Optimal. Leaf size=324 \[ -\frac{i \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2} \left (2 \left (-52 i a^2+118 a+61 i\right ) b x+112 i a^3-422 a^2-458 i a+163\right )}{40 b^5}-\frac{3 \left (8 i a^4-48 a^3-88 i a^2+68 a+19 i\right ) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^5}-\frac{3 \left (8 a^4+48 i a^3-88 a^2-68 i a+19\right ) \sinh ^{-1}(a+b x)}{8 b^5}-\frac{11 x^3 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{5 b^2}+\frac{3 (16 a+17 i) x^2 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{20 b^3}-\frac{2 i x^4 (i a+i b x+1)^{3/2}}{b \sqrt{-i a-i b x+1}} \]
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Rubi [A] time = 0.26693, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5095, 97, 153, 147, 50, 53, 619, 215} \[ -\frac{i \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2} \left (2 \left (-52 i a^2+118 a+61 i\right ) b x+112 i a^3-422 a^2-458 i a+163\right )}{40 b^5}-\frac{3 \left (8 i a^4-48 a^3-88 i a^2+68 a+19 i\right ) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^5}-\frac{3 \left (8 a^4+48 i a^3-88 a^2-68 i a+19\right ) \sinh ^{-1}(a+b x)}{8 b^5}-\frac{11 x^3 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{5 b^2}+\frac{3 (16 a+17 i) x^2 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{20 b^3}-\frac{2 i x^4 (i a+i b x+1)^{3/2}}{b \sqrt{-i a-i b x+1}} \]
Antiderivative was successfully verified.
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Rule 5095
Rule 97
Rule 153
Rule 147
Rule 50
Rule 53
Rule 619
Rule 215
Rubi steps
\begin{align*} \int e^{3 i \tan ^{-1}(a+b x)} x^4 \, dx &=\int \frac{x^4 (1+i a+i b x)^{3/2}}{(1-i a-i b x)^{3/2}} \, dx\\ &=-\frac{2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt{1-i a-i b x}}+\frac{(2 i) \int \frac{x^3 \sqrt{1+i a+i b x} \left (4 (1+i a)+\frac{11 i b x}{2}\right )}{\sqrt{1-i a-i b x}} \, dx}{b}\\ &=-\frac{2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt{1-i a-i b x}}-\frac{11 x^3 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac{(2 i) \int \frac{x^2 \sqrt{1+i a+i b x} \left (-\frac{33}{2} (i-a) (1-i a) b+\frac{3}{2} (17-16 i a) b^2 x\right )}{\sqrt{1-i a-i b x}} \, dx}{5 b^3}\\ &=-\frac{2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt{1-i a-i b x}}+\frac{3 (17 i+16 a) x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac{11 x^3 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac{i \int \frac{x \sqrt{1+i a+i b x} \left (3 (1+i a) (i+a) (17 i+16 a) b^2-\frac{3}{2} \left (118 a+i \left (61-52 a^2\right )\right ) b^3 x\right )}{\sqrt{1-i a-i b x}} \, dx}{10 b^5}\\ &=-\frac{2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt{1-i a-i b x}}+\frac{3 (17 i+16 a) x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac{11 x^3 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}-\frac{i \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2} \left (163-458 i a-422 a^2+112 i a^3+2 \left (61 i+118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac{\left (3 \left (19-68 i a-88 a^2+48 i a^3+8 a^4\right )\right ) \int \frac{\sqrt{1+i a+i b x}}{\sqrt{1-i a-i b x}} \, dx}{8 b^4}\\ &=-\frac{3 \left (19 i+68 a-88 i a^2-48 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{2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt{1-i a-i b x}}+\frac{3 (17 i+16 a) x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac{11 x^3 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}-\frac{i \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2} \left (163-458 i a-422 a^2+112 i a^3+2 \left (61 i+118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac{\left (3 \left (19-68 i a-88 a^2+48 i a^3+8 a^4\right )\right ) \int \frac{1}{\sqrt{1-i a-i b x} \sqrt{1+i a+i b x}} \, dx}{8 b^4}\\ &=-\frac{3 \left (19 i+68 a-88 i a^2-48 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{2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt{1-i a-i b x}}+\frac{3 (17 i+16 a) x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac{11 x^3 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}-\frac{i \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2} \left (163-458 i a-422 a^2+112 i a^3+2 \left (61 i+118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac{\left (3 \left (19-68 i a-88 a^2+48 i a^3+8 a^4\right )\right ) \int \frac{1}{\sqrt{(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^4}\\ &=-\frac{3 \left (19 i+68 a-88 i a^2-48 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{2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt{1-i a-i b x}}+\frac{3 (17 i+16 a) x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac{11 x^3 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}-\frac{i \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2} \left (163-458 i a-422 a^2+112 i a^3+2 \left (61 i+118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac{\left (3 \left (19-68 i a-88 a^2+48 i a^3+8 a^4\right )\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{3 \left (19 i+68 a-88 i a^2-48 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{2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt{1-i a-i b x}}+\frac{3 (17 i+16 a) x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac{11 x^3 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}-\frac{i \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2} \left (163-458 i a-422 a^2+112 i a^3+2 \left (61 i+118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac{3 \left (19-68 i a-88 a^2+48 i a^3+8 a^4\right ) \sinh ^{-1}(a+b x)}{8 b^5}\\ \end{align*}
Mathematica [A] time = 0.447303, size = 249, normalized size = 0.77 \[ \frac{3 (-1)^{3/4} \left (8 a^4+48 i a^3-88 a^2-68 i a+19\right ) \sinh ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right )}{4 \sqrt{-i b} b^{9/2}}-\frac{\sqrt{i a+i b x+1} \left (-i a^2 \left (52 b^2 x^2-422 i b x+2599\right )+14 i a^3 (8 b x+121 i)+8 a^5+418 i a^4+a \left (32 i b^3 x^3+118 b^2 x^2-458 i b x+1763\right )+8 b^5 x^5-22 i b^4 x^4-34 b^3 x^3+61 i b^2 x^2+163 b x+448 i\right )}{40 b^5 \sqrt{-i (a+b x+i)}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.131, size = 933, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05581, size = 822, normalized size = 2.54 \begin{align*} \frac{-62 i \, a^{6} + 2687 \, a^{5} + 11575 i \, a^{4} - 20350 \, a^{3} +{\left (-62 i \, a^{5} + 2625 \, a^{4} + 8950 i \, a^{3} - 11400 \, a^{2} - 6340 i \, a + 1280\right )} b x - 17740 i \, a^{2} +{\left (960 \, a^{5} + 6720 i \, a^{4} - 16320 \, a^{3} +{\left (960 \, a^{4} + 5760 i \, a^{3} - 10560 \, a^{2} - 8160 i \, a + 2280\right )} b x - 18720 i \, a^{2} + 10440 \, a + 2280 i\right )} \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) +{\left (-64 i \, b^{5} x^{5} - 176 \, b^{4} x^{4} +{\left (256 \, a + 272 i\right )} b^{3} x^{3} - 64 i \, a^{5} - 8 \,{\left (52 \, a^{2} + 118 i \, a - 61\right )} b^{2} x^{2} + 3344 \, a^{4} + 13552 i \, a^{3} +{\left (896 \, a^{3} + 3376 i \, a^{2} - 3664 \, a - 1304 i\right )} b x - 20792 \, a^{2} - 14104 i \, a + 3584\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 7620 \, a + 1280 i}{320 \, b^{6} x +{\left (320 \, a + 320 i\right )} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16915, size = 525, normalized size = 1.62 \begin{align*} -\frac{1}{40} \, \sqrt{{\left (b x + a\right )}^{2} + 1}{\left ({\left (2 \,{\left ({\left (\frac{4 \, i x}{b} - \frac{4 \, a b^{17} i - 15 \, b^{17}}{b^{19}}\right )} x + \frac{4 \, a^{2} b^{16} i - 35 \, a b^{16} - 32 \, b^{16} i}{b^{19}}\right )} x - \frac{8 \, a^{3} b^{15} i - 130 \, a^{2} b^{15} - 252 \, a b^{15} i + 125 \, b^{15}}{b^{19}}\right )} x + \frac{8 \, a^{4} b^{14} i - 250 \, a^{3} b^{14} - 804 \, a^{2} b^{14} i + 835 \, a b^{14} + 288 \, b^{14} i}{b^{19}}\right )} + \frac{{\left (8 \, a^{4} + 48 \, a^{3} i - 88 \, a^{2} - 68 \, a i + 19\right )} \log \left (3 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + 2 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}^{2} b i + 2 \, a^{2} b i +{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}^{3}{\left | b \right |} + 3 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )} a^{2}{\left | b \right |} + 4 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )} a i{\left | b \right |} - 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 |}} + \frac{{\left (8 \, a^{4}{\left | b \right |} + 48 \, a^{3} i{\left | b \right |} - 88 \, a^{2}{\left | b \right |} - 68 \, a i{\left | b \right |} + 19 \,{\left | b \right |}\right )} \log \left (96 \, b^{5}\right )}{4 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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