Optimal. Leaf size=264 \[ -\frac {3 (3 i+2 a) b^2 \sqrt {1-i a-i b x}}{(1+i a)^3 (i+a) \sqrt {1+i a+i b x}}+\frac {(3-2 i a) b (1-i a-i b x)^{3/2}}{2 (i-a)^2 (i+a) x \sqrt {1+i a+i b x}}-\frac {(1-i a-i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1+i a+i b x}}+\frac {3 (3-2 i a) b^2 \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{7/2} \sqrt {i+a}} \]
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Rubi [A]
time = 0.12, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {(-i a-i b x+1)^{5/2}}{2 \left (a^2+1\right ) x^2 \sqrt {i a+i b x+1}}-\frac {3 (2 a+3 i) b^2 \sqrt {-i a-i b x+1}}{(1+i a)^3 (a+i) \sqrt {i a+i b x+1}}+\frac {3 (3-2 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)^{7/2} \sqrt {a+i}}+\frac {(3-2 i a) b (-i a-i b x+1)^{3/2}}{2 (-a+i)^2 (a+i) x \sqrt {i a+i b x+1}} \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^{-3 i \tan ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac {(1-i a-i b x)^{3/2}}{x^3 (1+i a+i b x)^{3/2}} \, dx\\ &=-\frac {(1-i a-i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1+i a+i b x}}-\frac {((3 i+2 a) b) \int \frac {(1-i a-i b x)^{3/2}}{x^2 (1+i a+i b x)^{3/2}} \, dx}{2 \left (1+a^2\right )}\\ &=\frac {(3-2 i a) b (1-i a-i b x)^{3/2}}{2 (i-a)^2 (i+a) x \sqrt {1+i a+i b x}}-\frac {(1-i a-i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1+i a+i b x}}+\frac {\left (3 (3 i+2 a) b^2\right ) \int \frac {\sqrt {1-i a-i b x}}{x (1+i a+i b x)^{3/2}} \, dx}{2 (i-a)^2 (i+a)}\\ &=-\frac {3 (3-2 i a) b^2 \sqrt {1-i a-i b x}}{(i-a)^3 (i+a) \sqrt {1+i a+i b x}}+\frac {(3-2 i a) b (1-i a-i b x)^{3/2}}{2 (i-a)^2 (i+a) x \sqrt {1+i a+i b x}}-\frac {(1-i a-i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1+i a+i b x}}+\frac {\left (3 (3 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)^3}\\ &=-\frac {3 (3-2 i a) b^2 \sqrt {1-i a-i b x}}{(i-a)^3 (i+a) \sqrt {1+i a+i b x}}+\frac {(3-2 i a) b (1-i a-i b x)^{3/2}}{2 (i-a)^2 (i+a) x \sqrt {1+i a+i b x}}-\frac {(1-i a-i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1+i a+i b x}}+\frac {\left (3 (3 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)^3}\\ &=-\frac {3 (3-2 i a) b^2 \sqrt {1-i a-i b x}}{(i-a)^3 (i+a) \sqrt {1+i a+i b x}}+\frac {(3-2 i a) b (1-i a-i b x)^{3/2}}{2 (i-a)^2 (i+a) x \sqrt {1+i a+i b x}}-\frac {(1-i a-i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1+i a+i b x}}+\frac {3 (3-2 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)^{7/2} \sqrt {i+a}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 194, normalized size = 0.73 \begin {gather*} \frac {\frac {\sqrt {-i (i+a+b x)} \left (-i+a-i a^2+a^3-5 b x-5 i a b x-14 i b^2 x^2-a b^2 x^2\right )}{x^2 \sqrt {1+i a+i b x}}+\frac {6 i \sqrt {-1+i a} (3 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} (i+a)}}{2 (-i+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 2327 vs. \(2 (209 ) = 418\).
time = 0.22, size = 2328, normalized size = 8.82
method | result | size |
risch | \(-\frac {i \left (-a \,b^{3} x^{3}-6 i b^{3} x^{3}-a^{2} b^{2} x^{2}-12 i a \,b^{2} x^{2}+a^{3} b x -6 i x \,a^{2} b +a^{4}+b^{2} x^{2}+a b x -6 i b x +2 a^{2}+1\right )}{2 x^{2} \left (a -i\right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 i b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a}{2 \left (a^{3}-3 i a^{2}-3 a +i\right ) \left (i-a \right ) \sqrt {a^{2}+1}}-\frac {3 b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a^{2}}{\left (a^{3}-3 i a^{2}-3 a +i\right ) \left (i-a \right ) \sqrt {a^{2}+1}}-\frac {9 b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{3}-3 i a^{2}-3 a +i\right ) \left (i-a \right ) \sqrt {a^{2}+1}}+\frac {4 b \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a}{\left (a^{3}-3 i a^{2}-3 a +i\right ) \left (i-a \right ) \left (x -\frac {i}{b}+\frac {a}{b}\right )}-\frac {4 i b \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{\left (a^{3}-3 i a^{2}-3 a +i\right ) \left (i-a \right ) \left (x -\frac {i}{b}+\frac {a}{b}\right )}\) | \(529\) |
default | \(\text {Expression too large to display}\) | \(2328\) |
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 \begin {gather*} \text {Failed to integrate} \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. 574 vs. \(2 (180) = 360\).
time = 1.96, size = 574, normalized size = 2.17 \begin {gather*} \frac {{\left (i \, a - 14\right )} b^{3} x^{3} + {\left (i \, a^{2} - 13 \, a + 14 i\right )} b^{2} x^{2} - 3 \, {\left ({\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} b x^{3} + {\left (a^{4} - 4 i \, a^{3} - 6 \, a^{2} + 4 i \, a + 1\right )} x^{2}\right )} \sqrt {\frac {{\left (4 \, a^{2} + 12 i \, a - 9\right )} b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}} \log \left (-\frac {{\left (2 \, a + 3 i\right )} b^{3} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a + 3 i\right )} b^{2} + {\left (a^{5} - 3 i \, a^{4} - 2 \, a^{3} - 2 i \, a^{2} - 3 \, a + i\right )} \sqrt {\frac {{\left (4 \, a^{2} + 12 i \, a - 9\right )} b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}}}{{\left (2 \, a + 3 i\right )} b^{2}}\right ) + 3 \, {\left ({\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} b x^{3} + {\left (a^{4} - 4 i \, a^{3} - 6 \, a^{2} + 4 i \, a + 1\right )} x^{2}\right )} \sqrt {\frac {{\left (4 \, a^{2} + 12 i \, a - 9\right )} b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}} \log \left (-\frac {{\left (2 \, a + 3 i\right )} b^{3} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a + 3 i\right )} b^{2} - {\left (a^{5} - 3 i \, a^{4} - 2 \, a^{3} - 2 i \, a^{2} - 3 \, a + i\right )} \sqrt {\frac {{\left (4 \, a^{2} + 12 i \, a - 9\right )} b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}}}{{\left (2 \, a + 3 i\right )} b^{2}}\right ) + {\left ({\left (i \, a - 14\right )} b^{2} x^{2} - i \, a^{3} - 5 \, {\left (a - i\right )} b x - a^{2} - i \, a - 1\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{2 \, {\left ({\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} b x^{3} + {\left (a^{4} - 4 i \, a^{3} - 6 \, a^{2} + 4 i \, a + 1\right )} x^{2}\right )}} \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 \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} b x^{4} - 3 i a^{2} x^{3} + 3 a b^{2} x^{5} - 6 i a b x^{4} - 3 a x^{3} + b^{3} x^{6} - 3 i b^{2} x^{5} - 3 b x^{4} + i x^{3}}\, dx + \int \frac {a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} b x^{4} - 3 i a^{2} x^{3} + 3 a b^{2} x^{5} - 6 i a b x^{4} - 3 a x^{3} + b^{3} x^{6} - 3 i b^{2} x^{5} - 3 b x^{4} + i x^{3}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} b x^{4} - 3 i a^{2} x^{3} + 3 a b^{2} x^{5} - 6 i a b x^{4} - 3 a x^{3} + b^{3} x^{6} - 3 i b^{2} x^{5} - 3 b x^{4} + i x^{3}}\, dx + \int \frac {2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} b x^{4} - 3 i a^{2} x^{3} + 3 a b^{2} x^{5} - 6 i a b x^{4} - 3 a x^{3} + b^{3} x^{6} - 3 i b^{2} x^{5} - 3 b x^{4} + i x^{3}}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{x^3\,{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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