Optimal. Leaf size=70 \[ \frac{2 b \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{3 d}+\frac{2 b^2 \sin (c+d x)}{3 d \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0542509, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 3769, 3771, 2641} \[ \frac{2 b^2 \sin (c+d x)}{3 d \sqrt{b \sec (c+d x)}}+\frac{2 b \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (b \sec (c+d x))^{3/2} \, dx &=b^3 \int \frac{1}{(b \sec (c+d x))^{3/2}} \, dx\\ &=\frac{2 b^2 \sin (c+d x)}{3 d \sqrt{b \sec (c+d x)}}+\frac{1}{3} b \int \sqrt{b \sec (c+d x)} \, dx\\ &=\frac{2 b^2 \sin (c+d x)}{3 d \sqrt{b \sec (c+d x)}}+\frac{1}{3} \left (b \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d}+\frac{2 b^2 \sin (c+d x)}{3 d \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0493521, size = 52, normalized size = 0.74 \[ \frac{b \sqrt{b \sec (c+d x)} \left (2 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\sin (2 (c+d x))\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.16, size = 129, normalized size = 1.8 \begin{align*} -{\frac{2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) \cos \left ( dx+c \right ) }{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}} \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}} \left ( i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) - \left ( \cos \left ( dx+c \right ) \right ) ^{2}+\cos \left ( dx+c \right ) \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 \left (b \sec \left (d x + c\right )\right )^{\frac{3}{2}} \cos \left (d x + c\right )^{3}\,{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 (\sqrt{b \sec \left (d x + c\right )} b \cos \left (d x + c\right )^{3} \sec \left (d x + c\right ), 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 \left (b \sec \left (d x + c\right )\right )^{\frac{3}{2}} \cos \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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