Optimal. Leaf size=373 \[ -\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4} (452 a x+521 i)}{96 a^4}+\frac {475 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac {475 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}+\frac {475 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}-\frac {475 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}-\frac {17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}+\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {5062, 97, 153, 147, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac {17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4} (452 a x+521 i)}{96 a^4}+\frac {475 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac {475 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}+\frac {475 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}-\frac {475 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 97
Rule 147
Rule 153
Rule 204
Rule 211
Rule 240
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 5062
Rubi steps
\begin {align*} \int e^{-\frac {5}{2} i \tan ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (1-i a x)^{5/4}}{(1+i a x)^{5/4}} \, dx\\ &=\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}-\frac {(4 i) \int \frac {x^2 \sqrt [4]{1-i a x} \left (3-\frac {17 i a x}{4}\right )}{\sqrt [4]{1+i a x}} \, dx}{a}\\ &=\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}-\frac {17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {i \int \frac {x \sqrt [4]{1-i a x} \left (\frac {17 i a}{2}+\frac {113 a^2 x}{8}\right )}{\sqrt [4]{1+i a x}} \, dx}{a^3}\\ &=\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}-\frac {17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4} (521 i+452 a x)}{96 a^4}+\frac {(475 i) \int \frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}} \, dx}{64 a^3}\\ &=\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {475 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}-\frac {17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4} (521 i+452 a x)}{96 a^4}+\frac {(475 i) \int \frac {1}{(1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{128 a^3}\\ &=\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {475 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}-\frac {17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4} (521 i+452 a x)}{96 a^4}-\frac {475 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{32 a^4}\\ &=\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {475 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}-\frac {17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4} (521 i+452 a x)}{96 a^4}-\frac {475 \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{32 a^4}\\ &=\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {475 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}-\frac {17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4} (521 i+452 a x)}{96 a^4}-\frac {475 \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 a^4}-\frac {475 \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 a^4}\\ &=\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {475 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}-\frac {17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4} (521 i+452 a x)}{96 a^4}-\frac {475 \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 a^4}-\frac {475 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}\\ &=\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {475 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}-\frac {17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4} (521 i+452 a x)}{96 a^4}+\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}-\frac {475 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}\\ &=\frac {4 i x^3 (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {475 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}-\frac {17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}-\frac {i (1-i a x)^{5/4} (1+i a x)^{3/4} (521 i+452 a x)}{96 a^4}+\frac {475 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}-\frac {475 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.05, size = 100, normalized size = 0.27 \[ -\frac {\sqrt [4]{1-i a x} (a x+i)^2 \left (3 \left (6 a^2 x^2+5 i a x+59\right )-95\ 2^{3/4} \sqrt [4]{1+i a x} \, _2F_1\left (\frac {1}{4},\frac {9}{4};\frac {13}{4};\frac {1}{2} (1-i a x)\right )\right )}{72 a^4 \sqrt [4]{1+i a x}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 306, normalized size = 0.82 \[ \frac {{\left (96 \, a^{5} x - 96 i \, a^{4}\right )} \sqrt {\frac {225625 i}{4096 \, a^{8}}} \log \left (\frac {64}{475} i \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - {\left (96 \, a^{5} x - 96 i \, a^{4}\right )} \sqrt {\frac {225625 i}{4096 \, a^{8}}} \log \left (-\frac {64}{475} i \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - {\left (96 \, a^{5} x - 96 i \, a^{4}\right )} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} \log \left (\frac {64}{475} i \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + {\left (96 \, a^{5} x - 96 i \, a^{4}\right )} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} \log \left (-\frac {64}{475} i \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + {\left (48 i \, a^{4} x^{4} - 136 \, a^{3} x^{3} - 226 i \, a^{2} x^{2} + 521 \, a x - 2467 i\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{192 \, a^{5} x - 192 i \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3}{{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________