Optimal. Leaf size=72 \[ \frac{2 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{3 b^2 d}+\frac{2 \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 b^3 d} \]
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Rubi [A] time = 0.0377128, antiderivative size = 72, 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, 3768, 3771, 2641} \[ \frac{2 \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 b^3 d}+\frac{2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 b^2 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx &=\frac{\int (b \sec (c+d x))^{5/2} \, dx}{b^4}\\ &=\frac{2 (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 b^3 d}+\frac{\int \sqrt{b \sec (c+d x)} \, dx}{3 b^2}\\ &=\frac{2 (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 b^3 d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b^2}\\ &=\frac{2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 b^2 d}+\frac{2 (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 b^3 d}\\ \end{align*}
Mathematica [A] time = 0.0968243, size = 56, normalized size = 0.78 \[ \frac{2 \sec ^3(c+d x) \left (\cos ^{\frac{3}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\sin (c+d x)\right )}{3 d (b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.17, size = 125, normalized size = 1.7 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}{3\,d{b}^{3} \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 \frac{\sec \left (d x + c\right )^{4}}{\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{\sqrt{b \sec \left (d x + c\right )} \sec \left (d x + c\right )^{2}}{b^{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{\sec ^{4}{\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{\sec \left (d x + c\right )^{4}}{\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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