Optimal. Leaf size=100 \[ \frac{10 \sqrt{\cos (a+b x)} \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right ) \sqrt{c \sec (a+b x)}}{21 b c^4}+\frac{10 \sin (a+b x)}{21 b c^3 \sqrt{c \sec (a+b x)}}+\frac{2 \sin (a+b x)}{7 b c (c \sec (a+b x))^{5/2}} \]
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Rubi [A] time = 0.059252, 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 = {3769, 3771, 2641} \[ \frac{10 \sin (a+b x)}{21 b c^3 \sqrt{c \sec (a+b x)}}+\frac{10 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{c \sec (a+b x)}}{21 b c^4}+\frac{2 \sin (a+b x)}{7 b c (c \sec (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(c \sec (a+b x))^{7/2}} \, dx &=\frac{2 \sin (a+b x)}{7 b c (c \sec (a+b x))^{5/2}}+\frac{5 \int \frac{1}{(c \sec (a+b x))^{3/2}} \, dx}{7 c^2}\\ &=\frac{2 \sin (a+b x)}{7 b c (c \sec (a+b x))^{5/2}}+\frac{10 \sin (a+b x)}{21 b c^3 \sqrt{c \sec (a+b x)}}+\frac{5 \int \sqrt{c \sec (a+b x)} \, dx}{21 c^4}\\ &=\frac{2 \sin (a+b x)}{7 b c (c \sec (a+b x))^{5/2}}+\frac{10 \sin (a+b x)}{21 b c^3 \sqrt{c \sec (a+b x)}}+\frac{\left (5 \sqrt{\cos (a+b x)} \sqrt{c \sec (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{21 c^4}\\ &=\frac{10 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{c \sec (a+b x)}}{21 b c^4}+\frac{2 \sin (a+b x)}{7 b c (c \sec (a+b x))^{5/2}}+\frac{10 \sin (a+b x)}{21 b c^3 \sqrt{c \sec (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0960934, size = 66, normalized size = 0.66 \[ \frac{\sqrt{c \sec (a+b x)} \left (40 \sqrt{\cos (a+b x)} \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right )+26 \sin (2 (a+b x))+3 \sin (4 (a+b x))\right )}{84 b c^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.19, size = 153, normalized size = 1.5 \begin{align*} -{\frac{2\, \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( bx+a \right ) \right ) }{21\,b \left ( \cos \left ( bx+a \right ) \right ) ^{4} \left ( \sin \left ( bx+a \right ) \right ) ^{3}} \left ( 5\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }},i \right ) \sqrt{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( bx+a \right ) }{\cos \left ( bx+a \right ) +1}}}\sin \left ( bx+a \right ) -3\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}+3\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}-5\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+5\,\cos \left ( bx+a \right ) \right ) \left ({\frac{c}{\cos \left ( bx+a \right ) }} \right ) ^{-{\frac{7}{2}}}} \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 \sec \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 \sec \left (b x + a\right )}}{c^{4} \sec \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 \sec \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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