Optimal. Leaf size=97 \[ \frac{6 b^2 \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 d}-\frac{6 b^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d} \]
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Rubi [A] time = 0.0567826, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {16, 3768, 3771, 2639} \[ \frac{6 b^2 \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 d}-\frac{6 b^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \sec (c+d x) (b \sec (c+d x))^{5/2} \, dx &=\frac{\int (b \sec (c+d x))^{7/2} \, dx}{b}\\ &=\frac{2 (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac{1}{5} (3 b) \int (b \sec (c+d x))^{3/2} \, dx\\ &=\frac{6 b^2 \sqrt{b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac{1}{5} \left (3 b^3\right ) \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx\\ &=\frac{6 b^2 \sqrt{b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac{\left (3 b^3\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}\\ &=-\frac{6 b^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{6 b^2 \sqrt{b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.164128, size = 61, normalized size = 0.63 \[ \frac{(b \sec (c+d x))^{5/2} \left (7 \sin (c+d x)+3 \sin (3 (c+d x))-12 \cos ^{\frac{5}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{10 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.204, size = 348, normalized size = 3.6 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}} \left ( 3\,i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \left ( \cos \left ( dx+c \right ) \right ) ^{3}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) -3\,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 ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +3\,i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \left ( \cos \left ( dx+c \right ) \right ) ^{2}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) -3\,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 ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+1 \right ) \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{5}{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{5}{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^{2} \sec \left (d x + c\right )^{3}, 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{5}{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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