Optimal. Leaf size=83 \[ \frac {20}{63 f^2 \csc ^{\frac {3}{2}}(e+f x)}+\frac {4}{49 f^2 \csc ^{\frac {7}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{7 f \csc ^{\frac {5}{2}}(e+f x)}-\frac {10 x \cos (e+f x)}{21 f \sqrt {\csc (e+f x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4187, 4189} \[ \frac {20}{63 f^2 \csc ^{\frac {3}{2}}(e+f x)}+\frac {4}{49 f^2 \csc ^{\frac {7}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{7 f \csc ^{\frac {5}{2}}(e+f x)}-\frac {10 x \cos (e+f x)}{21 f \sqrt {\csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 4187
Rule 4189
Rubi steps
\begin {align*} \int \left (\frac {x}{\csc ^{\frac {7}{2}}(e+f x)}-\frac {5}{21} x \sqrt {\csc (e+f x)}\right ) \, dx &=-\left (\frac {5}{21} \int x \sqrt {\csc (e+f x)} \, dx\right )+\int \frac {x}{\csc ^{\frac {7}{2}}(e+f x)} \, dx\\ &=\frac {4}{49 f^2 \csc ^{\frac {7}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{7 f \csc ^{\frac {5}{2}}(e+f x)}+\frac {5}{7} \int \frac {x}{\csc ^{\frac {3}{2}}(e+f x)} \, dx-\frac {1}{21} \left (5 \sqrt {\csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {x}{\sqrt {\sin (e+f x)}} \, dx\\ &=\frac {4}{49 f^2 \csc ^{\frac {7}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{7 f \csc ^{\frac {5}{2}}(e+f x)}+\frac {20}{63 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {10 x \cos (e+f x)}{21 f \sqrt {\csc (e+f x)}}+\frac {5}{21} \int x \sqrt {\csc (e+f x)} \, dx-\frac {1}{21} \left (5 \sqrt {\csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {x}{\sqrt {\sin (e+f x)}} \, dx\\ &=\frac {4}{49 f^2 \csc ^{\frac {7}{2}}(e+f x)}-\frac {2 x \cos (e+f x)}{7 f \csc ^{\frac {5}{2}}(e+f x)}+\frac {20}{63 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {10 x \cos (e+f x)}{21 f \sqrt {\csc (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 2.40, size = 57, normalized size = 0.69 \[ \frac {-36 \cos (2 (e+f x))-483 f x \cot (e+f x)+63 f x \cos (3 (e+f x)) \csc (e+f x)+316}{882 f^2 \csc ^{\frac {3}{2}}(e+f x)} \]
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 \[ \int -\frac {5}{21} \, x \sqrt {\csc \left (f x + e\right )} + \frac {x}{\csc \left (f x + e\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {x}{\csc \left (f x +e \right )^{\frac {7}{2}}}-\frac {5 x \left (\sqrt {\csc }\left (f x +e \right )\right )}{21}\, dx \]
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 \[ \int -\frac {5}{21} \, x \sqrt {\csc \left (f x + e\right )} + \frac {x}{\csc \left (f x + e\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\left (\frac {1}{\sin \left (e+f\,x\right )}\right )}^{7/2}}-\frac {5\,x\,\sqrt {\frac {1}{\sin \left (e+f\,x\right )}}}{21} \,d x \]
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 \[ - \frac {\int \left (- \frac {21 x}{\csc ^{\frac {7}{2}}{\left (e + f x \right )}}\right )\, dx + \int 5 x \sqrt {\csc {\left (e + f x \right )}}\, dx}{21} \]
Verification of antiderivative is not currently implemented for this CAS.
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