Optimal. Leaf size=100 \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^5 d}+\frac{10 \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^3 d}+\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b^2 d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0601368, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 2635, 2642, 2641} \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^5 d}+\frac{10 \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^3 d}+\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b^2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{\cos ^6(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx &=\frac{\int (b \cos (c+d x))^{7/2} \, dx}{b^6}\\ &=\frac{2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^5 d}+\frac{5 \int (b \cos (c+d x))^{3/2} \, dx}{7 b^4}\\ &=\frac{10 \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 b^3 d}+\frac{2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^5 d}+\frac{5 \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx}{21 b^2}\\ &=\frac{10 \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 b^3 d}+\frac{2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^5 d}+\frac{\left (5 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b^2 d \sqrt{b \cos (c+d x)}}+\frac{10 \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 b^3 d}+\frac{2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^5 d}\\ \end{align*}
Mathematica [A] time = 0.0535276, size = 66, normalized size = 0.66 \[ \frac{26 \sin (2 (c+d x))+3 \sin (4 (c+d x))+40 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{84 b^2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.086, size = 210, normalized size = 2.1 \begin{align*} -{\frac{2}{21\,{b}^{2}d}\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 48\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}-120\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+128\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}-72\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+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 ) +16\,\cos \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/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 \frac{\cos \left (d x + c\right )^{6}}{\left (b \cos \left (d x + c\right )\right )^{\frac{5}{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{b \cos \left (d x + c\right )} \cos \left (d x + c\right )^{3}}{b^{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{6}}{\left (b \cos \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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