Optimal. Leaf size=144 \[ \frac{2 (7 A+6 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}-\frac{4 (7 A+6 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (7 A+6 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a B \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.264805, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {2981, 2759, 2751, 2646} \[ \frac{2 (7 A+6 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}-\frac{4 (7 A+6 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (7 A+6 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a B \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2981
Rule 2759
Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx &=\frac{2 a B \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{1}{7} (7 A+6 B) \int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a B \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (7 A+6 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 a d}+\frac{(2 (7 A+6 B)) \int \left (\frac{3 a}{2}-a \cos (c+d x)\right ) \sqrt{a+a \cos (c+d x)} \, dx}{35 a}\\ &=\frac{2 a B \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}-\frac{4 (7 A+6 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 (7 A+6 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 a d}+\frac{1}{15} (7 A+6 B) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a (7 A+6 B) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a B \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}-\frac{4 (7 A+6 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 (7 A+6 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 a d}\\ \end{align*}
Mathematica [A] time = 0.334778, size = 80, normalized size = 0.56 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} ((112 A+141 B) \cos (c+d x)+6 (7 A+6 B) \cos (2 (c+d x))+266 A+15 B \cos (3 (c+d x))+228 B)}{210 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.998, size = 102, normalized size = 0.7 \begin{align*}{\frac{2\,a\sqrt{2}}{105\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( -120\,B \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+ \left ( 84\,A+252\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -140\,A-210\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+105\,A+105\,B \right ){\frac{1}{\sqrt{ \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.93533, size = 159, normalized size = 1.1 \begin{align*} \frac{14 \,{\left (3 \, \sqrt{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, \sqrt{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 30 \, \sqrt{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} + 3 \,{\left (5 \, \sqrt{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 7 \, \sqrt{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 35 \, \sqrt{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 105 \, \sqrt{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} B \sqrt{a}}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.22406, size = 219, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (15 \, B \cos \left (d x + c\right )^{3} + 3 \,{\left (7 \, A + 6 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (7 \, A + 6 \, B\right )} \cos \left (d x + c\right ) + 56 \, A + 48 \, B\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 ) + d\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(-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]