Optimal. Leaf size=115 \[ \frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 b^2 d}+\frac{2 (7 A+5 C) \sin (c+d x)}{21 b d \sqrt{b \sec (c+d x)}}+\frac{2 b^2 C \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}} \]
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Rubi [A] time = 0.133047, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3238, 4045, 3769, 3771, 2641} \[ \frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 b^2 d}+\frac{2 (7 A+5 C) \sin (c+d x)}{21 b d \sqrt{b \sec (c+d x)}}+\frac{2 b^2 C \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 4045
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx &=b^2 \int \frac{C+A \sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx\\ &=\frac{2 b^2 C \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}+\frac{1}{7} (7 A+5 C) \int \frac{1}{(b \sec (c+d x))^{3/2}} \, dx\\ &=\frac{2 (7 A+5 C) \sin (c+d x)}{21 b d \sqrt{b \sec (c+d x)}}+\frac{2 b^2 C \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}+\frac{(7 A+5 C) \int \sqrt{b \sec (c+d x)} \, dx}{21 b^2}\\ &=\frac{2 (7 A+5 C) \sin (c+d x)}{21 b d \sqrt{b \sec (c+d x)}}+\frac{2 b^2 C \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}+\frac{\left ((7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 b^2}\\ &=\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 b^2 d}+\frac{2 (7 A+5 C) \sin (c+d x)}{21 b d \sqrt{b \sec (c+d x)}}+\frac{2 b^2 C \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.524948, size = 79, normalized size = 0.69 \[ \frac{2 \sin (c+d x) (14 A+3 C \cos (2 (c+d x))+13 C)+\frac{4 (7 A+5 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{\sqrt{\cos (c+d x)}}}{42 b d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.533, size = 241, normalized size = 2.1 \begin{align*}{\frac{2\, \left ( 1+\cos \left ( dx+c \right ) \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }{21\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}} \left ( -7\,iA\sqrt{ \left ( 1+\cos \left ( dx+c \right ) \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) -5\,iC\sqrt{ \left ( 1+\cos \left ( dx+c \right ) \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sin \left ( dx+c \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) +3\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}-3\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+7\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+5\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}-7\,A\cos \left ( dx+c \right ) -5\,C\cos \left ( dx+c \right ) \right ) \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{3}{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{C \cos \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac{3}{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 \sec \left (d x + c\right )}}{b^{2} \sec \left (d x + c\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + C \cos ^{2}{\left (c + d x \right )}}{\left (b \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \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 \sec \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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