Optimal. Leaf size=45 \[ i e^{\text {ArcCos}(a x)}-2 i e^{\text {ArcCos}(a x)} \, _2F_1\left (-\frac {i}{2},1;1-\frac {i}{2};-e^{2 i \text {ArcCos}(a x)}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4921, 12, 4527,
2225, 2283} \begin {gather*} i e^{\text {ArcCos}(a x)}-2 i e^{\text {ArcCos}(a x)} \, _2F_1\left (-\frac {i}{2},1;1-\frac {i}{2};-e^{2 i \text {ArcCos}(a x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2225
Rule 2283
Rule 4527
Rule 4921
Rubi steps
\begin {align*} \int \frac {e^{\cos ^{-1}(a x)}}{x} \, dx &=-\frac {\text {Subst}\left (\int a e^x \tan (x) \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=-\text {Subst}\left (\int e^x \tan (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\left (i \text {Subst}\left (\int \left (-e^x+\frac {2 e^x}{1+e^{2 i x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )\right )\\ &=i \text {Subst}\left (\int e^x \, dx,x,\cos ^{-1}(a x)\right )-2 i \text {Subst}\left (\int \frac {e^x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(a x)\right )\\ &=i e^{\cos ^{-1}(a x)}-2 i e^{\cos ^{-1}(a x)} \, _2F_1\left (-\frac {i}{2},1;1-\frac {i}{2};-e^{2 i \cos ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 79, normalized size = 1.76 \begin {gather*} i \left (-e^{\text {ArcCos}(a x)} \, _2F_1\left (-\frac {i}{2},1;1-\frac {i}{2};-e^{2 i \text {ArcCos}(a x)}\right )+\left (\frac {1}{5}-\frac {2 i}{5}\right ) e^{(1+2 i) \text {ArcCos}(a x)} \, _2F_1\left (1,1-\frac {i}{2};2-\frac {i}{2};-e^{2 i \text {ArcCos}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{\arccos \left (a x \right )}}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {e^{\operatorname {acos}{\left (a x \right )}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^{\mathrm {acos}\left (a\,x\right )}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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