Optimal. Leaf size=174 \[ \frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A b^2 \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A b E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.157485, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A b^2 \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A b E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{b \cos (c+d x)}} \]
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 ^5(c+d x) \, dx &=b^5 \int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{7/2}} \, dx\\ &=\left (A b^5\right ) \int \frac{1}{(b \cos (c+d x))^{7/2}} \, dx+\left (b^4 B\right ) \int \frac{1}{(b \cos (c+d x))^{5/2}} \, dx\\ &=\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^3 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{1}{5} \left (3 A b^3\right ) \int \frac{1}{(b \cos (c+d x))^{3/2}} \, dx+\frac{1}{3} \left (b^2 B\right ) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx\\ &=\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^3 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{6 A b^2 \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{1}{5} (3 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}{3 \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 )}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^3 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{6 A b^2 \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{\left (3 A b \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 \sqrt{\cos (c+d x)}}\\ &=-\frac{6 A b \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 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 )}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^3 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{6 A b^2 \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.264613, size = 107, normalized size = 0.61 \[ \frac{2 \sec ^3(c+d x) (b \cos (c+d x))^{3/2} \left (\frac{9}{2} A \sin (2 (c+d x))+3 A \tan (c+d x)-9 A \cos ^{\frac{3}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+5 B \sin (c+d x)+5 B \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{15 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.467, size = 576, normalized size = 3.3 \begin{align*} \text{result too large to display} \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 )^{5}\,{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 )^{5}, 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 )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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