Optimal. Leaf size=171 \[ -\frac {\left (i+2 a-2 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}-\frac {(i+4 a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}+\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{3 b^2}-\frac {\left (1-2 i a-2 a^2\right ) \sinh ^{-1}(a+b x)}{2 b^3} \]
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Rubi [A]
time = 0.09, 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 = {5203, 92, 81,
52, 55, 633, 221} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 55
Rule 81
Rule 92
Rule 221
Rule 633
Rule 5203
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 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{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) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}+\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{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 (i+2 a-2 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}-\frac {(i+4 a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}+\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{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 (i+2 a-2 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}-\frac {(i+4 a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}+\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{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 (i+2 a-2 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}-\frac {(i+4 a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}+\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{3 b^2}-\frac {\left (1-2 i a-2 a^2\right ) \text {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 (i+2 a-2 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}-\frac {(i+4 a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}+\frac {x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{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 = 0.12, size = 135, normalized size = 0.79 \begin {gather*} \frac {\sqrt {1+a^2+2 a b x+b^2 x^2} \left (-4 i+2 i a^2+3 b x+2 i b^2 x^2+a (-9-2 i b x)\right )}{6 b^3}+\frac {\sqrt [4]{-1} \left (-1+2 i a+2 a^2\right ) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{b^{7/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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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. 435 vs. \(2 (135 ) = 270\).
time = 0.09, size = 436, normalized size = 2.55
method | result | size |
risch | \(\frac {i \left (2 b^{2} x^{2}-2 a b x -3 i b x +2 a^{2}+9 i a -4\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{6 b^{3}}+\frac {i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a}{b^{2} \sqrt {b^{2}}}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a^{2}}{b^{2} \sqrt {b^{2}}}-\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}\) | \(198\) |
default | \(i b \left (\frac {x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 b^{2}}-\frac {5 a \left (\frac {x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {3 a \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{2 b}-\frac {\left (a^{2}+1\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{3 b}-\frac {2 \left (a^{2}+1\right ) \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{3 b^{2}}\right )+\left (i a +1\right ) \left (\frac {x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {3 a \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{2 b}-\frac {\left (a^{2}+1\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}\right )\) | \(436\) |
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. 351 vs. \(2 (119) = 238\).
time = 0.26, size = 351, normalized size = 2.05 \begin {gather*} \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x^{2}}{3 \, b} - \frac {5 i \, a^{3} \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^{3}} + \frac {3 \, a^{2} {\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 )}{2 \, b^{3}} - \frac {5 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{6 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-i \, a - 1\right )} x}{2 \, b^{2}} + \frac {3 i \, {\left (a^{2} + 1\right )} a \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^{3}} - \frac {{\left (a^{2} + 1\right )} {\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 )}{2 \, b^{3}} + \frac {5 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{2 \, b^{3}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a {\left (i \, a + 1\right )}}{2 \, b^{3}} - \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (i \, a^{2} + i\right )}}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.37, size = 106, normalized size = 0.62 \begin {gather*} \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 ) - 4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-2 i \, b^{2} x^{2} + {\left (2 i \, a - 3\right )} b x - 2 i \, a^{2} + 9 \, a + 4 i\right )} - 9 i \, a}{24 \, b^{3}} \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 \left (- \frac {i x^{2}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a x^{2}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {b x^{3}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 113, normalized size = 0.66 \begin {gather*} -\frac {1}{6} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (x {\left (-\frac {2 i \, x}{b} - \frac {-2 i \, a b^{3} + 3 \, b^{3}}{b^{5}}\right )} - \frac {2 i \, a^{2} b^{2} - 9 \, a b^{2} - 4 i \, b^{2}}{b^{5}}\right )} - \frac {{\left (2 \, a^{2} + 2 i \, a - 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 |}} \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 {x^2\,\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}{\sqrt {{\left (a+b\,x\right )}^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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