Optimal. Leaf size=97 \[ \frac{2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{14 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac{14 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 b d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0712229, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 3769, 3771, 2639} \[ \frac{2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{14 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac{14 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 b d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx &=b^3 \int \frac{1}{(b \sec (c+d x))^{9/2}} \, dx\\ &=\frac{2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{1}{9} (7 b) \int \frac{1}{(b \sec (c+d x))^{5/2}} \, dx\\ &=\frac{2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{14 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac{7 \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx}{15 b}\\ &=\frac{2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{14 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac{7 \int \sqrt{\cos (c+d x)} \, dx}{15 b \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}\\ &=\frac{14 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 b d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{14 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0770183, size = 73, normalized size = 0.75 \[ \frac{84 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+(33 \sin (c+d x)+5 \sin (3 (c+d x))) \cos ^{\frac{3}{2}}(c+d x)}{90 d \cos ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.19, size = 333, normalized size = 3.4 \begin{align*}{\frac{2}{45\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}} \left ( 21\,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 ) \sin \left ( dx+c \right ) \cos \left ( dx+c \right ) -21\,i\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) -5\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}+21\,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 ) \sin \left ( dx+c \right ) -21\,i\sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) -2\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-14\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+21\,\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{\cos \left (d x + c\right )^{3}}{\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 )} \cos \left (d x + c\right )^{3}}{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(-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 \frac{\cos \left (d x + c\right )^{3}}{\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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