Optimal. Leaf size=175 \[ \frac{2 a (6 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a (6 A+7 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (6 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.405595, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {2961, 2980, 2772, 2771} \[ \frac{2 a (6 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a (6 A+7 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (6 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2961
Rule 2980
Rule 2772
Rule 2771
Rubi steps
\begin{align*} \int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{1}{7} \left ((6 A+7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a (6 A+7 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a A \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{1}{35} \left (4 (6 A+7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{8 a (6 A+7 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (6 A+7 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a A \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{1}{105} \left (8 (6 A+7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{16 a (6 A+7 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{8 a (6 A+7 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (6 A+7 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a A \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.428378, size = 102, normalized size = 0.58 \[ \frac{2 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (9 (6 A+7 B) \cos (c+d x)+2 (6 A+7 B) \cos (2 (c+d x))+12 A \cos (3 (c+d x))+27 A+14 B \cos (3 (c+d x))+14 B)}{105 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.801, size = 116, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( 48\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+56\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+24\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+28\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+18\,A\cos \left ( dx+c \right ) +21\,B\cos \left ( dx+c \right ) +15\,A \right ) \cos \left ( dx+c \right ) }{105\,d\sin \left ( dx+c \right ) } \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{9}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 2.17877, size = 767, normalized size = 4.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.64265, size = 273, normalized size = 1.56 \begin{align*} \frac{2 \,{\left (8 \,{\left (6 \, A + 7 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (6 \, A + 7 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (6 \, A + 7 \, B\right )} \cos \left (d x + c\right ) + 15 \, A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )} \sqrt{\cos \left (d x + c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (d x + c\right ) + A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]