Optimal. Leaf size=65 \[ -\frac{6 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b}+\frac{2 \sin (a+b x)}{5 b \cos ^{\frac{5}{2}}(a+b x)}+\frac{6 \sin (a+b x)}{5 b \sqrt{\cos (a+b x)}} \]
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Rubi [A] time = 0.0292644, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2636, 2639} \[ -\frac{6 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b}+\frac{2 \sin (a+b x)}{5 b \cos ^{\frac{5}{2}}(a+b x)}+\frac{6 \sin (a+b x)}{5 b \sqrt{\cos (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{7}{2}}(a+b x)} \, dx &=\frac{2 \sin (a+b x)}{5 b \cos ^{\frac{5}{2}}(a+b x)}+\frac{3}{5} \int \frac{1}{\cos ^{\frac{3}{2}}(a+b x)} \, dx\\ &=\frac{2 \sin (a+b x)}{5 b \cos ^{\frac{5}{2}}(a+b x)}+\frac{6 \sin (a+b x)}{5 b \sqrt{\cos (a+b x)}}-\frac{3}{5} \int \sqrt{\cos (a+b x)} \, dx\\ &=-\frac{6 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b}+\frac{2 \sin (a+b x)}{5 b \cos ^{\frac{5}{2}}(a+b x)}+\frac{6 \sin (a+b x)}{5 b \sqrt{\cos (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0932566, size = 59, normalized size = 0.91 \[ \frac{3 \sin (2 (a+b x))+2 \tan (a+b x)-6 \cos ^{\frac{3}{2}}(a+b x) E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b \cos ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.014, size = 358, normalized size = 5.5 \begin{align*}{\frac{2}{5\,b}\sqrt{- \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\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \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} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-3}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}}}} \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}{\cos \left (b x + a\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{1}{\cos \left (b x + a\right )^{\frac{7}{2}}}, 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}{\cos \left (b x + a\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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