Optimal. Leaf size=110 \[ \frac {\sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{2 b^2}+\frac {(1-2 i a) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 b^2}-\frac {(2 a+i) \sinh ^{-1}(a+b x)}{2 b^2} \]
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Rubi [A] time = 0.08, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5095, 80, 50, 53, 619, 215} \[ \frac {\sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{2 b^2}+\frac {(1-2 i a) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 b^2}-\frac {(2 a+i) \sinh ^{-1}(a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 80
Rule 215
Rule 619
Rule 5095
Rubi steps
\begin {align*} \int e^{i \tan ^{-1}(a+b x)} x \, dx &=\int \frac {x \sqrt {1+i a+i b x}}{\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 b^2}-\frac {(i+2 a) \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx}{2 b}\\ &=\frac {(1-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(i+2 a) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 b}\\ &=\frac {(1-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(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 {(1-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(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 {(1-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(i+2 a) \sinh ^{-1}(a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 108, normalized size = 0.98 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2+1} (-i a+i b x+2)}{2 b^2}+\frac {(-1)^{3/4} (2 a+i) \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{\sqrt {-i b} b^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.43, size = 77, normalized size = 0.70 \[ \frac {-3 i \, a^{2} + {\left (8 \, a + 4 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (4 i \, b x - 4 i \, a + 8\right )} + 4 \, a}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 76, normalized size = 0.69 \[ \frac {1}{2} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (\frac {i x}{b} - \frac {a b i - 2 \, b}{b^{3}}\right )} + \frac {{\left (2 \, a + i\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, b {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 171, normalized size = 1.55 \[ \frac {i x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b}-\frac {i a \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b \sqrt {b^{2}}}+\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 209, normalized size = 1.90 \[ \frac {3 i \, a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{2}} - \frac {a {\left (i \, a + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{2}} + \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{2 \, b} - \frac {{\left (i \, a^{2} + i\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b^{2}} - \frac {3 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{2 \, b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (i \, a + 1\right )}}{b^{2}} \]
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 \[ \int \frac {x\,\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}{\sqrt {{\left (a+b\,x\right )}^2+1}} \,d x \]
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 \[ i \left (\int \left (- \frac {i x}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a x}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {b x^{2}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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