Optimal. Leaf size=42 \[ \frac {4}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}} \]
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Rubi [A] time = 0.12, 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 = {4187, 4189} \[ \frac {4}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 4187
Rule 4189
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*}
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Mathematica [A] time = 0.55, size = 29, normalized size = 0.69 \[ -\frac {2 (3 f x \cot (e+f x)-2)}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{3} \, x \sqrt {\csc \left (f x + e\right )} + \frac {x}{\csc \left (f x + e\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, 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.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{3} \, x \sqrt {\csc \left (f x + e\right )} + \frac {x}{\csc \left (f x + e\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x}{{\left (\frac {1}{\sin \left (e+f\,x\right )}\right )}^{3/2}}-\frac {x\,\sqrt {\frac {1}{\sin \left (e+f\,x\right )}}}{3} \,d x \]
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 \[ - \frac {\int \left (- \frac {3 x}{\csc ^{\frac {3}{2}}{\left (e + f x \right )}}\right )\, dx + \int x \sqrt {\csc {\left (e + f x \right )}}\, dx}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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