Optimal. Leaf size=337 \[ \frac {a^5 \sin ^9(c+d x)}{9 d}-\frac {4 a^5 \sin ^7(c+d x)}{7 d}+\frac {6 a^5 \sin ^5(c+d x)}{5 d}-\frac {4 a^5 \sin ^3(c+d x)}{3 d}+\frac {a^5 \sin (c+d x)}{d}-\frac {5 a^4 b \cos ^9(c+d x)}{9 d}-\frac {10 a^3 b^2 \sin ^9(c+d x)}{9 d}+\frac {30 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac {6 a^3 b^2 \sin ^5(c+d x)}{d}+\frac {10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac {10 a^2 b^3 \cos ^9(c+d x)}{9 d}-\frac {10 a^2 b^3 \cos ^7(c+d x)}{7 d}+\frac {5 a b^4 \sin ^9(c+d x)}{9 d}-\frac {10 a b^4 \sin ^7(c+d x)}{7 d}+\frac {a b^4 \sin ^5(c+d x)}{d}-\frac {b^5 \cos ^9(c+d x)}{9 d}+\frac {2 b^5 \cos ^7(c+d x)}{7 d}-\frac {b^5 \cos ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.30, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3090, 2633, 2565, 30, 2564, 270, 14} \[ -\frac {10 a^3 b^2 \sin ^9(c+d x)}{9 d}+\frac {30 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac {6 a^3 b^2 \sin ^5(c+d x)}{d}+\frac {10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac {10 a^2 b^3 \cos ^9(c+d x)}{9 d}-\frac {10 a^2 b^3 \cos ^7(c+d x)}{7 d}-\frac {5 a^4 b \cos ^9(c+d x)}{9 d}+\frac {a^5 \sin ^9(c+d x)}{9 d}-\frac {4 a^5 \sin ^7(c+d x)}{7 d}+\frac {6 a^5 \sin ^5(c+d x)}{5 d}-\frac {4 a^5 \sin ^3(c+d x)}{3 d}+\frac {a^5 \sin (c+d x)}{d}+\frac {5 a b^4 \sin ^9(c+d x)}{9 d}-\frac {10 a b^4 \sin ^7(c+d x)}{7 d}+\frac {a b^4 \sin ^5(c+d x)}{d}-\frac {b^5 \cos ^9(c+d x)}{9 d}+\frac {2 b^5 \cos ^7(c+d x)}{7 d}-\frac {b^5 \cos ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 270
Rule 2564
Rule 2565
Rule 2633
Rule 3090
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \cos ^9(c+d x)+5 a^4 b \cos ^8(c+d x) \sin (c+d x)+10 a^3 b^2 \cos ^7(c+d x) \sin ^2(c+d x)+10 a^2 b^3 \cos ^6(c+d x) \sin ^3(c+d x)+5 a b^4 \cos ^5(c+d x) \sin ^4(c+d x)+b^5 \cos ^4(c+d x) \sin ^5(c+d x)\right ) \, dx\\ &=a^5 \int \cos ^9(c+d x) \, dx+\left (5 a^4 b\right ) \int \cos ^8(c+d x) \sin (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \cos ^7(c+d x) \sin ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \cos ^5(c+d x) \sin ^4(c+d x) \, dx+b^5 \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx\\ &=-\frac {a^5 \operatorname {Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (5 a^4 b\right ) \operatorname {Subst}\left (\int x^8 \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (10 a^3 b^2\right ) \operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^3 \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (10 a^2 b^3\right ) \operatorname {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (5 a b^4\right ) \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}-\frac {b^5 \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {5 a^4 b \cos ^9(c+d x)}{9 d}+\frac {a^5 \sin (c+d x)}{d}-\frac {4 a^5 \sin ^3(c+d x)}{3 d}+\frac {6 a^5 \sin ^5(c+d x)}{5 d}-\frac {4 a^5 \sin ^7(c+d x)}{7 d}+\frac {a^5 \sin ^9(c+d x)}{9 d}+\frac {\left (10 a^3 b^2\right ) \operatorname {Subst}\left (\int \left (x^2-3 x^4+3 x^6-x^8\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (10 a^2 b^3\right ) \operatorname {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (5 a b^4\right ) \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac {b^5 \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {b^5 \cos ^5(c+d x)}{5 d}-\frac {10 a^2 b^3 \cos ^7(c+d x)}{7 d}+\frac {2 b^5 \cos ^7(c+d x)}{7 d}-\frac {5 a^4 b \cos ^9(c+d x)}{9 d}+\frac {10 a^2 b^3 \cos ^9(c+d x)}{9 d}-\frac {b^5 \cos ^9(c+d x)}{9 d}+\frac {a^5 \sin (c+d x)}{d}-\frac {4 a^5 \sin ^3(c+d x)}{3 d}+\frac {10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac {6 a^5 \sin ^5(c+d x)}{5 d}-\frac {6 a^3 b^2 \sin ^5(c+d x)}{d}+\frac {a b^4 \sin ^5(c+d x)}{d}-\frac {4 a^5 \sin ^7(c+d x)}{7 d}+\frac {30 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac {10 a b^4 \sin ^7(c+d x)}{7 d}+\frac {a^5 \sin ^9(c+d x)}{9 d}-\frac {10 a^3 b^2 \sin ^9(c+d x)}{9 d}+\frac {5 a b^4 \sin ^9(c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 1.02, size = 278, normalized size = 0.82 \[ \frac {420 a \left (21 a^4-5 b^4\right ) \sin (3 (c+d x))+252 b \left (b^4-25 a^4\right ) \cos (5 (c+d x))+630 a \left (63 a^4+70 a^2 b^2+15 b^4\right ) \sin (c+d x)+252 a \left (9 a^4-20 a^2 b^2-5 b^4\right ) \sin (5 (c+d x))+45 a \left (9 a^4-50 a^2 b^2+5 b^4\right ) \sin (7 (c+d x))+35 a \left (a^4-10 a^2 b^2+5 b^4\right ) \sin (9 (c+d x))-630 b \left (35 a^4+30 a^2 b^2+3 b^4\right ) \cos (c+d x)-420 b \left (35 a^4+20 a^2 b^2+b^4\right ) \cos (3 (c+d x))+45 b \left (-35 a^4+30 a^2 b^2+b^4\right ) \cos (7 (c+d x))-35 b \left (5 a^4-10 a^2 b^2+b^4\right ) \cos (9 (c+d x))}{80640 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 217, normalized size = 0.64 \[ -\frac {63 \, b^{5} \cos \left (d x + c\right )^{5} + 35 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{9} + 90 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{7} - {\left (35 \, {\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (4 \, a^{5} + 5 \, a^{3} b^{2} - 25 \, a b^{4}\right )} \cos \left (d x + c\right )^{6} + 128 \, a^{5} + 160 \, a^{3} b^{2} + 40 \, a b^{4} + 3 \, {\left (16 \, a^{5} + 20 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (16 \, a^{5} + 20 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.70, size = 313, normalized size = 0.93 \[ -\frac {{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {{\left (35 \, a^{4} b - 30 \, a^{2} b^{3} - b^{5}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {{\left (25 \, a^{4} b - b^{5}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (35 \, a^{4} b + 20 \, a^{2} b^{3} + b^{5}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {{\left (35 \, a^{4} b + 30 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right )}{128 \, d} + \frac {{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {{\left (9 \, a^{5} - 50 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (9 \, a^{5} - 20 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (21 \, a^{5} - 5 \, a b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \sin \left (d x + c\right )}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 291, normalized size = 0.86 \[ \frac {b^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+5 a \,b^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )+10 a^{2} b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {5 a^{4} b \left (\cos ^{9}\left (d x +c \right )\right )}{9}+\frac {a^{5} \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 224, normalized size = 0.66 \[ -\frac {175 \, a^{4} b \cos \left (d x + c\right )^{9} - {\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{5} + 10 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{3} b^{2} - 50 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{2} b^{3} - 5 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} a b^{4} + {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} b^{5}}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.34, size = 495, normalized size = 1.47 \[ \frac {2\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {152\,a^5}{5}-32\,a^3\,b^2+32\,a\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {152\,a^5}{5}-32\,a^3\,b^2+32\,a\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {1136\,a^5}{35}+\frac {1264\,a^3\,b^2}{7}-\frac {384\,a\,b^4}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {1136\,a^5}{35}+\frac {1264\,a^3\,b^2}{7}-\frac {384\,a\,b^4}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {21316\,a^5}{315}-\frac {5696\,a^3\,b^2}{63}+\frac {6976\,a\,b^4}{63}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (40\,a^4\,b-\frac {120\,a^2\,b^3}{7}+\frac {64\,b^5}{35}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (140\,a^4\,b-120\,a^2\,b^3+\frac {112\,b^5}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {280\,a^4\,b}{3}-\frac {200\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )-\frac {10\,a^4\,b}{9}-\frac {16\,b^5}{315}-\frac {40\,a^2\,b^3}{63}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {16\,a^5}{3}+\frac {80\,a^3\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {16\,a^5}{3}+\frac {80\,a^3\,b^2}{3}\right )+2\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {40\,a^2\,b^3}{7}+\frac {16\,b^5}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {32\,b^5}{5}-120\,a^2\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (16\,b^5-200\,a^2\,b^3\right )-40\,a^2\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.23, size = 440, normalized size = 1.31 \[ \begin {cases} \frac {128 a^{5} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {64 a^{5} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {16 a^{5} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {8 a^{5} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {a^{5} \sin {\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {5 a^{4} b \cos ^{9}{\left (c + d x \right )}}{9 d} + \frac {32 a^{3} b^{2} \sin ^{9}{\left (c + d x \right )}}{63 d} + \frac {16 a^{3} b^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{7 d} + \frac {4 a^{3} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {10 a^{3} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {10 a^{2} b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {20 a^{2} b^{3} \cos ^{9}{\left (c + d x \right )}}{63 d} + \frac {8 a b^{4} \sin ^{9}{\left (c + d x \right )}}{63 d} + \frac {4 a b^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{7 d} + \frac {a b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {b^{5} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {4 b^{5} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {8 b^{5} \cos ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{5} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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