Optimal. Leaf size=113 \[ \frac{2 b^2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}+\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}-\frac{2 b (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.126973, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {16, 3012, 2636, 2640, 2639} \[ \frac{2 b^2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}+\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}-\frac{2 b (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3012
Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx &=b^5 \int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{7/2}} \, dx\\ &=\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{1}{5} \left (b^3 (3 A+5 C)\right ) \int \frac{1}{(b \cos (c+d x))^{3/2}} \, dx\\ &=\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{1}{5} (b (3 A+5 C)) \int \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^2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{\left (b (3 A+5 C) \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 \sqrt{\cos (c+d x)}}\\ &=-\frac{2 b (3 A+5 C) \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 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.228633, size = 84, normalized size = 0.74 \[ -\frac{\sec ^3(c+d x) (b \cos (c+d x))^{3/2} \left (-(3 A+5 C) \sin (2 (c+d x))+2 (3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )-2 A \tan (c+d x)\right )}{5 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 8.875, size = 599, normalized size = 5.3 \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 (C \cos \left (d x + c\right )^{2} + 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 (C b \cos \left (d x + c\right )^{3} + 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 (C \cos \left (d x + c\right )^{2} + 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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