Optimal. Leaf size=249 \[ -\frac {i (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1} \left (-22 i a^2+2 (11+10 i a) b x-54 a+29 i\right )}{8 b^4}+\frac {3 \left (-8 i a^3-36 a^2+44 i a+17\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^4}+\frac {3 \left (8 a^3-36 i a^2-44 a+17 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}-\frac {9 x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{4 b^2}+\frac {2 i x^3 (-i a-i b x+1)^{3/2}}{b \sqrt {i a+i b x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 249, 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, 97, 153, 147, 50, 53, 619, 215} \[ -\frac {i (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1} \left (-22 i a^2+2 (11+10 i a) b x-54 a+29 i\right )}{8 b^4}+\frac {3 \left (-8 i a^3-36 a^2+44 i a+17\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^4}+\frac {3 \left (8 a^3-36 i a^2-44 a+17 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}-\frac {9 x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{4 b^2}+\frac {2 i x^3 (-i a-i b x+1)^{3/2}}{b \sqrt {i a+i b x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 53
Rule 97
Rule 147
Rule 153
Rule 215
Rule 619
Rule 5095
Rubi steps
\begin {align*} \int e^{-3 i \tan ^{-1}(a+b x)} x^3 \, dx &=\int \frac {x^3 (1-i a-i b x)^{3/2}}{(1+i a+i b x)^{3/2}} \, dx\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}-\frac {(2 i) \int \frac {x^2 \sqrt {1-i a-i b x} \left (3 (1-i a)-\frac {9 i b x}{2}\right )}{\sqrt {1+i a+i b x}} \, dx}{b}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}-\frac {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i \int \frac {x \sqrt {1-i a-i b x} \left (9 i \left (1+a^2\right ) b+\frac {3}{2} (11+10 i a) b^2 x\right )}{\sqrt {1+i a+i b x}} \, dx}{2 b^3}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}-\frac {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac {\left (3 \left (17 i-44 a-36 i a^2+8 a^3\right )\right ) \int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx}{8 b^3}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (17+44 i a-36 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 {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac {\left (3 \left (17 i-44 a-36 i a^2+8 a^3\right )\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{8 b^3}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (17+44 i a-36 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 {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac {\left (3 \left (17 i-44 a-36 i a^2+8 a^3\right )\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^3}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (17+44 i a-36 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 {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac {\left (3 \left (17 i-44 a-36 i a^2+8 a^3\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^5}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (17+44 i a-36 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 {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac {3 \left (17 i-44 a-36 i a^2+8 a^3\right ) \sinh ^{-1}(a+b x)}{8 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.39, size = 244, normalized size = 0.98 \[ \frac {3 \sqrt [4]{-1} \left (8 a^3-36 i a^2-44 a+17 i\right ) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{4 b^{9/2}}+\frac {-2 i a^5+a^4 (-76-2 i b x)-5 a^3 (20 b x-31 i)+a^2 \left (-12 b^2 x^2+265 i b x+4\right )+a \left (2 i b^4 x^4+4 b^3 x^3+53 i b^2 x^2+212 b x+157 i\right )+2 i b^5 x^5-8 b^4 x^4-17 i b^3 x^3+40 b^2 x^2-51 i b x+80}{8 b^4 \sqrt {a^2+2 a b x+b^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 217, normalized size = 0.87 \[ \frac {-15 i \, a^{5} - 495 \, a^{4} + 1664 i \, a^{3} + {\left (-15 i \, a^{4} - 480 \, a^{3} + 1184 i \, a^{2} + 968 \, a - 256 i\right )} b x + 2152 \, a^{2} - {\left (192 \, a^{4} - 1056 i \, a^{3} + {\left (192 \, a^{3} - 864 i \, a^{2} - 1056 \, a + 408 i\right )} b x - 1920 \, a^{2} + 1464 i \, a + 408\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (16 i \, b^{4} x^{4} - 48 \, b^{3} x^{3} + {\left (80 \, a - 88 i\right )} b^{2} x^{2} - 16 i \, a^{4} - 624 \, a^{3} - 8 \, {\left (22 \, a^{2} - 54 i \, a - 29\right )} b x + 1864 i \, a^{2} + 1896 \, a - 640 i\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 1224 i \, a - 256}{64 \, b^{5} x + {\left (64 \, a - 64 i\right )} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 303, normalized size = 1.22 \[ -\frac {1}{8} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, x {\left (\frac {x}{b i} - \frac {a b^{11} - 4 \, b^{11} i}{b^{13} i}\right )} + \frac {2 \, a^{2} b^{10} - 20 \, a b^{10} i - 19 \, b^{10}}{b^{13} i}\right )} x - \frac {2 \, a^{3} b^{9} - 44 \, a^{2} b^{9} i - 93 \, a b^{9} + 48 \, b^{9} i}{b^{13} i}\right )} - \frac {{\left (8 \, a^{3} - 36 \, a^{2} i - 44 \, a + 17 \, i\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b - 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b i - 2 \, a^{2} b i + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} - 4 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a i {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{3} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.19, size = 1529, normalized size = 6.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.47, size = 979, normalized size = 3.93 \[ -\frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{3}}{b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4} - 2 i \, b^{5} x - 2 i \, a b^{4} - b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4} - 2 i \, b^{5} x - 2 i \, a b^{4} - b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{2 i \, b^{5} x + 2 i \, a b^{4} + 2 \, b^{4}} - \frac {6 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3}}{i \, b^{5} x + i \, a b^{4} + b^{4}} + \frac {3 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4} - 2 i \, b^{5} x - 2 i \, a b^{4} - b^{4}} + \frac {6 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{2 i \, b^{5} x + 2 i \, a b^{4} + 2 \, b^{4}} - \frac {18 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{i \, b^{5} x + i \, a b^{4} + b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4} - 2 i \, b^{5} x - 2 i \, a b^{4} - b^{4}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{2 i \, b^{5} x + 2 i \, a b^{4} + 2 \, b^{4}} + \frac {18 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{i \, b^{5} x + i \, a b^{4} + b^{4}} + \frac {3 \, a^{3} \operatorname {arsinh}\left (b x + a\right )}{b^{4}} + \frac {6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{i \, b^{5} x + i \, a b^{4} + 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} + 4 i \, b x + 4 i \, a + 3} a x}{2 \, b^{3}} - \frac {27 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 {9 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{2 \, b^{4}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3} a^{2}}{2 \, b^{4}} + \frac {3 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{8 \, b^{3}} - \frac {3 i \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3} x}{2 \, b^{3}} - \frac {3 \, a \arcsin \left (i \, b x + i \, a + 2\right )}{2 \, b^{4}} - \frac {18 \, a \operatorname {arsinh}\left (b x + a\right )}{b^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{b^{4}} + \frac {75 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{8 \, b^{4}} - \frac {9 i \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3} a}{2 \, b^{4}} + \frac {3 i \, \arcsin \left (i \, b x + i \, a + 2\right )}{2 \, b^{4}} + \frac {63 i \, \operatorname {arsinh}\left (b x + a\right )}{8 \, b^{4}} + \frac {9 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{2 \, b^{4}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________