Optimal. Leaf size=299 \[ \frac {4 i (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {5 i (1+i a x)^{3/4} \sqrt [4]{1-i a x}}{a}+\frac {5 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2} a}-\frac {5 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2} a}+\frac {5 i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}-\frac {5 i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a} \]
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Rubi [A] time = 0.17, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5061, 47, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac {4 i (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {5 i (1+i a x)^{3/4} \sqrt [4]{1-i a x}}{a}+\frac {5 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2} a}-\frac {5 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2} a}+\frac {5 i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}-\frac {5 i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 204
Rule 211
Rule 240
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 5061
Rubi steps
\begin {align*} \int e^{-\frac {5}{2} i \tan ^{-1}(a x)} \, dx &=\int \frac {(1-i a x)^{5/4}}{(1+i a x)^{5/4}} \, dx\\ &=\frac {4 i (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}-5 \int \frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}} \, dx\\ &=\frac {4 i (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}-\frac {5}{2} \int \frac {1}{(1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=\frac {4 i (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}-\frac {(10 i) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{a}\\ &=\frac {4 i (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}-\frac {(10 i) \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 )}{a}\\ &=\frac {4 i (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}-\frac {(5 i) \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 )}{a}-\frac {(5 i) \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 )}{a}\\ &=\frac {4 i (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}-\frac {(5 i) \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 )}{2 a}-\frac {(5 i) \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 )}{2 a}+\frac {(5 i) \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 )}{2 \sqrt {2} a}+\frac {(5 i) \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 )}{2 \sqrt {2} a}\\ &=\frac {4 i (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}+\frac {5 i \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 )}{2 \sqrt {2} a}-\frac {5 i \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 )}{2 \sqrt {2} a}-\frac {(5 i) \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 )}{\sqrt {2} a}+\frac {(5 i) \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 )}{\sqrt {2} a}\\ &=\frac {4 i (1-i a x)^{5/4}}{a \sqrt [4]{1+i a x}}+\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}+\frac {5 i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}-\frac {5 i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {5 i \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 )}{2 \sqrt {2} a}-\frac {5 i \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 )}{2 \sqrt {2} a}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 39, normalized size = 0.13 \[ \frac {8 i e^{-\frac {1}{2} i \tan ^{-1}(a x)} \, _2F_1\left (-\frac {1}{4},2;\frac {3}{4};-e^{2 i \tan ^{-1}(a x)}\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 262, normalized size = 0.88 \[ -\frac {{\left (a^{2} x - i \, a\right )} \sqrt {\frac {25 i}{a^{2}}} \log \left (\frac {1}{5} \, a \sqrt {\frac {25 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - {\left (a^{2} x - i \, a\right )} \sqrt {\frac {25 i}{a^{2}}} \log \left (-\frac {1}{5} \, a \sqrt {\frac {25 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - {\left (a^{2} x - i \, a\right )} \sqrt {-\frac {25 i}{a^{2}}} \log \left (\frac {1}{5} \, a \sqrt {-\frac {25 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + {\left (a^{2} x - i \, a\right )} \sqrt {-\frac {25 i}{a^{2}}} \log \left (-\frac {1}{5} \, a \sqrt {-\frac {25 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 2 \, \sqrt {a^{2} x^{2} + 1} {\left (-i \, a x - 9\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{2 \, a^{2} x - 2 i \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {i a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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