Optimal. Leaf size=276 \[ -\frac {(-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1} \left (96 a^3+2 \left (-36 a^2+14 i a+13\right ) b x-86 i a^2-114 a+19 i\right )}{120 b^5}-\frac {\left (8 i a^4+16 a^3-24 i a^2-12 a+3 i\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^5}+\frac {\left (8 a^4-16 i a^3-24 a^2+12 i a+3\right ) \sinh ^{-1}(a+b x)}{8 b^5}+\frac {(-8 a+i) x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{20 b^3}+\frac {x^3 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{5 b^2} \]
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Rubi [A] time = 0.22, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5095, 100, 153, 147, 50, 53, 619, 215} \[ -\frac {(-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1} \left (2 \left (-36 a^2+14 i a+13\right ) b x+96 a^3-86 i a^2-114 a+19 i\right )}{120 b^5}-\frac {\left (8 i a^4+16 a^3-24 i a^2-12 a+3 i\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^5}+\frac {\left (8 a^4-16 i a^3-24 a^2+12 i a+3\right ) \sinh ^{-1}(a+b x)}{8 b^5}+\frac {x^3 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{5 b^2}+\frac {(-8 a+i) x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{20 b^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 100
Rule 147
Rule 153
Rule 215
Rule 619
Rule 5095
Rubi steps
\begin {align*} \int e^{-i \tan ^{-1}(a+b x)} x^4 \, dx &=\int \frac {x^4 \sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx\\ &=\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}+\frac {\int \frac {x^2 \sqrt {1-i a-i b x} \left (-3 \left (1+a^2\right )+(i-8 a) b x\right )}{\sqrt {1+i a+i b x}} \, dx}{5 b^2}\\ &=\frac {(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}+\frac {\int \frac {x \sqrt {1-i a-i b x} \left (2 (i-8 a) (i-a) (i+a) b-\left (13+14 i a-36 a^2\right ) b^2 x\right )}{\sqrt {1+i a+i b x}} \, dx}{20 b^4}\\ &=\frac {(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (19 i-114 a-86 i a^2+96 a^3+2 \left (13+14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3+12 i a-24 a^2-16 i a^3+8 a^4\right ) \int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx}{8 b^4}\\ &=-\frac {\left (3 i-12 a-24 i a^2+16 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}+\frac {(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (19 i-114 a-86 i a^2+96 a^3+2 \left (13+14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3+12 i a-24 a^2-16 i a^3+8 a^4\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{8 b^4}\\ &=-\frac {\left (3 i-12 a-24 i a^2+16 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}+\frac {(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (19 i-114 a-86 i a^2+96 a^3+2 \left (13+14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3+12 i a-24 a^2-16 i a^3+8 a^4\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^4}\\ &=-\frac {\left (3 i-12 a-24 i a^2+16 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}+\frac {(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (19 i-114 a-86 i a^2+96 a^3+2 \left (13+14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3+12 i a-24 a^2-16 i a^3+8 a^4\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^6}\\ &=-\frac {\left (3 i-12 a-24 i a^2+16 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}+\frac {(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (19 i-114 a-86 i a^2+96 a^3+2 \left (13+14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3+12 i a-24 a^2-16 i a^3+8 a^4\right ) \sinh ^{-1}(a+b x)}{8 b^5}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 248, normalized size = 0.90 \[ \frac {\sqrt [4]{-1} \left (-8 i a^4-16 a^3+24 i a^2+12 a-3 i\right ) \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{4 \sqrt {-i b} b^{9/2}}+\frac {i \sqrt {i a+i b x+1} \left (24 i a^5+226 a^4+2 a^3 (72 b x-41 i)+a^2 \left (-84 b^2 x^2-346 i b x+57\right )+a \left (64 b^3 x^3+154 i b^2 x^2-346 b x-211 i\right )+24 i b^5 x^5-54 b^4 x^4-62 i b^3 x^3+77 b^2 x^2+109 i b x-64\right )}{120 b^5 \sqrt {-i (a+b x+i)}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 175, normalized size = 0.63 \[ \frac {-186 i \, a^{5} - 1345 \, a^{4} + 1730 i \, a^{3} + 1320 \, a^{2} - {\left (960 \, a^{4} - 1920 i \, a^{3} - 2880 \, a^{2} + 1440 i \, a + 360\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (-192 i \, b^{4} x^{4} - 48 \, {\left (-4 i \, a - 5\right )} b^{3} x^{3} + {\left (-192 i \, a^{2} - 560 \, a + 256 i\right )} b^{2} x^{2} - 192 i \, a^{4} - 2000 \, a^{3} + {\left (192 i \, a^{3} + 1040 \, a^{2} - 928 i \, a - 360\right )} b x + 2656 i \, a^{2} + 2200 \, a - 512 i\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 300 i \, a}{960 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 214, normalized size = 0.78 \[ -\frac {1}{120} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, {\left (3 \, {\left (\frac {4 \, i x}{b} - \frac {4 \, a b^{7} i + 5 \, b^{7}}{b^{9}}\right )} x + \frac {12 \, a^{2} b^{6} i + 35 \, a b^{6} - 16 \, b^{6} i}{b^{9}}\right )} x - \frac {24 \, a^{3} b^{5} i + 130 \, a^{2} b^{5} - 116 \, a b^{5} i - 45 \, b^{5}}{b^{9}}\right )} x + \frac {24 \, a^{4} b^{4} i + 250 \, a^{3} b^{4} - 332 \, a^{2} b^{4} i - 275 \, a b^{4} + 64 \, b^{4} i}{b^{9}}\right )} - \frac {{\left (8 \, a^{4} - 16 \, a^{3} i - 24 \, a^{2} + 12 \, a i + 3\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{4} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 1208, normalized size = 4.38 \[ \frac {3 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 )}{b^{4} \sqrt {b^{2}}}+\frac {3 a^{2} x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{4}}-\frac {i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a^{4}}{b^{5}}+\frac {6 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^{5}}+\frac {2 i a^{3} x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{4}}-\frac {5 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 )}{2 b^{4} \sqrt {b^{2}}}-\frac {5 i a x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{4}}-\frac {4 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^{3}}{b^{4} \sqrt {b^{2}}}+\frac {4 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^{4} \sqrt {b^{2}}}+\frac {3 i a x \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{5 b^{4}}-\frac {5 i a^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{5}}-\frac {6 i a^{2} \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{5 b^{5}}-\frac {i x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{5 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^{4}}{b^{4} \sqrt {b^{2}}}-\frac {6 \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^{4} \sqrt {b^{2}}}+\frac {2 i a^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{5}}+\frac {2 i a^{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 )}{b^{4} \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 )}\, a^{3}}{b^{5}}+\frac {4 \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a}{b^{5}}+\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^{4} \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 )}}{b^{5}}+\frac {7 i \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{15 b^{5}}+\frac {x \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{4 b^{4}}-\frac {13 a \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{12 b^{5}}+\frac {3 a^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{5}}-\frac {5 x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{8 b^{4}}-\frac {5 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a}{8 b^{5}}-\frac {5 \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^{4} \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 456, normalized size = 1.65 \[ \frac {2 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3} x}{b^{4}} - \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{5 \, b^{3}} + \frac {a^{4} \operatorname {arsinh}\left (b x + a\right )}{b^{5}} + \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{4}}{b^{5}} + \frac {3 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a x}{5 \, b^{4}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} x}{b^{4}} - \frac {2 i \, a^{3} \operatorname {arsinh}\left (b x + a\right )}{b^{5}} - \frac {6 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{5 \, b^{5}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3}}{b^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, b^{4}} - \frac {5 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{2 \, b^{4}} - \frac {3 \, a^{2} \operatorname {arsinh}\left (b x + a\right )}{b^{5}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{12 \, b^{5}} + \frac {7 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{2 \, b^{5}} - \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{8 \, b^{4}} + \frac {3 i \, a \operatorname {arsinh}\left (b x + a\right )}{2 \, b^{5}} + \frac {7 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{15 \, b^{5}} + \frac {27 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{8 \, b^{5}} + \frac {3 \, \operatorname {arsinh}\left (b x + a\right )}{8 \, b^{5}} - \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{5}} \]
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 \[ \int \frac {x^4\,\sqrt {{\left (a+b\,x\right )}^2+1}}{1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}} \,d x \]
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 \[ - i \int \frac {x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a + b x - i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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