Optimal. Leaf size=261 \[ -\frac{(85 A-157 B) \sin (c+d x) \cos ^2(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(475 A-787 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 a^3 d}-\frac{(985 A-1729 B) \sin (c+d x)}{120 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(163 A-283 B) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(13 A-21 B) \sin (c+d x) \cos ^3(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.798846, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {2977, 2983, 2968, 3023, 2751, 2649, 206} \[ -\frac{(85 A-157 B) \sin (c+d x) \cos ^2(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(475 A-787 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 a^3 d}-\frac{(985 A-1729 B) \sin (c+d x)}{120 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(163 A-283 B) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(13 A-21 B) \sin (c+d x) \cos ^3(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2977
Rule 2983
Rule 2968
Rule 3023
Rule 2751
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{\int \frac{\cos ^3(c+d x) \left (4 a (A-B)-\frac{1}{2} a (5 A-13 B) \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{(13 A-21 B) \cos ^3(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\cos ^2(c+d x) \left (\frac{3}{2} a^2 (13 A-21 B)-\frac{1}{4} a^2 (85 A-157 B) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{8 a^4}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{(13 A-21 B) \cos ^3(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(85 A-157 B) \cos ^2(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{\cos (c+d x) \left (-\frac{1}{2} a^3 (85 A-157 B)+\frac{1}{8} a^3 (475 A-787 B) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{20 a^5}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{(13 A-21 B) \cos ^3(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(85 A-157 B) \cos ^2(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{-\frac{1}{2} a^3 (85 A-157 B) \cos (c+d x)+\frac{1}{8} a^3 (475 A-787 B) \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{20 a^5}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{(13 A-21 B) \cos ^3(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(85 A-157 B) \cos ^2(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{(475 A-787 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{240 a^3 d}+\frac{\int \frac{\frac{1}{16} a^4 (475 A-787 B)-\frac{1}{8} a^4 (985 A-1729 B) \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{30 a^6}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{(13 A-21 B) \cos ^3(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(985 A-1729 B) \sin (c+d x)}{120 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(85 A-157 B) \cos ^2(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{(475 A-787 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{240 a^3 d}+\frac{(163 A-283 B) \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx}{32 a^2}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{(13 A-21 B) \cos ^3(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(985 A-1729 B) \sin (c+d x)}{120 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(85 A-157 B) \cos ^2(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{(475 A-787 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{240 a^3 d}-\frac{(163 A-283 B) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{16 a^2 d}\\ &=\frac{(163 A-283 B) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \cos (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{(13 A-21 B) \cos ^3(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(985 A-1729 B) \sin (c+d x)}{120 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(85 A-157 B) \cos ^2(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{(475 A-787 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{240 a^3 d}\\ \end{align*}
Mathematica [A] time = 1.46063, size = 139, normalized size = 0.53 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right ) (-5 (479 A-887 B) \cos (c+d x)+(832 B-400 A) \cos (2 (c+d x))+40 A \cos (3 (c+d x))-1895 A-40 B \cos (3 (c+d x))+12 B \cos (4 (c+d x))+3491 B)+30 (163 A-283 B) \cos ^3\left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{240 a d (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 2.654, size = 467, normalized size = 1.8 \begin{align*}{\frac{1}{480\,d}\sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 768\,B\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+640\,A\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-2176\,B\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+2445\,A\ln \left ( 2\,{\frac{2\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+2\,a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) \sqrt{2} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a-4245\,B\ln \left ( 2\,{\frac{2\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+2\,a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) \sqrt{2} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a-2560\,A\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+5248\,B\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-435\,A\sqrt{a}\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+555\,B\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+30\,A\sqrt{a}\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}-30\,B\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}{a}^{-{\frac{7}{2}}} \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 [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]
________________________________________________________________________________________
Fricas [A] time = 1.72255, size = 741, normalized size = 2.84 \begin{align*} -\frac{15 \, \sqrt{2}{\left ({\left (163 \, A - 283 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (163 \, A - 283 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (163 \, A - 283 \, B\right )} \cos \left (d x + c\right ) + 163 \, A - 283 \, B\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} + 2 \, \sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - 4 \,{\left (96 \, B \cos \left (d x + c\right )^{4} + 160 \,{\left (A - B\right )} \cos \left (d x + c\right )^{3} - 32 \,{\left (25 \, A - 49 \, B\right )} \cos \left (d x + c\right )^{2} - 5 \,{\left (503 \, A - 911 \, B\right )} \cos \left (d x + c\right ) - 1495 \, A + 2671 \, B\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{960 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} 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 = 2.28102, size = 347, normalized size = 1.33 \begin{align*} -\frac{\frac{15 \,{\left (163 \, \sqrt{2} A - 283 \, \sqrt{2} B\right )} \log \left ({\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 |}\right )}{a^{\frac{5}{2}}} - \frac{{\left ({\left ({\left (15 \,{\left (\frac{2 \,{\left (\sqrt{2} A a^{2} - \sqrt{2} B a^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{2}} - \frac{21 \, \sqrt{2} A a^{2} - 29 \, \sqrt{2} B a^{2}}{a^{2}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \frac{3685 \, \sqrt{2} A a^{2} - 6733 \, \sqrt{2} B a^{2}}{a^{2}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \frac{5 \,{\left (1133 \, \sqrt{2} A a^{2} - 1973 \, \sqrt{2} B a^{2}\right )}}{a^{2}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \frac{15 \,{\left (155 \, \sqrt{2} A a^{2} - 291 \, \sqrt{2} B a^{2}\right )}}{a^{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}}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]