Optimal. Leaf size=104 \[ \frac{2 a (15 A+5 B+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (5 B-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.148088, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {3023, 2751, 2646} \[ \frac{2 a (15 A+5 B+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (5 B-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3023
Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \sqrt{a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac{2 \int \sqrt{a+a \cos (c+d x)} \left (\frac{1}{2} a (5 A+3 C)+\frac{1}{2} a (5 B-2 C) \cos (c+d x)\right ) \, dx}{5 a}\\ &=\frac{2 (5 B-2 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac{1}{15} (15 A+5 B+7 C) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a (15 A+5 B+7 C) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (5 B-2 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}\\ \end{align*}
Mathematica [A] time = 0.171688, size = 67, normalized size = 0.64 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} (30 A+2 (5 B+4 C) \cos (c+d x)+20 B+3 C \cos (2 (c+d x))+19 C)}{15 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.07, size = 86, normalized size = 0.8 \begin{align*}{\frac{2\,a\sqrt{2}}{15\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 12\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( -10\,B-20\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+15\,A+15\,B+15\,C \right ){\frac{1}{\sqrt{a \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.13371, size = 143, normalized size = 1.38 \begin{align*} \frac{60 \, \sqrt{2} A \sqrt{a} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 10 \,{\left (\sqrt{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 \, \sqrt{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} B \sqrt{a} +{\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 )} C \sqrt{a}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.85852, size = 180, normalized size = 1.73 \begin{align*} \frac{2 \,{\left (3 \, C \cos \left (d x + c\right )^{2} +{\left (5 \, B + 4 \, C\right )} \cos \left (d x + c\right ) + 15 \, A + 10 \, B + 8 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{15 \,{\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]