Optimal. Leaf size=81 \[ \frac {1}{10} a^4 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac {1}{34} a^4 e^{\csc ^{-1}(a x)} \cos \left (4 \csc ^{-1}(a x)\right )-\frac {1}{20} a^4 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )+\frac {1}{136} a^4 e^{\csc ^{-1}(a x)} \sin \left (4 \csc ^{-1}(a x)\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5375, 12, 4557,
4517} \begin {gather*} \frac {1}{10} a^4 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac {1}{34} a^4 e^{\csc ^{-1}(a x)} \cos \left (4 \csc ^{-1}(a x)\right )-\frac {1}{20} a^4 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )+\frac {1}{136} a^4 e^{\csc ^{-1}(a x)} \sin \left (4 \csc ^{-1}(a x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 4517
Rule 4557
Rule 5375
Rubi steps
\begin {align*} \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx &=-\frac {\text {Subst}\left (\int a^5 e^x \cos (x) \sin ^3(x) \, dx,x,\csc ^{-1}(a x)\right )}{a}\\ &=-\left (a^4 \text {Subst}\left (\int e^x \cos (x) \sin ^3(x) \, dx,x,\csc ^{-1}(a x)\right )\right )\\ &=-\left (a^4 \text {Subst}\left (\int \left (\frac {1}{4} e^x \sin (2 x)-\frac {1}{8} e^x \sin (4 x)\right ) \, dx,x,\csc ^{-1}(a x)\right )\right )\\ &=\frac {1}{8} a^4 \text {Subst}\left (\int e^x \sin (4 x) \, dx,x,\csc ^{-1}(a x)\right )-\frac {1}{4} a^4 \text {Subst}\left (\int e^x \sin (2 x) \, dx,x,\csc ^{-1}(a x)\right )\\ &=\frac {1}{10} a^4 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac {1}{34} a^4 e^{\csc ^{-1}(a x)} \cos \left (4 \csc ^{-1}(a x)\right )-\frac {1}{20} a^4 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )+\frac {1}{136} a^4 e^{\csc ^{-1}(a x)} \sin \left (4 \csc ^{-1}(a x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 50, normalized size = 0.62 \begin {gather*} -\frac {1}{680} a^4 e^{\csc ^{-1}(a x)} \left (-68 \cos \left (2 \csc ^{-1}(a x)\right )+20 \cos \left (4 \csc ^{-1}(a x)\right )+34 \sin \left (2 \csc ^{-1}(a x)\right )-5 \sin \left (4 \csc ^{-1}(a x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{\mathrm {arccsc}\left (a x \right )}}{x^{5}}\, dx\]
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 [A]
time = 0.40, size = 51, normalized size = 0.63 \begin {gather*} \frac {{\left (6 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - {\left (6 \, a^{2} x^{2} + 5\right )} \sqrt {a^{2} x^{2} - 1} - 20\right )} e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{85 \, x^{4}} \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 {acsc}{\left (a x \right )}}}{x^{5}}\, 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 = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x}\right )}}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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