3.55 \(\int \frac {a+b \text {sech}(c+d \sqrt {x})}{x^{3/2}} \, dx\)

Optimal. Leaf size=30 \[ b \text {Int}\left (\frac {\text {sech}\left (c+d \sqrt {x}\right )}{x^{3/2}},x\right )-\frac {2 a}{\sqrt {x}} \]

[Out]

-2*a/x^(1/2)+b*Unintegrable(sech(c+d*x^(1/2))/x^(3/2),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.

[In]

Int[(a + b*Sech[c + d*Sqrt[x]])/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + b*Defer[Int][Sech[c + d*Sqrt[x]]/x^(3/2), x]

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.

[In]

Integrate[(a + b*Sech[c + d*Sqrt[x]])/x^(3/2),x]

[Out]

Integrate[(a + b*Sech[c + d*Sqrt[x]])/x^(3/2), x]

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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.

[In]

integrate((a+b*sech(c+d*x^(1/2)))/x^(3/2),x, algorithm="fricas")

[Out]

integral((b*sqrt(x)*sech(d*sqrt(x) + c) + a*sqrt(x))/x^2, x)

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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.

[In]

integrate((a+b*sech(c+d*x^(1/2)))/x^(3/2),x, algorithm="giac")

[Out]

integrate((b*sech(d*sqrt(x) + c) + a)/x^(3/2), x)

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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.

[In]

int((a+b*sech(c+d*x^(1/2)))/x^(3/2),x)

[Out]

int((a+b*sech(c+d*x^(1/2)))/x^(3/2),x)

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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.

[In]

integrate((a+b*sech(c+d*x^(1/2)))/x^(3/2),x, algorithm="maxima")

[Out]

2*b*integrate(e^(d*sqrt(x) + c)/(x^(3/2)*e^(2*d*sqrt(x) + 2*c) + x^(3/2)), x) - 2*a/sqrt(x)

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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.

[In]

int((a + b/cosh(c + d*x^(1/2)))/x^(3/2),x)

[Out]

int((a + b/cosh(c + d*x^(1/2)))/x^(3/2), x)

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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.

[In]

integrate((a+b*sech(c+d*x**(1/2)))/x**(3/2),x)

[Out]

Integral((a + b*sech(c + d*sqrt(x)))/x**(3/2), x)

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