Optimal. Leaf size=62 \[ \frac {4 \sqrt {x+1}}{45 x^{3/2}}-\frac {\sqrt {x+1}}{15 x^{5/2}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{3 x^3}-\frac {8 \sqrt {x+1}}{45 \sqrt {x}} \]
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Rubi [A] time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5902, 12, 45, 37} \[ \frac {4 \sqrt {x+1}}{45 x^{3/2}}-\frac {\sqrt {x+1}}{15 x^{5/2}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{3 x^3}-\frac {8 \sqrt {x+1}}{45 \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 45
Rule 5902
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}\left (\sqrt {x}\right )}{x^4} \, dx &=-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {1}{3} \int \frac {1}{2 x^{7/2} \sqrt {1+x}} \, dx\\ &=-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {1}{6} \int \frac {1}{x^{7/2} \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1+x}}{15 x^{5/2}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{3 x^3}-\frac {2}{15} \int \frac {1}{x^{5/2} \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1+x}}{15 x^{5/2}}+\frac {4 \sqrt {1+x}}{45 x^{3/2}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {4}{45} \int \frac {1}{x^{3/2} \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1+x}}{15 x^{5/2}}+\frac {4 \sqrt {1+x}}{45 x^{3/2}}-\frac {8 \sqrt {1+x}}{45 \sqrt {x}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 0.63 \[ \frac {\sqrt {x} \sqrt {x+1} \left (-8 x^2+4 x-3\right )-15 \sinh ^{-1}\left (\sqrt {x}\right )}{45 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 37, normalized size = 0.60 \[ -\frac {{\left (8 \, x^{2} - 4 \, x + 3\right )} \sqrt {x + 1} \sqrt {x} + 15 \, \log \left (\sqrt {x + 1} + \sqrt {x}\right )}{45 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 67, normalized size = 1.08 \[ -\frac {\log \left (\sqrt {x + 1} + \sqrt {x}\right )}{3 \, x^{3}} + \frac {16 \, {\left (10 \, {\left (\sqrt {x + 1} - \sqrt {x}\right )}^{4} - 5 \, {\left (\sqrt {x + 1} - \sqrt {x}\right )}^{2} + 1\right )}}{45 \, {\left ({\left (\sqrt {x + 1} - \sqrt {x}\right )}^{2} - 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 0.66 \[ -\frac {\arcsinh \left (\sqrt {x}\right )}{3 x^{3}}-\frac {\sqrt {1+x}}{15 x^{\frac {5}{2}}}+\frac {4 \sqrt {1+x}}{45 x^{\frac {3}{2}}}-\frac {8 \sqrt {1+x}}{45 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 40, normalized size = 0.65 \[ -\frac {8 \, \sqrt {x + 1}}{45 \, \sqrt {x}} + \frac {4 \, \sqrt {x + 1}}{45 \, x^{\frac {3}{2}}} - \frac {\sqrt {x + 1}}{15 \, x^{\frac {5}{2}}} - \frac {\operatorname {arsinh}\left (\sqrt {x}\right )}{3 \, x^{3}} \]
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 \[ \int \frac {\mathrm {asinh}\left (\sqrt {x}\right )}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {asinh}{\left (\sqrt {x} \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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