Optimal. Leaf size=127 \[ -\frac {3 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x))^5}{5 d}+\frac {\left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x))^3}{d}-\frac {\left (a^2+b^2\right )^3 (b \cos (c+d x)-a \sin (c+d x))}{d}+\frac {(b \cos (c+d x)-a \sin (c+d x))^7}{7 d} \]
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Rubi [A] time = 0.08, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3072, 194} \[ -\frac {3 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x))^5}{5 d}+\frac {\left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x))^3}{d}-\frac {\left (a^2+b^2\right )^3 (b \cos (c+d x)-a \sin (c+d x))}{d}+\frac {(b \cos (c+d x)-a \sin (c+d x))^7}{7 d} \]
Antiderivative was successfully verified.
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Rule 194
Rule 3072
Rubi steps
\begin {align*} \int (a \cos (c+d x)+b \sin (c+d x))^7 \, dx &=-\frac {\operatorname {Subst}\left (\int \left (a^2+b^2-x^2\right )^3 \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (a^6 \left (1+\frac {3 a^4 b^2+3 a^2 b^4+b^6}{a^6}\right )-3 a^4 \left (1+\frac {2 a^2 b^2+b^4}{a^4}\right ) x^2+3 a^2 \left (1+\frac {b^2}{a^2}\right ) x^4-x^6\right ) \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{d}\\ &=-\frac {\left (a^2+b^2\right )^3 (b \cos (c+d x)-a \sin (c+d x))}{d}+\frac {\left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x))^3}{d}-\frac {3 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x))^5}{5 d}+\frac {(b \cos (c+d x)-a \sin (c+d x))^7}{7 d}\\ \end {align*}
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Mathematica [A] time = 1.02, size = 246, normalized size = 1.94 \[ \frac {1225 a \left (a^2+b^2\right )^3 \sin (c+d x)+245 a \left (a^2-3 b^2\right ) \left (a^2+b^2\right )^2 \sin (3 (c+d x))-1225 b \left (a^2+b^2\right )^3 \cos (c+d x)+245 b \left (b^2-3 a^2\right ) \left (a^2+b^2\right )^2 \cos (3 (c+d x))+49 a \left (a^6-9 a^4 b^2-5 a^2 b^4+5 b^6\right ) \sin (5 (c+d x))+5 a \left (a^6-21 a^4 b^2+35 a^2 b^4-7 b^6\right ) \sin (7 (c+d x))-49 b \left (5 a^6-5 a^4 b^2-9 a^2 b^4+b^6\right ) \cos (5 (c+d x))+5 b \left (-7 a^6+35 a^4 b^2-21 a^2 b^4+b^6\right ) \cos (7 (c+d x))}{2240 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 257, normalized size = 2.02 \[ -\frac {35 \, b^{7} \cos \left (d x + c\right ) + 5 \, {\left (7 \, a^{6} b - 35 \, a^{4} b^{3} + 21 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{7} + 7 \, {\left (35 \, a^{4} b^{3} - 42 \, a^{2} b^{5} + 3 \, b^{7}\right )} \cos \left (d x + c\right )^{5} + 35 \, {\left (7 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{3} - {\left (16 \, a^{7} + 56 \, a^{5} b^{2} + 70 \, a^{3} b^{4} + 35 \, a b^{6} + 5 \, {\left (a^{7} - 21 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 7 \, a b^{6}\right )} \cos \left (d x + c\right )^{6} + {\left (6 \, a^{7} + 21 \, a^{5} b^{2} - 280 \, a^{3} b^{4} + 105 \, a b^{6}\right )} \cos \left (d x + c\right )^{4} + {\left (8 \, a^{7} + 28 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 105 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{35 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 316, normalized size = 2.49 \[ -\frac {{\left (7 \, a^{6} b - 35 \, a^{4} b^{3} + 21 \, a^{2} b^{5} - b^{7}\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {7 \, {\left (5 \, a^{6} b - 5 \, a^{4} b^{3} - 9 \, a^{2} b^{5} + b^{7}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {7 \, {\left (3 \, a^{6} b + 5 \, a^{4} b^{3} + a^{2} b^{5} - b^{7}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {35 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )}{64 \, d} + \frac {{\left (a^{7} - 21 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 7 \, a b^{6}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {7 \, {\left (a^{7} - 9 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 5 \, a b^{6}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, {\left (a^{7} - a^{5} b^{2} - 5 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac {35 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 321, normalized size = 2.53 \[ \frac {-\frac {b^{7} \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}+a \,b^{6} \left (\sin ^{7}\left (d x +c \right )\right )+21 a^{2} b^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{7}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+35 a^{3} b^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{7}-\frac {3 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35}\right )+35 a^{4} b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+21 a^{5} b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-a^{6} b \left (\cos ^{7}\left (d x +c \right )\right )+\frac {a^{7} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 257, normalized size = 2.02 \[ -\frac {35 \, a^{6} b \cos \left (d x + c\right )^{7} - 35 \, a b^{6} \sin \left (d x + c\right )^{7} + {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{7} - 7 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{5} b^{2} - 35 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{4} b^{3} + 35 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} a^{3} b^{4} + 7 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{2} b^{5} - {\left (5 \, \cos \left (d x + c\right )^{7} - 21 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )\right )} b^{7}}{35 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.16, size = 422, normalized size = 3.32 \[ -\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (70\,a^6\,b-140\,a^4\,b^3+224\,a^2\,b^5\right )-2\,a^7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {424\,a^7}{35}+\frac {912\,a^5\,b^2}{5}-192\,a^3\,b^4+128\,a\,b^6\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (42\,a^6\,b-56\,a^4\,b^3+\frac {336\,a^2\,b^5}{5}+\frac {96\,b^7}{5}\right )+2\,a^6\,b+\frac {32\,b^7}{35}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {86\,a^7}{5}-\frac {224\,a^5\,b^2}{5}+224\,a^3\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {86\,a^7}{5}-\frac {224\,a^5\,b^2}{5}+224\,a^3\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (28\,a^4\,b^3+\frac {112\,a^2\,b^5}{5}+\frac {32\,b^7}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (280\,a^4\,b^3-112\,a^2\,b^5+32\,b^7\right )+\frac {16\,a^2\,b^5}{5}+4\,a^4\,b^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (4\,a^7+56\,a^5\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (4\,a^7+56\,a^5\,b^2\right )-2\,a^7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+140\,a^4\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+14\,a^6\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.56, size = 461, normalized size = 3.63 \[ \begin {cases} \frac {16 a^{7} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 a^{7} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 a^{7} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a^{7} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {a^{6} b \cos ^{7}{\left (c + d x \right )}}{d} + \frac {8 a^{5} b^{2} \sin ^{7}{\left (c + d x \right )}}{5 d} + \frac {28 a^{5} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {7 a^{5} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {7 a^{4} b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{d} - \frac {2 a^{4} b^{3} \cos ^{7}{\left (c + d x \right )}}{d} + \frac {2 a^{3} b^{4} \sin ^{7}{\left (c + d x \right )}}{d} + \frac {7 a^{3} b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {7 a^{2} b^{5} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac {28 a^{2} b^{5} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {8 a^{2} b^{5} \cos ^{7}{\left (c + d x \right )}}{5 d} + \frac {a b^{6} \sin ^{7}{\left (c + d x \right )}}{d} - \frac {b^{7} \sin ^{6}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} - \frac {2 b^{7} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac {8 b^{7} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {16 b^{7} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{7} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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