Optimal. Leaf size=98 \[ \frac{10 c^3 \sin (a+b x) \sqrt{c \cos (a+b x)}}{21 b}+\frac{10 c^4 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b \sqrt{c \cos (a+b x)}}+\frac{2 c \sin (a+b x) (c \cos (a+b x))^{5/2}}{7 b} \]
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Rubi [A] time = 0.0586644, antiderivative size = 98, 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 = {2635, 2642, 2641} \[ \frac{10 c^3 \sin (a+b x) \sqrt{c \cos (a+b x)}}{21 b}+\frac{10 c^4 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b \sqrt{c \cos (a+b x)}}+\frac{2 c \sin (a+b x) (c \cos (a+b x))^{5/2}}{7 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (c \cos (a+b x))^{7/2} \, dx &=\frac{2 c (c \cos (a+b x))^{5/2} \sin (a+b x)}{7 b}+\frac{1}{7} \left (5 c^2\right ) \int (c \cos (a+b x))^{3/2} \, dx\\ &=\frac{10 c^3 \sqrt{c \cos (a+b x)} \sin (a+b x)}{21 b}+\frac{2 c (c \cos (a+b x))^{5/2} \sin (a+b x)}{7 b}+\frac{1}{21} \left (5 c^4\right ) \int \frac{1}{\sqrt{c \cos (a+b x)}} \, dx\\ &=\frac{10 c^3 \sqrt{c \cos (a+b x)} \sin (a+b x)}{21 b}+\frac{2 c (c \cos (a+b x))^{5/2} \sin (a+b x)}{7 b}+\frac{\left (5 c^4 \sqrt{\cos (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{21 \sqrt{c \cos (a+b x)}}\\ &=\frac{10 c^4 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b \sqrt{c \cos (a+b x)}}+\frac{10 c^3 \sqrt{c \cos (a+b x)} \sin (a+b x)}{21 b}+\frac{2 c (c \cos (a+b x))^{5/2} \sin (a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.0920171, size = 76, normalized size = 0.78 \[ \frac{c^3 \sqrt{c \cos (a+b x)} \left (20 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )+(23 \sin (a+b x)+3 \sin (3 (a+b x))) \sqrt{\cos (a+b x)}\right )}{42 b \sqrt{\cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.047, size = 210, normalized size = 2.1 \begin{align*} -{\frac{2\,{c}^{4}}{21\,b}\sqrt{c \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( 48\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{9}-120\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{7}+128\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{5}-72\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}+5\,\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) +16\,\cos \left ( 1/2\,bx+a/2 \right ) \right ){\frac{1}{\sqrt{-c \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{c \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \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 \left (c \cos \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 (\sqrt{c \cos \left (b x + a\right )} c^{3} \cos \left (b x + a\right )^{3}, 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(-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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