Optimal. Leaf size=42 \[ \frac {4}{25 f^2 \csc ^{\frac {5}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{5 f \csc ^{\frac {3}{2}}(e+f x)} \]
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Rubi [A] time = 0.11, 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} \[ \frac {4}{25 f^2 \csc ^{\frac {5}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{5 f \csc ^{\frac {3}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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Rule 4187
Rule 4189
Rubi steps
\begin {align*} \int \left (\frac {x}{\csc ^{\frac {5}{2}}(e+f x)}-\frac {3 x}{5 \sqrt {\csc (e+f x)}}\right ) \, dx &=-\left (\frac {3}{5} \int \frac {x}{\sqrt {\csc (e+f x)}} \, dx\right )+\int \frac {x}{\csc ^{\frac {5}{2}}(e+f x)} \, dx\\ &=\frac {4}{25 f^2 \csc ^{\frac {5}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{5 f \csc ^{\frac {3}{2}}(e+f x)}+\frac {3}{5} \int \frac {x}{\sqrt {\csc (e+f x)}} \, dx-\frac {1}{5} \left (3 \sqrt {\csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int x \sqrt {\sin (e+f x)} \, dx\\ &=\frac {4}{25 f^2 \csc ^{\frac {5}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{5 f \csc ^{\frac {3}{2}}(e+f x)}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 29, normalized size = 0.69 \[ -\frac {2 (5 f x \cot (e+f x)-2)}{25 f^2 \csc ^{\frac {5}{2}}(e+f x)} \]
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 \[ \int -\frac {3 \, x}{5 \, \sqrt {\csc \left (f x + e\right )}} + \frac {x}{\csc \left (f x + e\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {x}{\csc \left (f x +e \right )^{\frac {5}{2}}}-\frac {3 x}{5 \sqrt {\csc \left (f x +e \right )}}\, dx \]
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 \[ \int -\frac {3 \, x}{5 \, \sqrt {\csc \left (f x + e\right )}} + \frac {x}{\csc \left (f x + e\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x}{{\left (\frac {1}{\sin \left (e+f\,x\right )}\right )}^{5/2}}-\frac {3\,x}{5\,\sqrt {\frac {1}{\sin \left (e+f\,x\right )}}} \,d x \]
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 \[ - \frac {\int \left (- \frac {5 x}{\csc ^{\frac {5}{2}}{\left (e + f x \right )}}\right )\, dx + \int \frac {3 x}{\sqrt {\csc {\left (e + f x \right )}}}\, dx}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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