Optimal. Leaf size=160 \[ -\frac{5 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{56 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac{3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{56 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.219112, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2765, 2977, 2748, 2635, 2641, 2639} \[ -\frac{5 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{56 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac{3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{56 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2765
Rule 2977
Rule 2748
Rule 2635
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx &=-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (\frac{7 a}{2}-\frac{11}{2} a \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{3 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{\int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{45 a^2}{2}-28 a^2 \cos (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{3 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{15 \int \cos ^{\frac{3}{2}}(c+d x) \, dx}{2 a^2}+\frac{28 \int \cos ^{\frac{5}{2}}(c+d x) \, dx}{3 a^2}\\ &=-\frac{5 \sqrt{\cos (c+d x)} \sin (c+d x)}{a^2 d}+\frac{56 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}-\frac{3 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{5 \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{2 a^2}+\frac{28 \int \sqrt{\cos (c+d x)} \, dx}{5 a^2}\\ &=\frac{56 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac{5 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac{5 \sqrt{\cos (c+d x)} \sin (c+d x)}{a^2 d}+\frac{56 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}-\frac{3 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{\cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}\\ \end{align*}
Mathematica [C] time = 2.50988, size = 367, normalized size = 2.29 \[ \frac{\cos ^4\left (\frac{1}{2} (c+d x)\right ) \left (-\frac{2 \csc (c) \sqrt{\cos (c+d x)} \left (40 \sin ^2(c) \cos (d x)-6 \sin (c) \sin (2 c) \cos (2 d x)+8 \cos (c) (5 \sin (c) \sin (d x)+27)-6 \sin (c) \cos (2 c) \sin (2 d x)-10 \sin \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right )+240 \sin \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right )-5 \sin (c) \tan \left (\frac{c}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )+120\right )}{3 d}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left (56 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )+25 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+56 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) d \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{5 a^2 (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 2.619, size = 283, normalized size = 1.8 \begin{align*} -{\frac{1}{30\,{a}^{2}d}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 96\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}-352\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+120\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-150\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-336\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +266\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-135\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+5 \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{\frac{9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (d x + c\right )^{\frac{9}{2}}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}}, x\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 \frac{\cos \left (d x + c\right )^{\frac{9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]