Optimal. Leaf size=30 \[ b \text {Int}\left (\frac {\text {sech}\left (c+d \sqrt {x}\right )}{x^{3/2}},x\right )-\frac {2 a}{\sqrt {x}} \]
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Rubi [A] time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx &=\int \left (\frac {a}{x^{3/2}}+\frac {b \text {sech}\left (c+d \sqrt {x}\right )}{x^{3/2}}\right ) \, dx\\ &=-\frac {2 a}{\sqrt {x}}+b \int \frac {\text {sech}\left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 8.92, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \sqrt {x} \operatorname {sech}\left (d \sqrt {x} + c\right ) + a \sqrt {x}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {sech}\left (c +d \sqrt {x}\right )}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, b \int \frac {e^{\left (d \sqrt {x} + c\right )}}{x^{\frac {3}{2}} e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} + x^{\frac {3}{2}}}\,{d x} - \frac {2 \, a}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}}{x^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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