Optimal. Leaf size=91 \[ \frac {(2+2 i) e^{(1+i) \sec ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a}-\frac {(1+i) e^{(1+i) \sec ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a} \]
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Rubi [A] time = 0.09, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5266, 4471, 2251} \[ \frac {(2+2 i) e^{(1+i) \sec ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a}-\frac {(1+i) e^{(1+i) \sec ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2251
Rule 4471
Rule 5266
Rubi steps
\begin {align*} \int e^{\sec ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int e^x \sec (x) \tan (x) \, dx,x,\sec ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {4 i e^{(1+i) x}}{\left (1+e^{2 i x}\right )^2}-\frac {2 i e^{(1+i) x}}{1+e^{2 i x}}\right ) \, dx,x,\sec ^{-1}(a x)\right )}{a}\\ &=-\frac {(2 i) \operatorname {Subst}\left (\int \frac {e^{(1+i) x}}{1+e^{2 i x}} \, dx,x,\sec ^{-1}(a x)\right )}{a}+\frac {(4 i) \operatorname {Subst}\left (\int \frac {e^{(1+i) x}}{\left (1+e^{2 i x}\right )^2} \, dx,x,\sec ^{-1}(a x)\right )}{a}\\ &=-\frac {(1+i) e^{(1+i) \sec ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a}+\frac {(2+2 i) e^{(1+i) \sec ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 54, normalized size = 0.59 \[ x e^{\sec ^{-1}(a x)}-\frac {(1-i) e^{(1+i) \sec ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (\operatorname {arcsec}\left (a x\right )\right )}, x\right ) \]
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 \[ \int e^{\left (\operatorname {arcsec}\left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\mathrm {arcsec}\left (a x \right )}\, dx \]
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 \[ \int e^{\left (\operatorname {arcsec}\left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{\mathrm {acos}\left (\frac {1}{a\,x}\right )} \,d x \]
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 \[ \int e^{\operatorname {asec}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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