Optimal. Leaf size=70 \[ \frac{10 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{21 b}-\frac{2 \sin ^{\frac{5}{2}}(a+b x) \cos (a+b x)}{7 b}-\frac{10 \sqrt{\sin (a+b x)} \cos (a+b x)}{21 b} \]
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Rubi [A] time = 0.0287382, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2635, 2641} \[ \frac{10 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{21 b}-\frac{2 \sin ^{\frac{5}{2}}(a+b x) \cos (a+b x)}{7 b}-\frac{10 \sqrt{\sin (a+b x)} \cos (a+b x)}{21 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \sin ^{\frac{7}{2}}(a+b x) \, dx &=-\frac{2 \cos (a+b x) \sin ^{\frac{5}{2}}(a+b x)}{7 b}+\frac{5}{7} \int \sin ^{\frac{3}{2}}(a+b x) \, dx\\ &=-\frac{10 \cos (a+b x) \sqrt{\sin (a+b x)}}{21 b}-\frac{2 \cos (a+b x) \sin ^{\frac{5}{2}}(a+b x)}{7 b}+\frac{5}{21} \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx\\ &=\frac{10 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{21 b}-\frac{10 \cos (a+b x) \sqrt{\sin (a+b x)}}{21 b}-\frac{2 \cos (a+b x) \sin ^{\frac{5}{2}}(a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.118821, size = 55, normalized size = 0.79 \[ \frac{\sqrt{\sin (a+b x)} (3 \cos (3 (a+b x))-23 \cos (a+b x))-20 F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )}{42 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 104, normalized size = 1.5 \begin{align*}{\frac{1}{b\cos \left ( bx+a \right ) } \left ({\frac{2\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{7}}+{\frac{5}{21}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }-{\frac{16\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sin \left ( bx+a \right ) }{21}} \right ){\frac{1}{\sqrt{\sin \left ( bx+a \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b x + a\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )^{\frac{3}{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b x + a\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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