Optimal. Leaf size=105 \[ -\frac{6 \cos (a+b x)}{5 b c^3 \sqrt{c \sin (a+b x)}}-\frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \sin (a+b x)}}{5 b c^4 \sqrt{\sin (a+b x)}}-\frac{2 \cos (a+b x)}{5 b c (c \sin (a+b x))^{5/2}} \]
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Rubi [A] time = 0.0510645, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2636, 2640, 2639} \[ -\frac{6 \cos (a+b x)}{5 b c^3 \sqrt{c \sin (a+b x)}}-\frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \sin (a+b x)}}{5 b c^4 \sqrt{\sin (a+b x)}}-\frac{2 \cos (a+b x)}{5 b c (c \sin (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(c \sin (a+b x))^{7/2}} \, dx &=-\frac{2 \cos (a+b x)}{5 b c (c \sin (a+b x))^{5/2}}+\frac{3 \int \frac{1}{(c \sin (a+b x))^{3/2}} \, dx}{5 c^2}\\ &=-\frac{2 \cos (a+b x)}{5 b c (c \sin (a+b x))^{5/2}}-\frac{6 \cos (a+b x)}{5 b c^3 \sqrt{c \sin (a+b x)}}-\frac{3 \int \sqrt{c \sin (a+b x)} \, dx}{5 c^4}\\ &=-\frac{2 \cos (a+b x)}{5 b c (c \sin (a+b x))^{5/2}}-\frac{6 \cos (a+b x)}{5 b c^3 \sqrt{c \sin (a+b x)}}-\frac{\left (3 \sqrt{c \sin (a+b x)}\right ) \int \sqrt{\sin (a+b x)} \, dx}{5 c^4 \sqrt{\sin (a+b x)}}\\ &=-\frac{2 \cos (a+b x)}{5 b c (c \sin (a+b x))^{5/2}}-\frac{6 \cos (a+b x)}{5 b c^3 \sqrt{c \sin (a+b x)}}-\frac{6 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{c \sin (a+b x)}}{5 b c^4 \sqrt{\sin (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.157936, size = 68, normalized size = 0.65 \[ -\frac{2 \left (\frac{3}{2} \sin (2 (a+b x))+\cot (a+b x)-3 \sin ^{\frac{3}{2}}(a+b x) E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{5 b c^2 (c \sin (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 168, normalized size = 1.6 \begin{align*}{\frac{1}{5\,{c}^{3} \left ( \sin \left ( bx+a \right ) \right ) ^{3}\cos \left ( bx+a \right ) b} \left ( 6\,\sqrt{-\sin \left ( bx+a \right ) +1}\sqrt{2\,\sin \left ( bx+a \right ) +2} \left ( \sin \left ( bx+a \right ) \right ) ^{7/2}{\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} \left ( \sin \left ( bx+a \right ) \right ) ^{7/2}{\it EllipticF} \left ( \sqrt{-\sin \left ( bx+a \right ) +1},1/2\,\sqrt{2} \right ) +6\, \left ( \sin \left ( bx+a \right ) \right ) ^{5}-4\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}-2\,\sin \left ( bx+a \right ) \right ){\frac{1}{\sqrt{c\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 \frac{1}{\left (c \sin \left (b x + a\right )\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{c \sin \left (b x + a\right )}}{c^{4} \cos \left (b x + a\right )^{4} - 2 \, c^{4} \cos \left (b x + a\right )^{2} + c^{4}}, 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}{\left (c \sin \left (b x + a\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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