Optimal. Leaf size=30 \[ b \text {Int}\left (\frac {\csc \left (c+d \sqrt {x}\right )}{x^{3/2}},x\right )-\frac {2 a}{\sqrt {x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, 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 = {} \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx &=\int \left (\frac {a}{x^{3/2}}+\frac {b \csc \left (c+d \sqrt {x}\right )}{x^{3/2}}\right ) \, dx\\ &=-\frac {2 a}{\sqrt {x}}+b \int \frac {\csc \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 6.35, size = 0, normalized size = 0.00 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \sqrt {x} \csc \left (d \sqrt {x} + c\right ) + a \sqrt {x}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \csc \left (d \sqrt {x} + c\right ) + a}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 2.10, size = 0, normalized size = 0.00 \[ \int \frac {a +b \csc \left (c +d \sqrt {x}\right )}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b \int \frac {\sin \left (d \sqrt {x} + c\right )}{{\left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} + 2 \, \cos \left (d \sqrt {x} + c\right ) + 1\right )} x^{\frac {3}{2}}}\,{d x} + b \int \frac {\sin \left (d \sqrt {x} + c\right )}{{\left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} - 2 \, \cos \left (d \sqrt {x} + c\right ) + 1\right )} x^{\frac {3}{2}}}\,{d x}\right )} \sqrt {x} - 2 \, a}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}}{x^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \csc {\left (c + d \sqrt {x} \right )}}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________