Optimal. Leaf size=134 \[ \frac {4 \sqrt {-i a-i b x+1}}{(1+i a) \sqrt {i a+i b x+1}}+i \sinh ^{-1}(a+b x)-\frac {2 (a+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(-a+i)^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5095, 98, 157, 53, 619, 215, 93, 208} \[ \frac {4 \sqrt {-i a-i b x+1}}{(1+i a) \sqrt {i a+i b x+1}}+i \sinh ^{-1}(a+b x)-\frac {2 (a+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(-a+i)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 53
Rule 93
Rule 98
Rule 157
Rule 208
Rule 215
Rule 619
Rule 5095
Rubi steps
\begin {align*} \int \frac {e^{-3 i \tan ^{-1}(a+b x)}}{x} \, dx &=\int \frac {(1-i a-i b x)^{3/2}}{x (1+i a+i b x)^{3/2}} \, dx\\ &=\frac {4 \sqrt {1-i a-i b x}}{(1+i a) \sqrt {1+i a+i b x}}+\frac {2 \int \frac {-\frac {1}{2} i (i+a)^2 b-\frac {1}{2} (1+i a) b^2 x}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{(i-a) b}\\ &=\frac {4 \sqrt {1-i a-i b x}}{(1+i a) \sqrt {1+i a+i b x}}-\frac {(i+a)^2 \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{1+i a}+(i b) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx\\ &=\frac {4 \sqrt {1-i a-i b x}}{(1+i a) \sqrt {1+i a+i b x}}-\frac {\left (2 (i+a)^2\right ) \operatorname {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 )}{1+i a}+(i b) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx\\ &=\frac {4 \sqrt {1-i a-i b x}}{(1+i a) \sqrt {1+i a+i b x}}-\frac {2 (i+a)^{3/2} \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}}+\frac {i \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{2 b}\\ &=\frac {4 \sqrt {1-i a-i b x}}{(1+i a) \sqrt {1+i a+i b x}}+i \sinh ^{-1}(a+b x)-\frac {2 (i+a)^{3/2} \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}}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 189, normalized size = 1.41 \[ \frac {2 \left (\frac {\sqrt {-1+i a} (a+i) \tanh ^{-1}\left (\frac {\sqrt {-1-i a} \sqrt {-i (a+b x+i)}}{\sqrt {-1+i a} \sqrt {i a+i b x+1}}\right )}{\sqrt {-1-i a}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2+1}}{a+b x-i}\right )}{a-i}+\frac {2 (-1)^{3/4} \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{\sqrt {b}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.49, size = 381, normalized size = 2.84 \[ \frac {{\left ({\left (a - i\right )} b x + a^{2} - 2 i \, a - 1\right )} \sqrt {-\frac {4 \, a^{3} + 12 i \, a^{2} - 12 \, a - 4 i}{a^{3} - 3 i \, a^{2} - 3 \, a + i}} \log \left (-\frac {{\left (2 \, a + 2 i\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a + 2 i\right )} - {\left (i \, a^{2} + 2 \, a - i\right )} \sqrt {-\frac {4 \, a^{3} + 12 i \, a^{2} - 12 \, a - 4 i}{a^{3} - 3 i \, a^{2} - 3 \, a + i}}}{2 \, a + 2 i}\right ) - {\left ({\left (a - i\right )} b x + a^{2} - 2 i \, a - 1\right )} \sqrt {-\frac {4 \, a^{3} + 12 i \, a^{2} - 12 \, a - 4 i}{a^{3} - 3 i \, a^{2} - 3 \, a + i}} \log \left (-\frac {{\left (2 \, a + 2 i\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a + 2 i\right )} - {\left (-i \, a^{2} - 2 \, a + i\right )} \sqrt {-\frac {4 \, a^{3} + 12 i \, a^{2} - 12 \, a - 4 i}{a^{3} - 3 i \, a^{2} - 3 \, a + i}}}{2 \, a + 2 i}\right ) - 8 \, b x - {\left (2 \, {\left (i \, a + 1\right )} b x + 2 i \, a^{2} + 4 \, a - 2 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 8 \, a - 8 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 8 i}{{\left (2 \, a - 2 i\right )} b x + 2 \, a^{2} - 4 i \, a - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 263, normalized size = 1.96 \[ \frac {{\left (a^{2} i - 2 \, a - i\right )} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{\sqrt {a^{2} + 1} {\left (a - i\right )}} + \frac {b \log \left (-3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b i - a^{3} b i - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} i {\left | b \right |} - 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} i {\left | b \right |} - 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b - 2 \, a^{2} b + a b i - 4 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} i {\left | b \right |}\right )}{3 \, i {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 1278, normalized size = 9.54 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left ({\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, b x + i \, a + 1\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{x\,{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
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 \[ i \left (\int \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x + 3 a^{2} b x^{2} - 3 i a^{2} x + 3 a b^{2} x^{3} - 6 i a b x^{2} - 3 a x + b^{3} x^{4} - 3 i b^{2} x^{3} - 3 b x^{2} + i x}\, dx + \int \frac {a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x + 3 a^{2} b x^{2} - 3 i a^{2} x + 3 a b^{2} x^{3} - 6 i a b x^{2} - 3 a x + b^{3} x^{4} - 3 i b^{2} x^{3} - 3 b x^{2} + i x}\, dx + \int \frac {b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x + 3 a^{2} b x^{2} - 3 i a^{2} x + 3 a b^{2} x^{3} - 6 i a b x^{2} - 3 a x + b^{3} x^{4} - 3 i b^{2} x^{3} - 3 b x^{2} + i x}\, dx + \int \frac {2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x + 3 a^{2} b x^{2} - 3 i a^{2} x + 3 a b^{2} x^{3} - 6 i a b x^{2} - 3 a x + b^{3} x^{4} - 3 i b^{2} x^{3} - 3 b x^{2} + i x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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