Optimal. Leaf size=176 \[ -\frac {6 i b \sqrt {1+i a+i b x}}{(i+a)^2 \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}+\frac {6 i \sqrt {i-a} b \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i+a)^{5/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {(i a+i b x+1)^{3/2}}{(1-i a) x \sqrt {-i a-i b x+1}}-\frac {6 i b \sqrt {i a+i b x+1}}{(a+i)^2 \sqrt {-i a-i b x+1}}+\frac {6 i \sqrt {-a+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 )}{(a+i)^{5/2}} \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^{3 i \tan ^{-1}(a+b x)}}{x^2} \, dx &=\int \frac {(1+i a+i b x)^{3/2}}{x^2 (1-i a-i b x)^{3/2}} \, dx\\ &=-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}-\frac {(3 b) \int \frac {\sqrt {1+i a+i b x}}{x (1-i a-i b x)^{3/2}} \, dx}{i+a}\\ &=-\frac {6 i b \sqrt {1+i a+i b x}}{(i+a)^2 \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}-\frac {(3 (i-a) b) \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{(i+a)^2}\\ &=-\frac {6 i b \sqrt {1+i a+i b x}}{(i+a)^2 \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}-\frac {(6 (i-a) 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)^2}\\ &=-\frac {6 i b \sqrt {1+i a+i b x}}{(i+a)^2 \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}+\frac {6 i \sqrt {i-a} 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 )}{(i+a)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 145, normalized size = 0.82 \begin {gather*} \frac {\frac {\sqrt {1+i a+i b x} \left (1+a^2-5 i b x+a b x\right )}{x \sqrt {-i (i+a+b x)}}+\frac {6 i \sqrt {-1-i a} 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}}}{(i+a)^2} \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. 630 vs. \(2 (136 ) = 272\).
time = 0.20, size = 631, normalized size = 3.59
method | result | size |
risch | \(\frac {i \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \left (a -i\right )}{\left (i+a \right )^{2} x}-\frac {3 b \sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+2 i a -1\right ) \left (i+a \right )}+\frac {4 \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}\, a}{\left (a^{2}+2 i a -1\right ) \left (i+a \right ) \left (x +\frac {i}{b}+\frac {a}{b}\right )}+\frac {4 i \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{\left (a^{2}+2 i a -1\right ) \left (i+a \right ) \left (x +\frac {i}{b}+\frac {a}{b}\right )}\) | \(245\) |
default | \(-i b^{3} \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )-\frac {6 i a \,b^{2} \left (2 b^{2} x +2 a b \right )}{\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {6 b^{2} \left (2 b^{2} x +2 a b \right )}{\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-3 b \left (i a^{2}+2 a -i\right ) \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )+\left (-i a^{3}-3 a^{2}+3 i a +1\right ) \left (-\frac {1}{\left (a^{2}+1\right ) x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a b \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{a^{2}+1}-\frac {4 b^{2} \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )\) | \(631\) |
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. 992 vs. \(2 (116) = 232\).
time = 0.27, size = 992, normalized size = 5.64 \begin {gather*} -\frac {i \, a b^{4} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {3 \, {\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} a^{2} b^{4} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}^{2}} - \frac {i \, a^{2} b^{3}}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {3 \, {\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} a^{3} b^{3}}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}^{2}} - \frac {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )} a b^{3} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}} + \frac {2 \, {\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} b^{4} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}} - \frac {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )} a^{2} b^{2}}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}} + \frac {2 \, {\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} a b^{3}}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}} + \frac {3 \, {\left (i \, a b^{2} + b^{2}\right )} b^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {3 \, {\left (i \, a b^{2} + b^{2}\right )} a b}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {3 \, {\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} a 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 {5}{2}}} + \frac {i \, b}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {3 \, {\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} a b}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}^{2}} + \frac {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )} \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 {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}} - \frac {-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\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. 389 vs. \(2 (116) = 232\).
time = 2.39, size = 389, normalized size = 2.21 \begin {gather*} -\frac {{\left (-i \, a - 5\right )} b^{2} x^{2} + {\left (-i \, a^{2} - 4 \, a - 5 i\right )} b x - 3 \, {\left ({\left (a^{2} + 2 i \, a - 1\right )} b x^{2} + {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} x\right )} \sqrt {\frac {{\left (a - i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} \log \left (-\frac {b^{2} x + {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (a - i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b}{b}\right ) + 3 \, {\left ({\left (a^{2} + 2 i \, a - 1\right )} b x^{2} + {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} x\right )} \sqrt {\frac {{\left (a - i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} \log \left (-\frac {b^{2} x - {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (a - i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b}{b}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left ({\left (-i \, a - 5\right )} b x - i \, a^{2} - i\right )}}{{\left (a^{2} + 2 i \, a - 1\right )} b x^{2} + {\left (a^{3} + 3 i \, a^{2} - 3 \, 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 \frac {i}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3}}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2}}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \left (- \frac {3 b x}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {b^{3} x^{3}}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i b^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {3 a b^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} b x}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {6 i a b x}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, 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 = -0.01 \begin {gather*} \int \frac {{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{x^2\,{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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