Optimal. Leaf size=84 \[ -\frac {1}{8} a^3 e^{\csc ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^2 e^{\csc ^{-1}(a x)}}{8 x}+\frac {1}{40} a^3 e^{\csc ^{-1}(a x)} \cos \left (3 \csc ^{-1}(a x)\right )+\frac {3}{40} a^3 e^{\csc ^{-1}(a x)} \sin \left (3 \csc ^{-1}(a x)\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 84, 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,
4518} \begin {gather*} \frac {1}{40} a^3 e^{\csc ^{-1}(a x)} \cos \left (3 \csc ^{-1}(a x)\right )+\frac {3}{40} a^3 e^{\csc ^{-1}(a x)} \sin \left (3 \csc ^{-1}(a x)\right )-\frac {a^2 e^{\csc ^{-1}(a x)}}{8 x}-\frac {1}{8} a^3 \sqrt {1-\frac {1}{a^2 x^2}} e^{\csc ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 4518
Rule 4557
Rule 5375
Rubi steps
\begin {align*} \int \frac {e^{\csc ^{-1}(a x)}}{x^4} \, dx &=-\frac {\text {Subst}\left (\int a^4 e^x \cos (x) \sin ^2(x) \, dx,x,\csc ^{-1}(a x)\right )}{a}\\ &=-\left (a^3 \text {Subst}\left (\int e^x \cos (x) \sin ^2(x) \, dx,x,\csc ^{-1}(a x)\right )\right )\\ &=-\left (a^3 \text {Subst}\left (\int \left (\frac {1}{4} e^x \cos (x)-\frac {1}{4} e^x \cos (3 x)\right ) \, dx,x,\csc ^{-1}(a x)\right )\right )\\ &=-\left (\frac {1}{4} a^3 \text {Subst}\left (\int e^x \cos (x) \, dx,x,\csc ^{-1}(a x)\right )\right )+\frac {1}{4} a^3 \text {Subst}\left (\int e^x \cos (3 x) \, dx,x,\csc ^{-1}(a x)\right )\\ &=-\frac {1}{8} a^3 e^{\csc ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^2 e^{\csc ^{-1}(a x)}}{8 x}+\frac {1}{40} a^3 e^{\csc ^{-1}(a x)} \cos \left (3 \csc ^{-1}(a x)\right )+\frac {3}{40} a^3 e^{\csc ^{-1}(a x)} \sin \left (3 \csc ^{-1}(a x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 54, normalized size = 0.64 \begin {gather*} \frac {1}{40} a^3 e^{\csc ^{-1}(a x)} \left (-5 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {5}{a x}+\cos \left (3 \csc ^{-1}(a x)\right )+3 \sin \left (3 \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 \[\int \frac {{\mathrm e}^{\mathrm {arccsc}\left (a x \right )}}{x^{4}}\, dx\]
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 = 0.42, size = 41, normalized size = 0.49 \begin {gather*} \frac {{\left (a^{2} x^{2} - {\left (a^{2} x^{2} + 1\right )} \sqrt {a^{2} x^{2} - 1} - 3\right )} e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{10 \, x^{3}} \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^{4}}\, 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^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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