Optimal. Leaf size=100 \[ \frac{6 \sin (a+b x)}{5 b c^3 \sqrt{c \cos (a+b x)}}-\frac{6 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{c \cos (a+b x)}}{5 b c^4 \sqrt{\cos (a+b x)}}+\frac{2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}} \]
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Rubi [A] time = 0.0556972, antiderivative size = 100, 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} \[ \frac{6 \sin (a+b x)}{5 b c^3 \sqrt{c \cos (a+b x)}}-\frac{6 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{c \cos (a+b x)}}{5 b c^4 \sqrt{\cos (a+b x)}}+\frac{2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(c \cos (a+b x))^{7/2}} \, dx &=\frac{2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac{3 \int \frac{1}{(c \cos (a+b x))^{3/2}} \, dx}{5 c^2}\\ &=\frac{2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac{6 \sin (a+b x)}{5 b c^3 \sqrt{c \cos (a+b x)}}-\frac{3 \int \sqrt{c \cos (a+b x)} \, dx}{5 c^4}\\ &=\frac{2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac{6 \sin (a+b x)}{5 b c^3 \sqrt{c \cos (a+b x)}}-\frac{\left (3 \sqrt{c \cos (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \, dx}{5 c^4 \sqrt{\cos (a+b x)}}\\ &=-\frac{6 \sqrt{c \cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b c^4 \sqrt{\cos (a+b x)}}+\frac{2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac{6 \sin (a+b x)}{5 b c^3 \sqrt{c \cos (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0947585, size = 68, normalized size = 0.68 \[ \frac{6 \sin (a+b x)-6 \sqrt{\cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )+2 \tan (a+b x) \sec (a+b x)}{5 b c^3 \sqrt{c \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.175, size = 366, normalized size = 3.7 \begin{align*}{\frac{2}{5\,{c}^{4}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 ( 12\,\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}-24\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}\cos \left ( 1/2\,bx+a/2 \right ) -12\,\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+24\,\cos \left ( 1/2\,bx+a/2 \right ) \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+3\,\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) -8\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}\cos \left ( 1/2\,bx+a/2 \right ) \right ) \sqrt{-2\,c \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+c \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-3} \left ( 8\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}-12\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+6\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \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 \frac{1}{\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 (\frac{\sqrt{c \cos \left (b x + a\right )}}{c^{4} \cos \left (b x + a\right )^{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 \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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