Optimal. Leaf size=110 \[ \frac{2 A b^2 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A b E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.120401, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {16, 2748, 2636, 2640, 2639, 2642, 2641} \[ \frac{2 A b^2 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A b E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2748
Rule 2636
Rule 2640
Rule 2639
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx &=b^3 \int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{3/2}} \, dx\\ &=\left (A b^3\right ) \int \frac{1}{(b \cos (c+d x))^{3/2}} \, dx+\left (b^2 B\right ) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx\\ &=\frac{2 A b^2 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-(A b) \int \sqrt{b \cos (c+d x)} \, dx+\frac{\left (b^2 B \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{\sqrt{b \cos (c+d x)}}\\ &=\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{b \cos (c+d x)}}+\frac{2 A b^2 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{\left (A b \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{\sqrt{\cos (c+d x)}}\\ &=-\frac{2 A b \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{b \cos (c+d x)}}+\frac{2 A b^2 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.158263, size = 73, normalized size = 0.66 \[ \frac{2 (b \cos (c+d x))^{3/2} \left (-A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{A \sin (c+d x)}{\sqrt{\cos (c+d x)}}+B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.645, size = 215, normalized size = 2. \begin{align*} -2\,{\frac{{b}^{2}\sqrt{-2\,b \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b} \left ( A\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -2\,A\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+B\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) }{\sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) }d}} \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 \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}} \sec \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 ({\left (B b \cos \left (d x + c\right )^{2} + A b \cos \left (d x + c\right )\right )} \sqrt{b \cos \left (d x + c\right )} \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 \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}} \sec \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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