Optimal. Leaf size=171 \[ \frac {\left (-2 i a^2-2 a+i\right ) \sqrt {i a+i b x+1} \sqrt {-i a-i b x+1}}{2 b^3}-\frac {\left (-2 a^2+2 i a+1\right ) \sinh ^{-1}(a+b x)}{2 b^3}+\frac {(-4 a+i) \sqrt {i a+i b x+1} (-i a-i b x+1)^{3/2}}{6 b^3}+\frac {x \sqrt {i a+i b x+1} (-i a-i b x+1)^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.13, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5095, 90, 80, 50, 53, 619, 215} \[ \frac {\left (-2 i a^2-2 a+i\right ) \sqrt {i a+i b x+1} \sqrt {-i a-i b x+1}}{2 b^3}-\frac {\left (-2 a^2+2 i a+1\right ) \sinh ^{-1}(a+b x)}{2 b^3}+\frac {x \sqrt {i a+i b x+1} (-i a-i b x+1)^{3/2}}{3 b^2}+\frac {(-4 a+i) \sqrt {i a+i b x+1} (-i a-i b x+1)^{3/2}}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 80
Rule 90
Rule 215
Rule 619
Rule 5095
Rubi steps
\begin {align*} \int e^{-i \tan ^{-1}(a+b x)} x^2 \, dx &=\int \frac {x^2 \sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx\\ &=\frac {x (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{3 b^2}+\frac {\int \frac {\sqrt {1-i a-i b x} \left (-1-a^2+(i-4 a) b x\right )}{\sqrt {1+i a+i b x}} \, dx}{3 b^2}\\ &=\frac {(i-4 a) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}+\frac {x (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{3 b^2}-\frac {\left (1+2 i a-2 a^2\right ) \int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx}{2 b^2}\\ &=-\frac {\left (2 a-i \left (1-2 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {(i-4 a) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}+\frac {x (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{3 b^2}-\frac {\left (1+2 i a-2 a^2\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 b^2}\\ &=-\frac {\left (2 a-i \left (1-2 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {(i-4 a) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}+\frac {x (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{3 b^2}-\frac {\left (1+2 i a-2 a^2\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{2 b^2}\\ &=-\frac {\left (2 a-i \left (1-2 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {(i-4 a) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}+\frac {x (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{3 b^2}-\frac {\left (1+2 i a-2 a^2\right ) \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^4}\\ &=-\frac {\left (2 a-i \left (1-2 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {(i-4 a) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}+\frac {x (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{3 b^2}-\frac {\left (1+2 i a-2 a^2\right ) \sinh ^{-1}(a+b x)}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 162, normalized size = 0.95 \[ \frac {\sqrt [4]{-1} \left (2 a^2-2 i a-1\right ) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{b^{7/2}}+\frac {i \sqrt {i a+i b x+1} \left (2 i a^3+7 a^2+a (8 b x+5 i)+2 i b^3 x^3-5 b^2 x^2-7 i b x+4\right )}{6 b^3 \sqrt {-i (a+b x+i)}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.45, size = 106, normalized size = 0.62 \[ \frac {-7 i \, a^{3} - 21 \, a^{2} - 12 \, {\left (2 \, a^{2} - 2 i \, a - 1\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-8 i \, b^{2} x^{2} - 4 \, {\left (-2 i \, a - 3\right )} b x - 8 i \, a^{2} - 36 \, a + 16 i\right )} + 9 i \, a}{24 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 117, normalized size = 0.68 \[ -\frac {1}{6} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (\frac {2 \, i x}{b} - \frac {2 \, a b^{3} i + 3 \, b^{3}}{b^{5}}\right )} x + \frac {2 \, a^{2} b^{2} i + 9 \, a b^{2} - 4 \, b^{2} i}{b^{5}}\right )} - \frac {{\left (2 \, a^{2} - 2 \, a i - 1\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, b^{2} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 605, normalized size = 3.54 \[ -\frac {i \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{3}}+\frac {i a x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}+\frac {i a^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{3}}+\frac {i a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{2} \sqrt {b^{2}}}+\frac {x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}+\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a}{2 b^{3}}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}-\frac {i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a^{2}}{b^{3}}+\frac {i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{b^{3}}-\frac {2 \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a}{b^{3}}+\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right ) a^{2}}{b^{2} \sqrt {b^{2}}}-\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{b^{2} \sqrt {b^{2}}}-\frac {2 i \ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right ) a}{b^{2} \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 161, normalized size = 0.94 \[ \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{b^{2}} + \frac {a^{2} \operatorname {arsinh}\left (b x + a\right )}{b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{2 \, b^{2}} - \frac {i \, a \operatorname {arsinh}\left (b x + a\right )}{b^{3}} - \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{3}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{2 \, b^{3}} - \frac {\operatorname {arsinh}\left (b x + a\right )}{2 \, b^{3}} + \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{3}} \]
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 \[ \int \frac {x^2\,\sqrt {{\left (a+b\,x\right )}^2+1}}{1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}} \,d x \]
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 \[ - i \int \frac {x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a + b x - i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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