Optimal. Leaf size=42 \[ \frac {4}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4272, 4274}
\begin {gather*} \frac {4}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 4272
Rule 4274
Rubi steps
\begin {align*} \int \left (\frac {x}{\csc ^{\frac {3}{2}}(e+f x)}-\frac {1}{3} x \sqrt {\csc (e+f x)}\right ) \, dx &=-\left (\frac {1}{3} \int x \sqrt {\csc (e+f x)} \, dx\right )+\int \frac {x}{\csc ^{\frac {3}{2}}(e+f x)} \, dx\\ &=\frac {4}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}}+\frac {1}{3} \int x \sqrt {\csc (e+f x)} \, dx-\frac {1}{3} \left (\sqrt {\csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {x}{\sqrt {\sin (e+f x)}} \, dx\\ &=\frac {4}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.26, size = 29, normalized size = 0.69 \begin {gather*} -\frac {2 (-2+3 f x \cot (e+f x))}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {x}{\csc \left (f x +e \right )^{\frac {3}{2}}}-\frac {x \left (\sqrt {\csc }\left (f x +e \right )\right )}{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \left (- \frac {3 x}{\csc ^{\frac {3}{2}}{\left (e + f x \right )}}\right )\, dx + \int x \sqrt {\csc {\left (e + f x \right )}}\, dx}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{{\left (\frac {1}{\sin \left (e+f\,x\right )}\right )}^{3/2}}-\frac {x\,\sqrt {\frac {1}{\sin \left (e+f\,x\right )}}}{3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________