Optimal. Leaf size=126 \[ \frac{5 a \left (a^2+b^2\right ) \sin ^2(c+d x) (a \cot (c+d x)+b) (a-b \cot (c+d x))}{16 d}+\frac{5}{16} a x \left (a^2+b^2\right )^2+\frac{\sin ^6(c+d x) (a \cot (c+d x)+b)^5}{6 d}+\frac{5 a \sin ^4(c+d x) (a \cot (c+d x)+b)^3 (a-b \cot (c+d x))}{24 d} \]
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Rubi [A] time = 0.0898664, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3088, 805, 723, 203} \[ \frac{5 a \left (a^2+b^2\right ) \sin ^2(c+d x) (a \cot (c+d x)+b) (a-b \cot (c+d x))}{16 d}+\frac{5}{16} a x \left (a^2+b^2\right )^2+\frac{\sin ^6(c+d x) (a \cot (c+d x)+b)^5}{6 d}+\frac{5 a \sin ^4(c+d x) (a \cot (c+d x)+b)^3 (a-b \cot (c+d x))}{24 d} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 805
Rule 723
Rule 203
Rubi steps
\begin{align*} \int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x (b+a x)^5}{\left (1+x^2\right )^4} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{(b+a \cot (c+d x))^5 \sin ^6(c+d x)}{6 d}-\frac{(5 a) \operatorname{Subst}\left (\int \frac{(b+a x)^4}{\left (1+x^2\right )^3} \, dx,x,\cot (c+d x)\right )}{6 d}\\ &=\frac{5 a (b+a \cot (c+d x))^3 (a-b \cot (c+d x)) \sin ^4(c+d x)}{24 d}+\frac{(b+a \cot (c+d x))^5 \sin ^6(c+d x)}{6 d}-\frac{\left (5 a \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{(b+a x)^2}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{8 d}\\ &=\frac{5 a \left (a^2+b^2\right ) (b+a \cot (c+d x)) (a-b \cot (c+d x)) \sin ^2(c+d x)}{16 d}+\frac{5 a (b+a \cot (c+d x))^3 (a-b \cot (c+d x)) \sin ^4(c+d x)}{24 d}+\frac{(b+a \cot (c+d x))^5 \sin ^6(c+d x)}{6 d}-\frac{\left (5 a \left (a^2+b^2\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{16 d}\\ &=\frac{5}{16} a \left (a^2+b^2\right )^2 x+\frac{5 a \left (a^2+b^2\right ) (b+a \cot (c+d x)) (a-b \cot (c+d x)) \sin ^2(c+d x)}{16 d}+\frac{5 a (b+a \cot (c+d x))^3 (a-b \cot (c+d x)) \sin ^4(c+d x)}{24 d}+\frac{(b+a \cot (c+d x))^5 \sin ^6(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.617089, size = 188, normalized size = 1.49 \[ \frac{60 a \left (a^2+b^2\right )^2 (c+d x)+15 a \left (2 a^2 b^2+3 a^4-b^4\right ) \sin (2 (c+d x))+3 a \left (-10 a^2 b^2+3 a^4-5 b^4\right ) \sin (4 (c+d x))+a \left (-10 a^2 b^2+a^4+5 b^4\right ) \sin (6 (c+d x))-15 b \left (6 a^2 b^2+5 a^4+b^4\right ) \cos (2 (c+d x))+6 b \left (b^4-5 a^4\right ) \cos (4 (c+d x))-b \left (-10 a^2 b^2+5 a^4+b^4\right ) \cos (6 (c+d x))}{192 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.18, size = 236, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6}}+5\,a{b}^{4} \left ( -1/6\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}-1/8\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+1/16\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/16\,dx+c/16 \right ) +10\,{a}^{2}{b}^{3} \left ( -1/6\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}-1/12\, \left ( \cos \left ( dx+c \right ) \right ) ^{4} \right ) +10\,{a}^{3}{b}^{2} \left ( -1/6\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1/24\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +1/16\,dx+c/16 \right ) -{\frac{5\,{a}^{4}b \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6}}+{a}^{5} \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09425, size = 252, normalized size = 2. \begin{align*} -\frac{160 \, a^{4} b \cos \left (d x + c\right )^{6} - 32 \, b^{5} \sin \left (d x + c\right )^{6} +{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{5} - 10 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} b^{2} + 160 \,{\left (2 \, \sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4}\right )} a^{2} b^{3} + 5 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{4}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.536616, size = 417, normalized size = 3.31 \begin{align*} -\frac{24 \, b^{5} \cos \left (d x + c\right )^{2} + 8 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{6} + 24 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{4} - 15 \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d x -{\left (8 \,{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 10 \,{\left (a^{5} + 2 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.86799, size = 663, normalized size = 5.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32448, size = 285, normalized size = 2.26 \begin{align*} \frac{5}{16} \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} x - \frac{{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac{{\left (5 \, a^{4} b - b^{5}\right )} \cos \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac{5 \,{\left (5 \, a^{4} b + 6 \, a^{2} b^{3} + b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{{\left (3 \, a^{5} - 10 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{5 \,{\left (3 \, a^{5} + 2 \, a^{3} b^{2} - a b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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