Optimal. Leaf size=142 \[ \frac{2 a^3 (35 A+32 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (5 A+8 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d}+\frac{2 a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.414053, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {2976, 2981, 2773, 206} \[ \frac{2 a^3 (35 A+32 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (5 A+8 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d}+\frac{2 a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2976
Rule 2981
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx &=\frac{2 a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{2}{5} \int (a+a \cos (c+d x))^{3/2} \left (\frac{5 a A}{2}+\frac{1}{2} a (5 A+8 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{2 a^2 (5 A+8 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{4}{15} \int \sqrt{a+a \cos (c+d x)} \left (\frac{15 a^2 A}{4}+\frac{1}{4} a^2 (35 A+32 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{2 a^3 (35 A+32 B) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (5 A+8 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\left (a^2 A\right ) \int \sqrt{a+a \cos (c+d x)} \sec (c+d x) \, dx\\ &=\frac{2 a^3 (35 A+32 B) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (5 A+8 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}-\frac{\left (2 a^3 A\right ) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{d}\\ &=\frac{2 a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{d}+\frac{2 a^3 (35 A+32 B) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (5 A+8 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.380174, size = 104, normalized size = 0.73 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right ) (2 (5 A+14 B) \cos (c+d x)+80 A+3 B \cos (2 (c+d x))+89 B)+15 \sqrt{2} A \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{15 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 4.168, size = 311, normalized size = 2.2 \begin{align*}{\frac{1}{15\,d}{a}^{{\frac{3}{2}}}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 24\,B\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-20\,\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a}\sqrt{2} \left ( A+4\,B \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+90\,A\sqrt{a}\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+15\,A\ln \left ( -4\,{\frac{\sqrt{a}\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}-a\sqrt{2}\cos \left ( 1/2\,dx+c/2 \right ) +2\,a}{-2\,\cos \left ( 1/2\,dx+c/2 \right ) +\sqrt{2}}} \right ) a+15\,A\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a\sqrt{2}\cos \left ( 1/2\,dx+c/2 \right ) +2\,a}{2\,\cos \left ( 1/2\,dx+c/2 \right ) +\sqrt{2}}} \right ) a+120\,B\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\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 = 2.64634, size = 82, normalized size = 0.58 \begin{align*} \frac{{\left (3 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 25 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 150 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} B \sqrt{a}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.7643, size = 462, normalized size = 3.25 \begin{align*} \frac{15 \,{\left (A a^{2} \cos \left (d x + c\right ) + A a^{2}\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a}{\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \,{\left (3 \, B a^{2} \cos \left (d x + c\right )^{2} +{\left (5 \, A + 14 \, B\right )} a^{2} \cos \left (d x + c\right ) +{\left (40 \, A + 43 \, B\right )} a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{30 \,{\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 [A] time = 3.8328, size = 306, normalized size = 2.15 \begin{align*} \frac{\frac{15 \, A a^{\frac{7}{2}} \log \left (\frac{{\left | 2 \,{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} - 4 \, \sqrt{2}{\left | a \right |} - 6 \, a \right |}}{{\left | 2 \,{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} + 4 \, \sqrt{2}{\left | a \right |} - 6 \, a \right |}}\right )}{{\left | a \right |}} + \frac{2 \,{\left (45 \, \sqrt{2} A a^{5} + 60 \, \sqrt{2} B a^{5} +{\left (80 \, \sqrt{2} A a^{5} + 80 \, \sqrt{2} B a^{5} +{\left (35 \, \sqrt{2} A a^{5} + 32 \, \sqrt{2} B a^{5}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{5}{2}}}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]