Optimal. Leaf size=201 \[ -\frac {(-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1} \left (-18 a^2-2 (-6 a+i) b x+10 i a+7\right )}{24 b^4}-\frac {\left (-8 i a^3-12 a^2+12 i a+3\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^4}-\frac {\left (8 a^3-12 i a^2-12 a+3 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}+\frac {x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{4 b^2} \]
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Rubi [A] time = 0.19, antiderivative size = 201, 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, 100, 147, 50, 53, 619, 215} \[ -\frac {(-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1} \left (-18 a^2-2 (-6 a+i) b x+10 i a+7\right )}{24 b^4}-\frac {\left (-8 i a^3-12 a^2+12 i a+3\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^4}-\frac {\left (8 a^3-12 i a^2-12 a+3 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}+\frac {x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 100
Rule 147
Rule 215
Rule 619
Rule 5095
Rubi steps
\begin {align*} \int e^{-i \tan ^{-1}(a+b x)} x^3 \, dx &=\int \frac {x^3 \sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx\\ &=\frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}+\frac {\int \frac {x \sqrt {1-i a-i b x} \left (-2 \left (1+a^2\right )+(i-6 a) b x\right )}{\sqrt {1+i a+i b x}} \, dx}{4 b^2}\\ &=\frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (7+10 i a-18 a^2-2 (i-6 a) b x\right )}{24 b^4}-\frac {\left (3 i-12 a-12 i a^2+8 a^3\right ) \int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx}{8 b^3}\\ &=-\frac {\left (3+12 i a-12 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (7+10 i a-18 a^2-2 (i-6 a) b x\right )}{24 b^4}-\frac {\left (3 i-12 a-12 i a^2+8 a^3\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{8 b^3}\\ &=-\frac {\left (3+12 i a-12 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (7+10 i a-18 a^2-2 (i-6 a) b x\right )}{24 b^4}-\frac {\left (3 i-12 a-12 i a^2+8 a^3\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^3}\\ &=-\frac {\left (3+12 i a-12 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (7+10 i a-18 a^2-2 (i-6 a) b x\right )}{24 b^4}-\frac {\left (3 i-12 a-12 i a^2+8 a^3\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 )}{16 b^5}\\ &=-\frac {\left (3+12 i a-12 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (7+10 i a-18 a^2-2 (i-6 a) b x\right )}{24 b^4}-\frac {\left (3 i-12 a-12 i a^2+8 a^3\right ) \sinh ^{-1}(a+b x)}{8 b^4}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 202, normalized size = 1.00 \[ \frac {(-1)^{3/4} \left (8 i a^3+12 a^2-12 i a-3\right ) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{4 b^{9/2}}+\frac {\sqrt {i a+i b x+1} \left (6 a^4-38 i a^3+5 a^2 (1-6 i b x)+i a \left (18 b^2 x^2+50 i b x-23\right )-6 b^4 x^4-14 i b^3 x^3+17 b^2 x^2+25 i b x-16\right )}{24 b^4 \sqrt {-i (a+b x+i)}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 137, normalized size = 0.68 \[ \frac {45 i \, a^{4} + 224 \, a^{3} - 192 i \, a^{2} + {\left (192 \, a^{3} - 288 i \, a^{2} - 288 \, a + 72 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (-48 i \, b^{3} x^{3} - 16 \, {\left (-3 i \, a - 4\right )} b^{2} x^{2} + 48 i \, a^{3} + {\left (-48 i \, a^{2} - 160 \, a + 72 i\right )} b x + 352 \, a^{2} - 312 i \, a - 128\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 72 \, a}{192 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 163, normalized size = 0.81 \[ -\frac {1}{24} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, {\left (\frac {3 \, i x}{b} - \frac {3 \, a b^{5} i + 4 \, b^{5}}{b^{7}}\right )} x + \frac {6 \, a^{2} b^{4} i + 20 \, a b^{4} - 9 \, b^{4} i}{b^{7}}\right )} x - \frac {6 \, a^{3} b^{3} i + 44 \, a^{2} b^{3} - 39 \, a b^{3} i - 16 \, b^{3}}{b^{7}}\right )} + \frac {{\left (8 \, a^{3} - 12 \, a^{2} i - 12 \, a + 3 \, i\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{3} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 894, normalized size = 4.45 \[ \frac {i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a^{3}}{b^{4}}+\frac {5 i a \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{8 b^{4}}-\frac {3 i a^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{4}}-\frac {3 a x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{3}}-\frac {3 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 )}{2 b^{3} \sqrt {b^{2}}}-\frac {3 i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a}{b^{4}}+\frac {5 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 )}{8 b^{3} \sqrt {b^{2}}}-\frac {i x \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{4 b^{3}}+\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{4}}-\frac {3 i a^{2} \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^{3} \sqrt {b^{2}}}-\frac {\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{b^{4}}+\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^{4}}+\frac {3 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^{2}}{b^{3} \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 ) a^{3}}{b^{3} \sqrt {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^{3} \sqrt {b^{2}}}-\frac {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 )}{b^{3} \sqrt {b^{2}}}-\frac {3 i a^{2} x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{3}}+\frac {5 i x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{8 b^{3}}+\frac {3 i a \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{4 b^{4}}-\frac {3 a^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 308, normalized size = 1.53 \[ -\frac {3 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} x}{2 \, b^{3}} - \frac {a^{3} \operatorname {arsinh}\left (b x + a\right )}{b^{4}} - \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3}}{2 \, b^{4}} - \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, b^{3}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{2 \, b^{3}} + \frac {3 i \, a^{2} \operatorname {arsinh}\left (b x + a\right )}{2 \, b^{4}} + \frac {3 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{4 \, b^{4}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{2 \, b^{4}} + \frac {5 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{8 \, b^{3}} + \frac {3 \, a \operatorname {arsinh}\left (b x + a\right )}{2 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{4}} - \frac {19 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{8 \, b^{4}} - \frac {3 i \, \operatorname {arsinh}\left (b x + a\right )}{8 \, b^{4}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\sqrt {{\left (a+b\,x\right )}^2+1}}{1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}} \,d x \]
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 \[ - i \int \frac {x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a + b x - i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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