Optimal. Leaf size=135 \[ \frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.130198, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {16, 2748, 2636, 2642, 2641, 2640, 2639} \[ \frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2748
Rule 2636
Rule 2642
Rule 2641
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx &=b^2 \int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx\\ &=\left (A b^2\right ) \int \frac{1}{(b \cos (c+d x))^{5/2}} \, dx+(b B) \int \frac{1}{(b \cos (c+d x))^{3/2}} \, dx\\ &=\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}+\frac{1}{3} A \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx-\frac{B \int \sqrt{b \cos (c+d x)} \, dx}{b}\\ &=\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}+\frac{\left (A \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 \sqrt{b \cos (c+d x)}}-\frac{\left (B \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{b \sqrt{\cos (c+d x)}}\\ &=-\frac{2 B \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d \sqrt{\cos (c+d x)}}+\frac{2 A \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.146592, size = 84, normalized size = 0.62 \[ \frac{2 \left (A \tan (c+d x)+A \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+3 B \sin (c+d x)-3 B \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{3 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 6.625, size = 405, normalized size = 3. \begin{align*}{\frac{2}{3\,bd}\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 2\,A{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \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} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+6\,B{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \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} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-12\,B\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-A\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{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}-3\,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 EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +6\,B \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) \right ) \sqrt{-2\,b \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}b} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3} \left ( 4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{-1}{\frac{1}{\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) }}}} \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{{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{2}}{\sqrt{b \cos \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 (\frac{{\left (B \cos \left (d x + c\right ) + A\right )} \sqrt{b \cos \left (d x + c\right )} \sec \left (d x + c\right )^{2}}{b \cos \left (d x + c\right )}, 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{{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{2}}{\sqrt{b \cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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