Optimal. Leaf size=67 \[ \frac{2 b \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{3 d}+\frac{2 \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.0407184, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {16, 3768, 3771, 2641} \[ \frac{2 \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}+\frac{2 b \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \sec (c+d x) (b \sec (c+d x))^{3/2} \, dx &=\frac{\int (b \sec (c+d x))^{5/2} \, dx}{b}\\ &=\frac{2 (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{1}{3} b \int \sqrt{b \sec (c+d x)} \, dx\\ &=\frac{2 (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\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 \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0721231, size = 49, normalized size = 0.73 \[ \frac{2 b \sqrt{b \sec (c+d x)} \left (\sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\tan (c+d x)\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.146, size = 122, 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 ) }{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}} \left ( i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -\cos \left ( dx+c \right ) +1 \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 \left (b \sec \left (d x + c\right )\right )^{\frac{3}{2}} \sec \left (d x + c\right )\,{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 \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 \left (b \sec{\left (c + d x \right )}\right )^{\frac{3}{2}} \sec{\left (c + d x \right )}\, 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 \left (b \sec \left (d x + c\right )\right )^{\frac{3}{2}} \sec \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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