Optimal. Leaf size=161 \[ -\frac {5 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac {5 \left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{16 d}+\frac {5}{16} x \left (a^2+b^2\right )^3-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d} \]
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Rubi [A] time = 0.08, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3073, 8} \[ -\frac {5 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac {5 \left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{16 d}+\frac {5}{16} x \left (a^2+b^2\right )^3-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3073
Rubi steps
\begin {align*} \int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx &=-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}+\frac {1}{6} \left (5 \left (a^2+b^2\right )\right ) \int (a \cos (c+d x)+b \sin (c+d x))^4 \, dx\\ &=-\frac {5 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}+\frac {1}{8} \left (5 \left (a^2+b^2\right )^2\right ) \int (a \cos (c+d x)+b \sin (c+d x))^2 \, dx\\ &=-\frac {5 \left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{16 d}-\frac {5 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}+\frac {1}{16} \left (5 \left (a^2+b^2\right )^3\right ) \int 1 \, dx\\ &=\frac {5}{16} \left (a^2+b^2\right )^3 x-\frac {5 \left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{16 d}-\frac {5 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 192, normalized size = 1.19 \[ \frac {-36 a b \left (a^4-b^4\right ) \cos (4 (c+d x))+60 \left (a^2+b^2\right )^3 (c+d x)+45 \left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \sin (2 (c+d x))-90 a b \left (a^2+b^2\right )^2 \cos (2 (c+d x))-2 a b \left (3 a^4-10 a^2 b^2+3 b^4\right ) \cos (6 (c+d x))+9 \left (a^6-5 a^4 b^2-5 a^2 b^4+b^6\right ) \sin (4 (c+d x))+\left (a^6-15 a^4 b^2+15 a^2 b^4-b^6\right ) \sin (6 (c+d x))}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 219, normalized size = 1.36 \[ -\frac {144 \, a b^{5} \cos \left (d x + c\right )^{2} + 16 \, {\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{6} + 48 \, {\left (5 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 15 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d x - {\left (8 \, {\left (a^{6} - 15 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (5 \, a^{6} + 15 \, a^{4} b^{2} - 105 \, a^{2} b^{4} + 13 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (5 \, a^{6} + 15 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - 11 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 235, normalized size = 1.46 \[ \frac {5}{16} \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} x - \frac {{\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {3 \, {\left (a^{5} b - a b^{5}\right )} \cos \left (4 \, d x + 4 \, c\right )}{16 \, d} - \frac {15 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right )}{32 \, d} + \frac {{\left (a^{6} - 15 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - b^{6}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, {\left (a^{6} - 5 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + b^{6}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {15 \, {\left (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 285, normalized size = 1.77 \[ \frac {b^{6} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a \,b^{5} \left (\sin ^{6}\left (d x +c \right )\right )+15 a^{2} b^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+20 a^{3} b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )+15 a^{4} b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-a^{5} b \left (\cos ^{6}\left (d x +c \right )\right )+a^{6} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 238, normalized size = 1.48 \[ -\frac {192 \, a^{5} b \cos \left (d x + c\right )^{6} - 192 \, a b^{5} \sin \left (d x + c\right )^{6} + {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{6} - 15 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} b^{2} + 320 \, {\left (2 \, \sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4}\right )} a^{3} b^{3} + 15 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b^{4} - {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 60 \, d x + 60 \, c + 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b^{6}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.12, size = 519, normalized size = 3.22 \[ \frac {5\,\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\left (a^2+b^2\right )}^3}{8\,\left (\frac {5\,a^6}{8}+\frac {15\,a^4\,b^2}{8}+\frac {15\,a^2\,b^4}{8}+\frac {5\,b^6}{8}\right )}\right )\,{\left (a^2+b^2\right )}^3}{8\,d}-\frac {5\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,{\left (a^2+b^2\right )}^3}{8\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (40\,a^5\,b-\frac {160\,a^3\,b^3}{3}+64\,a\,b^5\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-\frac {11\,a^6}{8}+\frac {15\,a^4\,b^2}{8}+\frac {15\,a^2\,b^4}{8}+\frac {5\,b^6}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-\frac {11\,a^6}{8}+\frac {15\,a^4\,b^2}{8}+\frac {15\,a^2\,b^4}{8}+\frac {5\,b^6}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,a^6}{24}-\frac {235\,a^4\,b^2}{8}+\frac {85\,a^2\,b^4}{8}+\frac {85\,b^6}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {5\,a^6}{24}-\frac {235\,a^4\,b^2}{8}+\frac {85\,a^2\,b^4}{8}+\frac {85\,b^6}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {15\,a^6}{4}-\frac {195\,a^4\,b^2}{4}+\frac {285\,a^2\,b^4}{4}-\frac {33\,b^6}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {15\,a^6}{4}-\frac {195\,a^4\,b^2}{4}+\frac {285\,a^2\,b^4}{4}-\frac {33\,b^6}{4}\right )+80\,a^3\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+80\,a^3\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,a^5\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+12\,a^5\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.62, size = 770, normalized size = 4.78 \[ \begin {cases} \frac {5 a^{6} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{6} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{6} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 a^{6} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a^{6} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a^{6} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 a^{6} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {a^{5} b \cos ^{6}{\left (c + d x \right )}}{d} + \frac {15 a^{4} b^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {45 a^{4} b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {45 a^{4} b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {15 a^{4} b^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{4} b^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a^{4} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {15 a^{4} b^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {5 a^{3} b^{3} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac {5 a^{3} b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {15 a^{2} b^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {45 a^{2} b^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {45 a^{2} b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {15 a^{2} b^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{2} b^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {5 a^{2} b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {15 a^{2} b^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {a b^{5} \sin ^{6}{\left (c + d x \right )}}{d} + \frac {5 b^{6} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 b^{6} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 b^{6} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 b^{6} x \cos ^{6}{\left (c + d x \right )}}{16} - \frac {11 b^{6} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {5 b^{6} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {5 b^{6} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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