Optimal. Leaf size=94 \[ \frac {2 i (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}-\frac {3 \sinh ^{-1}(a+b x)}{b} \]
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Rubi [A]
time = 0.03, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5201, 49, 52,
55, 633, 221} \begin {gather*} \frac {2 i (-i a-i b x+1)^{3/2}}{b \sqrt {i a+i b x+1}}+\frac {3 i \sqrt {i a+i b x+1} \sqrt {-i a-i b x+1}}{b}-\frac {3 \sinh ^{-1}(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 55
Rule 221
Rule 633
Rule 5201
Rubi steps
\begin {align*} \int e^{-3 i \tan ^{-1}(a+b x)} \, dx &=\int \frac {(1-i a-i b x)^{3/2}}{(1+i a+i b x)^{3/2}} \, dx\\ &=\frac {2 i (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}-3 \int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx\\ &=\frac {2 i (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}-3 \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx\\ &=\frac {2 i (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}-3 \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx\\ &=\frac {2 i (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}-\frac {3 \text {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^2}\\ &=\frac {2 i (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}-\frac {3 \sinh ^{-1}(a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 45, normalized size = 0.48 \begin {gather*} \frac {\sqrt {1+(a+b x)^2} \left (i+\frac {4}{-i+a+b x}\right )}{b}-\frac {3 \sinh ^{-1}(a+b x)}{b} \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. 326 vs. \(2 (76 ) = 152\).
time = 0.14, size = 327, normalized size = 3.48
method | result | size |
risch | \(\frac {i \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b}-\frac {3 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {4 \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{b^{2} \left (x -\frac {i-a}{b}\right )}\) | \(129\) |
default | \(\frac {i \left (\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{3}}-2 i b \left (-\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{2}}+3 i b \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )\right )}{b^{3}}\) | \(327\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 103, normalized size = 1.10 \begin {gather*} \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b - 2 i \, b^{2} x - 2 i \, a b - b} - \frac {3 \, \operatorname {arsinh}\left (b x + a\right )}{b} + \frac {6 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{i \, b^{2} x + i \, a b + b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.16, size = 99, normalized size = 1.05 \begin {gather*} \frac {{\left (i \, a + 8\right )} b x + i \, a^{2} + 6 \, {\left (b x + a - i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-i \, b x - i \, a - 5\right )} + 9 \, a - 8 i}{2 \, {\left (b^{2} x + {\left (a - i\right )} b\right )}} \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 \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} + 3 a^{2} b x - 3 i a^{2} + 3 a b^{2} x^{2} - 6 i a b x - 3 a + b^{3} x^{3} - 3 i b^{2} x^{2} - 3 b x + i}\, dx + \int \frac {a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} + 3 a^{2} b x - 3 i a^{2} + 3 a b^{2} x^{2} - 6 i a b x - 3 a + b^{3} x^{3} - 3 i b^{2} x^{2} - 3 b x + i}\, dx + \int \frac {b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} + 3 a^{2} b x - 3 i a^{2} + 3 a b^{2} x^{2} - 6 i a b x - 3 a + b^{3} x^{3} - 3 i b^{2} x^{2} - 3 b x + i}\, dx + \int \frac {2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} + 3 a^{2} b x - 3 i a^{2} + 3 a b^{2} x^{2} - 6 i a b x - 3 a + b^{3} x^{3} - 3 i b^{2} x^{2} - 3 b x + i}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 182 vs. \(2 (66) = 132\).
time = 0.49, size = 182, normalized size = 1.94 \begin {gather*} \frac {\log \left (3 i \, {\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} {\left | b \right |} - 3 \, {\left (-i \, x {\left | b \right |} + i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} + 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b + 2 \, a^{2} b + 4 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - i \, a b + {\left (-i \, x {\left | b \right |} + i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{{\left | b \right |}} + \frac {i \, \sqrt {{\left (b x + a\right )}^{2} + 1}}{b} \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 {{\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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