Optimal. Leaf size=70 \[ -\frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b}-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}-\frac{6 \cos (a+b x)}{5 b \sqrt{\sin (a+b x)}} \]
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Rubi [A] time = 0.0268507, 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 = {2636, 2639} \[ -\frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b}-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}-\frac{6 \cos (a+b x)}{5 b \sqrt{\sin (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sin ^{\frac{7}{2}}(a+b x)} \, dx &=-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}+\frac{3}{5} \int \frac{1}{\sin ^{\frac{3}{2}}(a+b x)} \, dx\\ &=-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}-\frac{6 \cos (a+b x)}{5 b \sqrt{\sin (a+b x)}}-\frac{3}{5} \int \sqrt{\sin (a+b x)} \, dx\\ &=-\frac{6 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{5 b}-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}-\frac{6 \cos (a+b x)}{5 b \sqrt{\sin (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.284052, size = 55, normalized size = 0.79 \[ \frac{2 \left (3 E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )-\frac{\left (3 \sin ^2(a+b x)+1\right ) \cos (a+b x)}{\sin ^{\frac{5}{2}}(a+b x)}\right )}{5 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 160, normalized size = 2.3 \begin{align*}{\frac{1}{5\,b\cos \left ( bx+a \right ) } \left ( 6\,\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) } \left ( \sin \left ( bx+a \right ) \right ) ^{2}{\it EllipticE} \left ( \sqrt{\sin \left ( bx+a \right ) +1},1/2\,\sqrt{2} \right ) -3\,\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) } \left ( \sin \left ( bx+a \right ) \right ) ^{2}{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},1/2\,\sqrt{2} \right ) +6\, \left ( \sin \left ( bx+a \right ) \right ) ^{4}-4\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}-2 \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{-{\frac{5}{2}}}} \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 \frac{1}{\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 (\frac{\sqrt{\sin \left (b x + a\right )}}{\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1}, 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 \frac{1}{\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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