Optimal. Leaf size=175 \[ -\frac {a^3 \sin ^7(c+d x)}{7 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}-\frac {a^3 \sin ^3(c+d x)}{d}+\frac {a^3 \sin (c+d x)}{d}-\frac {3 a^2 b \cos ^7(c+d x)}{7 d}+\frac {3 a b^2 \sin ^7(c+d x)}{7 d}-\frac {6 a b^2 \sin ^5(c+d x)}{5 d}+\frac {a b^2 \sin ^3(c+d x)}{d}+\frac {b^3 \cos ^7(c+d x)}{7 d}-\frac {b^3 \cos ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.18, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ -\frac {3 a^2 b \cos ^7(c+d x)}{7 d}-\frac {a^3 \sin ^7(c+d x)}{7 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}-\frac {a^3 \sin ^3(c+d x)}{d}+\frac {a^3 \sin (c+d x)}{d}+\frac {3 a b^2 \sin ^7(c+d x)}{7 d}-\frac {6 a b^2 \sin ^5(c+d x)}{5 d}+\frac {a b^2 \sin ^3(c+d x)}{d}+\frac {b^3 \cos ^7(c+d x)}{7 d}-\frac {b^3 \cos ^5(c+d x)}{5 d} \]
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))^3 \, dx &=\int \left (a^3 \cos ^7(c+d x)+3 a^2 b \cos ^6(c+d x) \sin (c+d x)+3 a b^2 \cos ^5(c+d x) \sin ^2(c+d x)+b^3 \cos ^4(c+d x) \sin ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^7(c+d x) \, dx+\left (3 a^2 b\right ) \int \cos ^6(c+d x) \sin (c+d x) \, dx+\left (3 a b^2\right ) \int \cos ^5(c+d x) \sin ^2(c+d x) \, dx+b^3 \int \cos ^4(c+d x) \sin ^3(c+d x) \, dx\\ &=-\frac {a^3 \operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (3 a^2 b\right ) \operatorname {Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}-\frac {b^3 \operatorname {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {3 a^2 b \cos ^7(c+d x)}{7 d}+\frac {a^3 \sin (c+d x)}{d}-\frac {a^3 \sin ^3(c+d x)}{d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}-\frac {a^3 \sin ^7(c+d x)}{7 d}+\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac {b^3 \operatorname {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {b^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^2 b \cos ^7(c+d x)}{7 d}+\frac {b^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \sin (c+d x)}{d}-\frac {a^3 \sin ^3(c+d x)}{d}+\frac {a b^2 \sin ^3(c+d x)}{d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}-\frac {6 a b^2 \sin ^5(c+d x)}{5 d}-\frac {a^3 \sin ^7(c+d x)}{7 d}+\frac {3 a b^2 \sin ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 204, normalized size = 1.17 \[ \frac {1225 a^3 \sin (c+d x)+245 a^3 \sin (3 (c+d x))+49 a^3 \sin (5 (c+d x))+5 a^3 \sin (7 (c+d x))-35 \left (9 a^2 b+b^3\right ) \cos (3 (c+d x))-105 b \left (5 a^2+b^2\right ) \cos (c+d x)-105 a^2 b \cos (5 (c+d x))-15 a^2 b \cos (7 (c+d x))+525 a b^2 \sin (c+d x)-35 a b^2 \sin (3 (c+d x))-63 a b^2 \sin (5 (c+d x))-15 a b^2 \sin (7 (c+d x))+7 b^3 \cos (5 (c+d x))+5 b^3 \cos (7 (c+d x))}{2240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 123, normalized size = 0.70 \[ -\frac {7 \, b^{3} \cos \left (d x + c\right )^{5} + 5 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{7} - {\left (5 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (2 \, a^{3} + a b^{2}\right )} \cos \left (d x + c\right )^{4} + 16 \, a^{3} + 8 \, a b^{2} + 4 \, {\left (2 \, a^{3} + a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{35 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 197, normalized size = 1.13 \[ -\frac {{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {{\left (15 \, a^{2} b - b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (9 \, a^{2} b + b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {3 \, {\left (5 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )}{64 \, d} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {{\left (7 \, a^{3} - 9 \, a b^{2}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (7 \, a^{3} - a b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac {5 \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 10.38, size = 145, normalized size = 0.83 \[ \frac {b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+3 b^{2} a \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {3 a^{2} b \left (\cos ^{7}\left (d x +c \right )\right )}{7}+\frac {a^{3} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 126, normalized size = 0.72 \[ -\frac {15 \, a^{2} b \cos \left (d x + c\right )^{7} + {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{3} - {\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a b^{2} - {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} b^{3}}{35 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 214, normalized size = 1.22 \[ \frac {16\,a^3\,\sin \left (c+d\,x\right )}{35\,d}-\frac {b^3\,{\cos \left (c+d\,x\right )}^5}{5\,d}+\frac {b^3\,{\cos \left (c+d\,x\right )}^7}{7\,d}-\frac {3\,a^2\,b\,{\cos \left (c+d\,x\right )}^7}{7\,d}+\frac {8\,a^3\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{35\,d}+\frac {6\,a^3\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{35\,d}+\frac {a^3\,{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )}{7\,d}+\frac {8\,a\,b^2\,\sin \left (c+d\,x\right )}{35\,d}+\frac {4\,a\,b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{35\,d}+\frac {3\,a\,b^2\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{35\,d}-\frac {3\,a\,b^2\,{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )}{7\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.53, size = 233, normalized size = 1.33 \[ \begin {cases} \frac {16 a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {3 a^{2} b \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac {8 a b^{2} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {4 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {2 b^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{3} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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