Optimal. Leaf size=301 \[ \frac {a^4 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5 a^4 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {5 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {5 a^4 x}{16}-\frac {2 a^3 b \cos ^6(c+d x)}{3 d}-\frac {a^2 b^2 \sin (c+d x) \cos ^5(c+d x)}{d}+\frac {a^2 b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 a^2 b^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3}{8} a^2 b^2 x-\frac {2 a b^3 \sin ^6(c+d x)}{3 d}+\frac {a b^3 \sin ^4(c+d x)}{d}-\frac {b^4 \sin ^3(c+d x) \cos ^3(c+d x)}{6 d}-\frac {b^4 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac {b^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {b^4 x}{16} \]
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Rubi [A] time = 0.30, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 19, 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, 2564, 14} \[ -\frac {a^2 b^2 \sin (c+d x) \cos ^5(c+d x)}{d}+\frac {a^2 b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 a^2 b^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3}{8} a^2 b^2 x-\frac {2 a^3 b \cos ^6(c+d x)}{3 d}+\frac {a^4 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5 a^4 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {5 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {5 a^4 x}{16}-\frac {2 a b^3 \sin ^6(c+d x)}{3 d}+\frac {a b^3 \sin ^4(c+d x)}{d}-\frac {b^4 \sin ^3(c+d x) \cos ^3(c+d x)}{6 d}-\frac {b^4 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac {b^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {b^4 x}{16} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 30
Rule 2564
Rule 2565
Rule 2568
Rule 2635
Rule 3090
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=\int \left (a^4 \cos ^6(c+d x)+4 a^3 b \cos ^5(c+d x) \sin (c+d x)+6 a^2 b^2 \cos ^4(c+d x) \sin ^2(c+d x)+4 a b^3 \cos ^3(c+d x) \sin ^3(c+d x)+b^4 \cos ^2(c+d x) \sin ^4(c+d x)\right ) \, dx\\ &=a^4 \int \cos ^6(c+d x) \, dx+\left (4 a^3 b\right ) \int \cos ^5(c+d x) \sin (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \cos ^3(c+d x) \sin ^3(c+d x) \, dx+b^4 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx\\ &=\frac {a^4 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{d}-\frac {b^4 \cos ^3(c+d x) \sin ^3(c+d x)}{6 d}+\frac {1}{6} \left (5 a^4\right ) \int \cos ^4(c+d x) \, dx+\left (a^2 b^2\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{2} b^4 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (4 a^3 b\right ) \operatorname {Subst}\left (\int x^5 \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (4 a b^3\right ) \operatorname {Subst}\left (\int x^3 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {2 a^3 b \cos ^6(c+d x)}{3 d}+\frac {5 a^4 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 b^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^3(c+d x) \sin (c+d x)}{8 d}+\frac {a^4 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{d}-\frac {b^4 \cos ^3(c+d x) \sin ^3(c+d x)}{6 d}+\frac {1}{8} \left (5 a^4\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{4} \left (3 a^2 b^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{8} b^4 \int \cos ^2(c+d x) \, dx+\frac {\left (4 a b^3\right ) \operatorname {Subst}\left (\int \left (x^3-x^5\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {2 a^3 b \cos ^6(c+d x)}{3 d}+\frac {5 a^4 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {3 a^2 b^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {b^4 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a^4 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 b^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^3(c+d x) \sin (c+d x)}{8 d}+\frac {a^4 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{d}-\frac {b^4 \cos ^3(c+d x) \sin ^3(c+d x)}{6 d}+\frac {a b^3 \sin ^4(c+d x)}{d}-\frac {2 a b^3 \sin ^6(c+d x)}{3 d}+\frac {1}{16} \left (5 a^4\right ) \int 1 \, dx+\frac {1}{8} \left (3 a^2 b^2\right ) \int 1 \, dx+\frac {1}{16} b^4 \int 1 \, dx\\ &=\frac {5 a^4 x}{16}+\frac {3}{8} a^2 b^2 x+\frac {b^4 x}{16}-\frac {2 a^3 b \cos ^6(c+d x)}{3 d}+\frac {5 a^4 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {3 a^2 b^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {b^4 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a^4 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 b^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^3(c+d x) \sin (c+d x)}{8 d}+\frac {a^4 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{d}-\frac {b^4 \cos ^3(c+d x) \sin ^3(c+d x)}{6 d}+\frac {a b^3 \sin ^4(c+d x)}{d}-\frac {2 a b^3 \sin ^6(c+d x)}{3 d}\\ \end {align*}
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Mathematica [C] time = 0.43, size = 178, normalized size = 0.59 \[ \frac {-24 a^3 b \cos (4 (c+d x))+12 (a-i b) (a+i b) \left (5 a^2+b^2\right ) (c+d x)-12 a b \left (5 a^2+3 b^2\right ) \cos (2 (c+d x))-4 a b \left (a^2-b^2\right ) \cos (6 (c+d x))+3 \left (15 a^4+6 a^2 b^2-b^4\right ) \sin (2 (c+d x))+3 \left (3 a^4-6 a^2 b^2-b^4\right ) \sin (4 (c+d x))+\left (a^4-6 a^2 b^2+b^4\right ) \sin (6 (c+d x))}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 151, normalized size = 0.50 \[ -\frac {48 \, a b^{3} \cos \left (d x + c\right )^{4} + 32 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (5 \, a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} d x - {\left (8 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (5 \, a^{4} + 6 \, a^{2} b^{2} - 7 \, b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (5 \, a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.02, size = 187, normalized size = 0.62 \[ -\frac {a^{3} b \cos \left (4 \, d x + 4 \, c\right )}{8 \, d} + \frac {1}{16} \, {\left (5 \, a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} x - \frac {{\left (a^{3} b - a b^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{48 \, d} - \frac {{\left (5 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{16 \, d} + \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (3 \, a^{4} - 6 \, a^{2} b^{2} - b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (15 \, a^{4} + 6 \, a^{2} b^{2} - b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 28.97, size = 219, normalized size = 0.73 \[ \frac {b^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+4 a \,b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )+6 a^{2} b^{2} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {2 a^{3} b \left (\cos ^{6}\left (d x +c \right )\right )}{3}+a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 170, normalized size = 0.56 \[ -\frac {128 \, a^{3} b \cos \left (d x + c\right )^{6} + {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 6 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b^{2} + 64 \, {\left (2 \, \sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4}\right )} a b^{3} + {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{4}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.31, size = 471, normalized size = 1.56 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-\frac {11\,a^4}{8}+\frac {3\,a^2\,b^2}{4}+\frac {b^4}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {15\,a^4}{4}-\frac {39\,a^2\,b^2}{2}+\frac {19\,b^4}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {15\,a^4}{4}-\frac {39\,a^2\,b^2}{2}+\frac {19\,b^4}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,a^4}{24}-\frac {47\,a^2\,b^2}{4}+\frac {17\,b^4}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {5\,a^4}{24}-\frac {47\,a^2\,b^2}{4}+\frac {17\,b^4}{24}\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-\frac {11\,a^4}{8}+\frac {3\,a^2\,b^2}{4}+\frac {b^4}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {32\,a\,b^3}{3}-\frac {80\,a^3\,b}{3}\right )+8\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+16\,a\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+16\,a\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,a^3\,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 )}-\frac {\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (5\,a^4+6\,a^2\,b^2+b^4\right )}{8\,d}+\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,a^2+b^2\right )\,\left (a^2+b^2\right )}{8\,\left (\frac {5\,a^4}{8}+\frac {3\,a^2\,b^2}{4}+\frac {b^4}{8}\right )}\right )\,\left (5\,a^2+b^2\right )\,\left (a^2+b^2\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.99, size = 563, normalized size = 1.87 \[ \begin {cases} \frac {5 a^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 a^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 a^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {2 a^{3} b \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {3 a^{2} b^{2} x \sin ^{6}{\left (c + d x \right )}}{8} + \frac {9 a^{2} b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {9 a^{2} b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{2} b^{2} x \cos ^{6}{\left (c + d x \right )}}{8} + \frac {3 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac {3 a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{8 d} + \frac {a b^{3} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac {a b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {b^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 b^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {b^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {b^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {b^{4} \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 )^{4} \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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