Optimal. Leaf size=163 \[ -\frac {(1+i a) (-i a-i b x+1)^{5/2}}{b^2 \sqrt {i a+i b x+1}}-\frac {(3+2 i a) \sqrt {i a+i b x+1} (-i a-i b x+1)^{3/2}}{2 b^2}-\frac {3 (3+2 i a) \sqrt {i a+i b x+1} \sqrt {-i a-i b x+1}}{2 b^2}-\frac {3 (-2 a+3 i) \sinh ^{-1}(a+b x)}{2 b^2} \]
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Rubi [A] time = 0.12, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5095, 78, 50, 53, 619, 215} \[ -\frac {(1+i a) (-i a-i b x+1)^{5/2}}{b^2 \sqrt {i a+i b x+1}}-\frac {(3+2 i a) \sqrt {i a+i b x+1} (-i a-i b x+1)^{3/2}}{2 b^2}-\frac {3 (3+2 i a) \sqrt {i a+i b x+1} \sqrt {-i a-i b x+1}}{2 b^2}-\frac {3 (-2 a+3 i) \sinh ^{-1}(a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 78
Rule 215
Rule 619
Rule 5095
Rubi steps
\begin {align*} \int e^{-3 i \tan ^{-1}(a+b x)} x \, dx &=\int \frac {x (1-i a-i b x)^{3/2}}{(1+i a+i b x)^{3/2}} \, dx\\ &=-\frac {(1+i a) (1-i a-i b x)^{5/2}}{b^2 \sqrt {1+i a+i b x}}-\frac {(3 i-2 a) \int \frac {(1-i a-i b x)^{3/2}}{\sqrt {1+i a+i b x}} \, dx}{b}\\ &=-\frac {(1+i a) (1-i a-i b x)^{5/2}}{b^2 \sqrt {1+i a+i b x}}-\frac {(3+2 i a) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3 (3 i-2 a)) \int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx}{2 b}\\ &=-\frac {(1+i a) (1-i a-i b x)^{5/2}}{b^2 \sqrt {1+i a+i b x}}-\frac {3 (3+2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3+2 i a) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3 (3 i-2 a)) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 b}\\ &=-\frac {(1+i a) (1-i a-i b x)^{5/2}}{b^2 \sqrt {1+i a+i b x}}-\frac {3 (3+2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3+2 i a) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3 (3 i-2 a)) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{2 b}\\ &=-\frac {(1+i a) (1-i a-i b x)^{5/2}}{b^2 \sqrt {1+i a+i b x}}-\frac {3 (3+2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3+2 i a) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3 (3 i-2 a)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{4 b^3}\\ &=-\frac {(1+i a) (1-i a-i b x)^{5/2}}{b^2 \sqrt {1+i a+i b x}}-\frac {3 (3+2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3+2 i a) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{2 b^2}-\frac {3 (3 i-2 a) \sinh ^{-1}(a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 157, normalized size = 0.96 \[ \frac {i \left (-a^3+a^2 (-b x+14 i)+a \left (b^2 x^2+20 i b x-1\right )+b^3 x^3+6 i b^2 x^2+9 b x+14 i\right )}{2 b^2 \sqrt {a^2+2 a b x+b^2 x^2+1}}+\frac {3 \sqrt [4]{-1} (2 a-3 i) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{b^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 137, normalized size = 0.84 \[ \frac {-3 i \, a^{3} + {\left (-3 i \, a^{2} - 44 \, a + 32 i\right )} b x - 47 \, a^{2} - {\left ({\left (24 \, a - 36 i\right )} b x + 24 \, a^{2} - 60 i \, a - 36\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (4 i \, b^{2} x^{2} - 4 i \, a^{2} - 20 \, b x - 60 \, a + 56 i\right )} + 76 i \, a + 32}{8 \, b^{3} x + {\left (8 \, a - 8 i\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 219, normalized size = 1.34 \[ -\frac {1}{2} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (\frac {x}{b i} - \frac {a b^{2} - 6 \, b^{2} i}{b^{4} i}\right )} - \frac {{\left (2 \, a - 3 \, i\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b - 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b i - 2 \, a^{2} b i + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} - 4 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a i {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, b {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 676, normalized size = 4.15 \[ \frac {2 i a \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b^{4} \left (x -\frac {i}{b}+\frac {a}{b}\right )^{2}}+\frac {3 \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b^{4} \left (x -\frac {i}{b}+\frac {a}{b}\right )^{2}}-\frac {3 \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{b^{2}}-\frac {2 i a \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{b^{2}}-\frac {9 i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a}{2 b^{2}}-\frac {9 i \ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{2 b \sqrt {b^{2}}}+\frac {a \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b^{5} \left (x -\frac {i}{b}+\frac {a}{b}\right )^{3}}-\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b^{5} \left (x -\frac {i}{b}+\frac {a}{b}\right )^{3}}-\frac {9 i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, x}{2 b}+\frac {3 a \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, x}{b}+\frac {3 \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a^{2}}{b^{2}}+\frac {3 \ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right ) a}{b \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 293, normalized size = 1.80 \[ -\frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2} - 2 i \, b^{3} x - 2 i \, a b^{2} - b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2} - 2 i \, b^{3} x - 2 i \, a b^{2} - b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{2 i \, b^{3} x + 2 i \, a b^{2} + 2 \, b^{2}} - \frac {6 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{i \, b^{3} x + i \, a b^{2} + b^{2}} + \frac {3 \, a \operatorname {arsinh}\left (b x + a\right )}{b^{2}} - \frac {6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{i \, b^{3} x + i \, a b^{2} + b^{2}} - \frac {9 i \, \operatorname {arsinh}\left (b x + a\right )}{2 \, b^{2}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{2 \, b^{2}} \]
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 \[ \int \frac {x\,{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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