Optimal. Leaf size=115 \[ \frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}+\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b^4 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}} \]
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Rubi [A] time = 0.0896272, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3012, 2636, 2642, 2641} \[ \frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}+\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b^4 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3012
Rule 2636
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{9/2}} \, dx &=\frac{2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}}+\frac{(5 A+7 C) \int \frac{1}{(b \cos (c+d x))^{5/2}} \, dx}{7 b^2}\\ &=\frac{2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}+\frac{(5 A+7 C) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx}{21 b^4}\\ &=\frac{2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}+\frac{\left ((5 A+7 C) \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 b^4 \sqrt{b \cos (c+d x)}}\\ &=\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b^4 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.375375, size = 77, normalized size = 0.67 \[ \frac{2 \left ((5 A+7 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\tan (c+d x) \left (3 A \sec ^2(c+d x)+5 A+7 C\right )\right )}{21 b^4 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.382, size = 413, normalized size = 3.6 \begin{align*} -2\,{\frac{\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}{{b}^{4}\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) }d} \left ( C \left ( -1/6\,{\frac{\cos \left ( 1/2\,dx+c/2 \right ) \sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}{b \left ( \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1/2 \right ) ^{2}}}+1/3\,{\frac{\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) }{\sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}} \right ) +A \left ( -{\frac{\cos \left ( 1/2\,dx+c/2 \right ) \sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}{56\,b \left ( \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1/2 \right ) ^{4}}}-{\frac{5\,\cos \left ( 1/2\,dx+c/2 \right ) \sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}{42\,b \left ( \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1/2 \right ) ^{2}}}+{\frac{5\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) }{21\,\sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}} \right ) \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{C \cos \left (d x + c\right )^{2} + A}{\left (b \cos \left (d x + c\right )\right )^{\frac{9}{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{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt{b \cos \left (d x + c\right )}}{b^{5} \cos \left (d x + c\right )^{5}}, 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{C \cos \left (d x + c\right )^{2} + A}{\left (b \cos \left (d x + c\right )\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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