Optimal. Leaf size=426 \[ \frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac{5 a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac{5 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac{25 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac{25 a^3 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{25}{64} a^3 b^2 x-\frac{5 a^4 b \cos ^8(c+d x)}{8 d}+\frac{a^5 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^5 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^5 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^5 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a^5 x}{128}-\frac{5 a b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{5 a b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{15 a b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{15}{128} a b^4 x-\frac{b^5 \sin ^8(c+d x)}{8 d}+\frac{b^5 \sin ^6(c+d x)}{6 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.412553, antiderivative size = 426, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3090, 2635, 8, 2565, 30, 2568, 14, 2564} \[ \frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac{5 a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac{5 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac{25 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac{25 a^3 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{25}{64} a^3 b^2 x-\frac{5 a^4 b \cos ^8(c+d x)}{8 d}+\frac{a^5 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^5 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^5 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^5 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a^5 x}{128}-\frac{5 a b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{5 a b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{15 a b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{15}{128} a b^4 x-\frac{b^5 \sin ^8(c+d x)}{8 d}+\frac{b^5 \sin ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3090
Rule 2635
Rule 8
Rule 2565
Rule 30
Rule 2568
Rule 14
Rule 2564
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \cos ^8(c+d x)+5 a^4 b \cos ^7(c+d x) \sin (c+d x)+10 a^3 b^2 \cos ^6(c+d x) \sin ^2(c+d x)+10 a^2 b^3 \cos ^5(c+d x) \sin ^3(c+d x)+5 a b^4 \cos ^4(c+d x) \sin ^4(c+d x)+b^5 \cos ^3(c+d x) \sin ^5(c+d x)\right ) \, dx\\ &=a^5 \int \cos ^8(c+d x) \, dx+\left (5 a^4 b\right ) \int \cos ^7(c+d x) \sin (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \cos ^5(c+d x) \sin ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+b^5 \int \cos ^3(c+d x) \sin ^5(c+d x) \, dx\\ &=\frac{a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac{5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac{1}{8} \left (7 a^5\right ) \int \cos ^6(c+d x) \, dx+\frac{1}{4} \left (5 a^3 b^2\right ) \int \cos ^6(c+d x) \, dx+\frac{1}{8} \left (15 a b^4\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac{\left (5 a^4 b\right ) \operatorname{Subst}\left (\int x^7 \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{b^5 \operatorname{Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{5 a^4 b \cos ^8(c+d x)}{8 d}+\frac{7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac{5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac{5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac{1}{48} \left (35 a^5\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{24} \left (25 a^3 b^2\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{16} \left (5 a b^4\right ) \int \cos ^4(c+d x) \, dx-\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{b^5 \operatorname{Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac{5 a^4 b \cos ^8(c+d x)}{8 d}+\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}+\frac{35 a^5 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{25 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac{5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac{7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac{5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac{5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}-\frac{b^5 \sin ^8(c+d x)}{8 d}+\frac{1}{64} \left (35 a^5\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{32} \left (25 a^3 b^2\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{64} \left (15 a b^4\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac{5 a^4 b \cos ^8(c+d x)}{8 d}+\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}+\frac{35 a^5 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{25 a^3 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac{15 a b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{35 a^5 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{25 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac{5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac{7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac{5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac{5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}-\frac{b^5 \sin ^8(c+d x)}{8 d}+\frac{1}{128} \left (35 a^5\right ) \int 1 \, dx+\frac{1}{64} \left (25 a^3 b^2\right ) \int 1 \, dx+\frac{1}{128} \left (15 a b^4\right ) \int 1 \, dx\\ &=\frac{35 a^5 x}{128}+\frac{25}{64} a^3 b^2 x+\frac{15}{128} a b^4 x-\frac{5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac{5 a^4 b \cos ^8(c+d x)}{8 d}+\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}+\frac{35 a^5 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{25 a^3 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac{15 a b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{35 a^5 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{25 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac{5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac{7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac{5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac{5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}-\frac{b^5 \sin ^8(c+d x)}{8 d}\\ \end{align*}
Mathematica [C] time = 0.877272, size = 259, normalized size = 0.61 \[ \frac{120 a (a-i b) (a+i b) \left (7 a^2+3 b^2\right ) (c+d x)+96 a^3 \left (7 a^2+5 b^2\right ) \sin (2 (c+d x))+32 a^3 \left (a^2-5 b^2\right ) \sin (6 (c+d x))+24 a \left (-10 a^2 b^2+7 a^4-5 b^4\right ) \sin (4 (c+d x))+3 a \left (-10 a^2 b^2+a^4+5 b^4\right ) \sin (8 (c+d x))-24 b \left (30 a^2 b^2+35 a^4+3 b^4\right ) \cos (2 (c+d x))+12 b \left (-10 a^2 b^2-35 a^4+b^4\right ) \cos (4 (c+d x))+8 b \left (10 a^2 b^2-15 a^4+b^4\right ) \cos (6 (c+d x))-3 b \left (-10 a^2 b^2+5 a^4+b^4\right ) \cos (8 (c+d x))}{3072 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.197, size = 305, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({b}^{5} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{8}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{12}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{24}} \right ) +5\,a{b}^{4} \left ( -1/8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-1/16\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{ \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) }{64}}+{\frac{3\,dx}{128}}+{\frac{3\,c}{128}} \right ) +10\,{a}^{2}{b}^{3} \left ( -1/8\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}-1/24\, \left ( \cos \left ( dx+c \right ) \right ) ^{6} \right ) +10\,{a}^{3}{b}^{2} \left ( -1/8\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}+1/48\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{128}}+{\frac{5\,c}{128}} \right ) -{\frac{5\,{a}^{4}b \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{8}}+{a}^{5} \left ({\frac{\sin \left ( dx+c \right ) }{8} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( dx+c \right ) }{16}} \right ) }+{\frac{35\,dx}{128}}+{\frac{35\,c}{128}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.25331, size = 308, normalized size = 0.72 \begin{align*} -\frac{1920 \, a^{4} b \cos \left (d x + c\right )^{8} +{\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{5} - 10 \,{\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} b^{2} - 1280 \,{\left (3 \, \sin \left (d x + c\right )^{8} - 8 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4}\right )} a^{2} b^{3} - 15 \,{\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{4} + 128 \,{\left (3 \, \sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6}\right )} b^{5}}{3072 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.56612, size = 509, normalized size = 1.19 \begin{align*} -\frac{96 \, b^{5} \cos \left (d x + c\right )^{4} + 48 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{8} + 128 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{6} - 15 \,{\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} d x -{\left (48 \,{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 8 \,{\left (7 \, a^{5} + 10 \, a^{3} b^{2} - 45 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 10 \,{\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 27.4128, size = 821, normalized size = 1.93 \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 = 1.41208, size = 375, normalized size = 0.88 \begin{align*} \frac{5}{128} \,{\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} x - \frac{{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{{\left (15 \, a^{4} b - 10 \, a^{2} b^{3} - b^{5}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac{{\left (35 \, a^{4} b + 10 \, a^{2} b^{3} - b^{5}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac{{\left (35 \, a^{4} b + 30 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac{{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac{{\left (a^{5} - 5 \, a^{3} b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac{{\left (7 \, a^{5} - 10 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{{\left (7 \, a^{5} + 5 \, a^{3} b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]