Optimal. Leaf size=118 \[ \frac{4 \sin ^3(c+d x)}{3 a d}-\frac{4 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{15 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{15 x}{8 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.106884, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2767, 2748, 2633, 2635, 8} \[ \frac{4 \sin ^3(c+d x)}{3 a d}-\frac{4 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{15 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{15 x}{8 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2767
Rule 2748
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{a+a \cos (c+d x)} \, dx &=-\frac{\cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{\int \cos ^3(c+d x) (4 a-5 a \cos (c+d x)) \, dx}{a^2}\\ &=-\frac{\cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{4 \int \cos ^3(c+d x) \, dx}{a}+\frac{5 \int \cos ^4(c+d x) \, dx}{a}\\ &=\frac{5 \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{\cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{15 \int \cos ^2(c+d x) \, dx}{4 a}+\frac{4 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d}\\ &=-\frac{4 \sin (c+d x)}{a d}+\frac{15 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac{5 \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{\cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{4 \sin ^3(c+d x)}{3 a d}+\frac{15 \int 1 \, dx}{8 a}\\ &=\frac{15 x}{8 a}-\frac{4 \sin (c+d x)}{a d}+\frac{15 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac{5 \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{\cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{4 \sin ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.318477, size = 173, normalized size = 1.47 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \left (-168 \sin \left (c+\frac{d x}{2}\right )-120 \sin \left (c+\frac{3 d x}{2}\right )-120 \sin \left (2 c+\frac{3 d x}{2}\right )+40 \sin \left (2 c+\frac{5 d x}{2}\right )+40 \sin \left (3 c+\frac{5 d x}{2}\right )-5 \sin \left (3 c+\frac{7 d x}{2}\right )-5 \sin \left (4 c+\frac{7 d x}{2}\right )+3 \sin \left (4 c+\frac{9 d x}{2}\right )+3 \sin \left (5 c+\frac{9 d x}{2}\right )+360 d x \cos \left (c+\frac{d x}{2}\right )-552 \sin \left (\frac{d x}{2}\right )+360 d x \cos \left (\frac{d x}{2}\right )\right )}{384 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 171, normalized size = 1.5 \begin{align*} -{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{25}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{115}{12\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{109}{12\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{7}{4\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{15}{4\,da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.7319, size = 293, normalized size = 2.48 \begin{align*} -\frac{\frac{\frac{21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{109 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{115 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{75 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac{4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac{45 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{12 \, \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.65273, size = 213, normalized size = 1.81 \begin{align*} \frac{45 \, d x \cos \left (d x + c\right ) + 45 \, d x +{\left (6 \, \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} + 13 \, \cos \left (d x + c\right )^{2} - 19 \, \cos \left (d x + c\right ) - 64\right )} \sin \left (d x + c\right )}{24 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 11.5405, size = 882, normalized size = 7.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.35848, size = 136, normalized size = 1.15 \begin{align*} \frac{\frac{45 \,{\left (d x + c\right )}}{a} - \frac{24 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a} - \frac{2 \,{\left (75 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 115 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 109 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]