Optimal. Leaf size=83 \[ -\frac {4}{15 f^2 \sin ^{\frac {3}{2}}(e+f x)}+\frac {12 \sqrt {\sin (e+f x)}}{5 f^2}-\frac {2 x \cos (e+f x)}{5 f \sin ^{\frac {5}{2}}(e+f x)}-\frac {6 x \cos (e+f x)}{5 f \sqrt {\sin (e+f x)}} \]
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Rubi [A] time = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {3315} \[ -\frac {4}{15 f^2 \sin ^{\frac {3}{2}}(e+f x)}+\frac {12 \sqrt {\sin (e+f x)}}{5 f^2}-\frac {2 x \cos (e+f x)}{5 f \sin ^{\frac {5}{2}}(e+f x)}-\frac {6 x \cos (e+f x)}{5 f \sqrt {\sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin {align*} \int \left (\frac {x}{\sin ^{\frac {7}{2}}(e+f x)}+\frac {3}{5} x \sqrt {\sin (e+f x)}\right ) \, dx &=\frac {3}{5} \int x \sqrt {\sin (e+f x)} \, dx+\int \frac {x}{\sin ^{\frac {7}{2}}(e+f x)} \, dx\\ &=-\frac {2 x \cos (e+f x)}{5 f \sin ^{\frac {5}{2}}(e+f x)}-\frac {4}{15 f^2 \sin ^{\frac {3}{2}}(e+f x)}+\frac {3}{5} \int \frac {x}{\sin ^{\frac {3}{2}}(e+f x)} \, dx+\frac {3}{5} \int x \sqrt {\sin (e+f x)} \, dx\\ &=-\frac {2 x \cos (e+f x)}{5 f \sin ^{\frac {5}{2}}(e+f x)}-\frac {4}{15 f^2 \sin ^{\frac {3}{2}}(e+f x)}-\frac {6 x \cos (e+f x)}{5 f \sqrt {\sin (e+f x)}}+\frac {12 \sqrt {\sin (e+f x)}}{5 f^2}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 58, normalized size = 0.70 \[ \frac {46 \sin (e+f x)-18 \sin (3 (e+f x))-21 f x \cos (e+f x)+9 f x \cos (3 (e+f x))}{30 f^2 \sin ^{\frac {5}{2}}(e+f x)} \]
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 \[ \int \frac {3}{5} \, x \sqrt {\sin \left (f x + e\right )} + \frac {x}{\sin \left (f x + e\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sin \left (f x +e \right )^{\frac {7}{2}}}+\frac {3 x \left (\sqrt {\sin }\left (f x +e \right )\right )}{5}\, dx \]
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 \[ \int \frac {3}{5} \, x \sqrt {\sin \left (f x + e\right )} + \frac {x}{\sin \left (f x + e\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.49, size = 253, normalized size = 3.05 \[ \left (\frac {12}{5\,f^2}+\frac {x\,6{}\mathrm {i}}{5\,f}\right )\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}-\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,\left (\frac {x\,3{}\mathrm {i}}{5\,f}-\frac {32+f\,x\,66{}\mathrm {i}}{30\,f^2}\right )}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^2}-\frac {x\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,12{}\mathrm {i}}{5\,f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}+\frac {x\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,16{}\mathrm {i}}{5\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3} \]
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 \[ \frac {\int \frac {5 x}{\sin ^{\frac {7}{2}}{\left (e + f x \right )}}\, dx + \int 3 x \sqrt {\sin {\left (e + f x \right )}}\, dx}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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