Optimal. Leaf size=165 \[ \frac {a^4 \sin ^5(c+d x)}{5 d}-\frac {2 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^3 b \cos ^5(c+d x)}{5 d}-\frac {6 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac {4 a b^3 \cos ^5(c+d x)}{5 d}-\frac {4 a b^3 \cos ^3(c+d x)}{3 d}+\frac {b^4 \sin ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.18, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3090, 2633, 2565, 30, 2564, 14} \[ -\frac {6 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac {4 a^3 b \cos ^5(c+d x)}{5 d}+\frac {a^4 \sin ^5(c+d x)}{5 d}-\frac {2 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin (c+d x)}{d}+\frac {4 a b^3 \cos ^5(c+d x)}{5 d}-\frac {4 a b^3 \cos ^3(c+d x)}{3 d}+\frac {b^4 \sin ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2564
Rule 2565
Rule 2633
Rule 3090
Rubi steps
\begin {align*} \int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=\int \left (a^4 \cos ^5(c+d x)+4 a^3 b \cos ^4(c+d x) \sin (c+d x)+6 a^2 b^2 \cos ^3(c+d x) \sin ^2(c+d x)+4 a b^3 \cos ^2(c+d x) \sin ^3(c+d x)+b^4 \cos (c+d x) \sin ^4(c+d x)\right ) \, dx\\ &=a^4 \int \cos ^5(c+d x) \, dx+\left (4 a^3 b\right ) \int \cos ^4(c+d x) \sin (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \cos ^3(c+d x) \sin ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \cos ^2(c+d x) \sin ^3(c+d x) \, dx+b^4 \int \cos (c+d x) \sin ^4(c+d x) \, dx\\ &=-\frac {a^4 \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (4 a^3 b\right ) \operatorname {Subst}\left (\int x^4 \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (6 a^2 b^2\right ) \operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^4 \operatorname {Subst}\left (\int x^4 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {4 a^3 b \cos ^5(c+d x)}{5 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {2 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin ^5(c+d x)}{5 d}+\frac {b^4 \sin ^5(c+d x)}{5 d}+\frac {\left (6 a^2 b^2\right ) \operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {4 a b^3 \cos ^3(c+d x)}{3 d}-\frac {4 a^3 b \cos ^5(c+d x)}{5 d}+\frac {4 a b^3 \cos ^5(c+d x)}{5 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {2 a^4 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac {a^4 \sin ^5(c+d x)}{5 d}-\frac {6 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {b^4 \sin ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 146, normalized size = 0.88 \[ \frac {-120 a b \left (a^2+b^2\right ) \cos (c+d x)-20 a b \left (3 a^2+b^2\right ) \cos (3 (c+d x))-12 a b \left (a^2-b^2\right ) \cos (5 (c+d x))+30 \left (5 a^4+6 a^2 b^2+b^4\right ) \sin (c+d x)+5 \left (5 a^4-6 a^2 b^2-3 b^4\right ) \sin (3 (c+d x))+3 \left (a^4-6 a^2 b^2+b^4\right ) \sin (5 (c+d x))}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 123, normalized size = 0.75 \[ -\frac {20 \, a b^{3} \cos \left (d x + c\right )^{3} + 12 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{5} - {\left (3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 8 \, a^{4} + 12 \, a^{2} b^{2} + 3 \, b^{4} + 2 \, {\left (2 \, a^{4} + 3 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.94, size = 165, normalized size = 1.00 \[ -\frac {{\left (a^{3} b - a b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{20 \, d} - \frac {{\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {{\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )}{2 \, d} + \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (5 \, a^{4} - 6 \, a^{2} b^{2} - 3 \, b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (5 \, a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 39.38, size = 142, normalized size = 0.86 \[ \frac {\frac {b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}+4 a \,b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+6 a^{2} b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{15}\right )-\frac {4 a^{3} b \left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 123, normalized size = 0.75 \[ -\frac {12 \, a^{3} b \cos \left (d x + c\right )^{5} - 3 \, b^{4} \sin \left (d x + c\right )^{5} - {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{4} + 6 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{3}\right )} a^{2} b^{2} - 4 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a b^{3}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 204, normalized size = 1.24 \[ \frac {2\,\left (\frac {3\,\sin \left (c+d\,x\right )\,a^4\,{\cos \left (c+d\,x\right )}^4}{2}+2\,\sin \left (c+d\,x\right )\,a^4\,{\cos \left (c+d\,x\right )}^2+4\,\sin \left (c+d\,x\right )\,a^4-6\,a^3\,b\,{\cos \left (c+d\,x\right )}^5-9\,\sin \left (c+d\,x\right )\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^4+3\,\sin \left (c+d\,x\right )\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^2+6\,\sin \left (c+d\,x\right )\,a^2\,b^2+6\,a\,b^3\,{\cos \left (c+d\,x\right )}^5-10\,a\,b^3\,{\cos \left (c+d\,x\right )}^3+\frac {3\,\sin \left (c+d\,x\right )\,b^4\,{\cos \left (c+d\,x\right )}^4}{2}-3\,\sin \left (c+d\,x\right )\,b^4\,{\cos \left (c+d\,x\right )}^2+\frac {3\,\sin \left (c+d\,x\right )\,b^4}{2}\right )}{15\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.99, size = 206, normalized size = 1.25 \[ \begin {cases} \frac {8 a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {4 a^{3} b \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {4 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {2 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {4 a b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {8 a b^{3} \cos ^{5}{\left (c + d x \right )}}{15 d} + \frac {b^{4} \sin ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{4} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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