}+a b x +2 i b x +2 a^{2}+1\right )}{2 x^{2} \left (i+a \right )^{2} \left (a -i\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {i b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}} \left (i+a \right )}-\frac {b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a}{\left (a^{2}+1\right )^{\frac {3}{2}} \left (i+a \right )}\) | \(246\) |
default | \(\left (i a +1\right ) \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {3 a b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )+i b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(287\) |
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. 424 vs. \(2 (135) = 270\).
time = 0.28, size = 424, normalized size = 2.11 \begin {gather*} -\frac {3 \, a^{2} {\left (i \, a + 1\right )} b^{2} \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 )}{2 \, {\left (a^{2} + 1\right )}^{\frac {5}{2}}} + \frac {i \, a b^{2} \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 {{\left (-i \, a - 1\right )} b^{2} \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 )}{2 \, {\left (a^{2} + 1\right )}^{\frac {3}{2}}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a {\left (i \, a + 1\right )} b}{2 \, {\left (a^{2} + 1\right )}^{2} x} - \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b}{{\left (a^{2} + 1\right )} x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (i \, a + 1\right )}}{2 \, {\left (a^{2} + 1\right )} x^{2}} \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. 452 vs. \(2 (135) = 270\).
time = 2.09, size = 452, normalized size = 2.25 \begin {gather*} \frac {{\left (i \, a + 2\right )} b^{2} x^{2} + \sqrt {\frac {{\left (4 \, a^{2} - 4 i \, a - 1\right )} b^{4}}{a^{8} + 2 i \, a^{7} + 2 \, a^{6} + 6 i \, a^{5} + 6 i \, a^{3} - 2 \, a^{2} + 2 i \, a - 1}} {\left (a^{3} + i \, a^{2} + a + i\right )} x^{2} \log \left (-\frac {{\left (2 \, a - i\right )} b^{3} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a - i\right )} b^{2} + {\left (a^{5} + i \, a^{4} + 2 \, a^{3} + 2 i \, a^{2} + a + i\right )} \sqrt {\frac {{\left (4 \, a^{2} - 4 i \, a - 1\right )} b^{4}}{a^{8} + 2 i \, a^{7} + 2 \, a^{6} + 6 i \, a^{5} + 6 i \, a^{3} - 2 \, a^{2} + 2 i \, a - 1}}}{{\left (2 \, a - i\right )} b^{2}}\right ) - \sqrt {\frac {{\left (4 \, a^{2} - 4 i \, a - 1\right )} b^{4}}{a^{8} + 2 i \, a^{7} + 2 \, a^{6} + 6 i \, a^{5} + 6 i \, a^{3} - 2 \, a^{2} + 2 i \, a - 1}} {\left (a^{3} + i \, a^{2} + a + i\right )} x^{2} \log \left (-\frac {{\left (2 \, a - i\right )} b^{3} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a - i\right )} b^{2} - {\left (a^{5} + i \, a^{4} + 2 \, a^{3} + 2 i \, a^{2} + a + i\right )} \sqrt {\frac {{\left (4 \, a^{2} - 4 i \, a - 1\right )} b^{4}}{a^{8} + 2 i \, a^{7} + 2 \, a^{6} + 6 i \, a^{5} + 6 i \, a^{3} - 2 \, a^{2} + 2 i \, a - 1}}}{{\left (2 \, a - i\right )} b^{2}}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left ({\left (i \, a + 2\right )} b x - i \, a^{2} - i\right )}}{2 \, {\left (a^{3} + i \, a^{2} + a + i\right )} x^{2}} \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^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a}{x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {b}{x^{2} \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 [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 471 vs. \(2 (135) = 270\).
time = 0.53, size = 471, normalized size = 2.34 \begin {gather*} -\frac {{\left (2 \, a b^{2} - i \, b^{2}\right )} \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 )}{2 \, {\left (a^{3} + i \, a^{2} + a + i\right )} \sqrt {a^{2} + 1}} - \frac {4 \, {\left (-i \, x {\left | b \right |} + i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{4} b^{2} - 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a^{3} b {\left | b \right |} - 2 i \, a^{5} b {\left | b \right |} + 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} a b^{2} - 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{3} b^{2} + 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a^{2} b {\left | b \right |} - 2 \, a^{4} b {\left | b \right |} - i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} b^{2} + 5 \, {\left (-i \, x {\left | b \right |} + i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} b^{2} - 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b {\left | b \right |} - 4 i \, a^{3} b {\left | b \right |} - 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a b^{2} + 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b {\left | b \right |} - 4 \, a^{2} b {\left | b \right |} - {\left (i \, x {\left | b \right |} - i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} b^{2} - 2 i \, a b {\left | b \right |} - 2 \, b {\left | b \right |}}{{\left (a^{3} + i \, a^{2} + a + i\right )} {\left ({\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} - a^{2} - 1\right )}^{2}} \end {gather*}
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 \begin {gather*} \int \frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{x^3\,\sqrt {{\left (a+b\,x\right )}^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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