Optimal. Leaf size=137 \[ -\frac {a^2 \sin ^7(c+d x)}{7 d}+\frac {3 a^2 \sin ^5(c+d x)}{5 d}-\frac {a^2 \sin ^3(c+d x)}{d}+\frac {a^2 \sin (c+d x)}{d}-\frac {2 a b \cos ^7(c+d x)}{7 d}+\frac {b^2 \sin ^7(c+d x)}{7 d}-\frac {2 b^2 \sin ^5(c+d x)}{5 d}+\frac {b^2 \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.14, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3090, 2633, 2565, 30, 2564, 270} \[ -\frac {a^2 \sin ^7(c+d x)}{7 d}+\frac {3 a^2 \sin ^5(c+d x)}{5 d}-\frac {a^2 \sin ^3(c+d x)}{d}+\frac {a^2 \sin (c+d x)}{d}-\frac {2 a b \cos ^7(c+d x)}{7 d}+\frac {b^2 \sin ^7(c+d x)}{7 d}-\frac {2 b^2 \sin ^5(c+d x)}{5 d}+\frac {b^2 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 270
Rule 2564
Rule 2565
Rule 2633
Rule 3090
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx &=\int \left (a^2 \cos ^7(c+d x)+2 a b \cos ^6(c+d x) \sin (c+d x)+b^2 \cos ^5(c+d x) \sin ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cos ^7(c+d x) \, dx+(2 a b) \int \cos ^6(c+d x) \sin (c+d x) \, dx+b^2 \int \cos ^5(c+d x) \sin ^2(c+d x) \, dx\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {(2 a b) \operatorname {Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^2 \operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {2 a b \cos ^7(c+d x)}{7 d}+\frac {a^2 \sin (c+d x)}{d}-\frac {a^2 \sin ^3(c+d x)}{d}+\frac {3 a^2 \sin ^5(c+d x)}{5 d}-\frac {a^2 \sin ^7(c+d x)}{7 d}+\frac {b^2 \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {2 a b \cos ^7(c+d x)}{7 d}+\frac {a^2 \sin (c+d x)}{d}-\frac {a^2 \sin ^3(c+d x)}{d}+\frac {b^2 \sin ^3(c+d x)}{3 d}+\frac {3 a^2 \sin ^5(c+d x)}{5 d}-\frac {2 b^2 \sin ^5(c+d x)}{5 d}-\frac {a^2 \sin ^7(c+d x)}{7 d}+\frac {b^2 \sin ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 154, normalized size = 1.12 \[ -\frac {-3675 a^2 \sin (c+d x)-735 a^2 \sin (3 (c+d x))-147 a^2 \sin (5 (c+d x))-15 a^2 \sin (7 (c+d x))+1050 a b \cos (c+d x)+630 a b \cos (3 (c+d x))+210 a b \cos (5 (c+d x))+30 a b \cos (7 (c+d x))-525 b^2 \sin (c+d x)+35 b^2 \sin (3 (c+d x))+63 b^2 \sin (5 (c+d x))+15 b^2 \sin (7 (c+d x))}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 94, normalized size = 0.69 \[ -\frac {30 \, a b \cos \left (d x + c\right )^{7} - {\left (15 \, {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (6 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (6 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} + 48 \, a^{2} + 8 \, b^{2}\right )} \sin \left (d x + c\right )}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 155, normalized size = 1.13 \[ -\frac {a b \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} - \frac {a b \cos \left (5 \, d x + 5 \, c\right )}{32 \, d} - \frac {3 \, a b \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac {5 \, a b \cos \left (d x + c\right )}{32 \, d} + \frac {{\left (a^{2} - b^{2}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {{\left (7 \, a^{2} - 3 \, b^{2}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (21 \, a^{2} - b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {5 \, {\left (7 \, a^{2} + b^{2}\right )} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.67, size = 108, normalized size = 0.79 \[ \frac {b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {2 a b \left (\cos ^{7}\left (d x +c \right )\right )}{7}+\frac {a^{2} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 98, normalized size = 0.72 \[ -\frac {30 \, a b \cos \left (d x + c\right )^{7} + 3 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{2} - {\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} b^{2}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 176, normalized size = 1.28 \[ \frac {16\,a^2\,\sin \left (c+d\,x\right )}{35\,d}+\frac {8\,b^2\,\sin \left (c+d\,x\right )}{105\,d}+\frac {8\,a^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{35\,d}+\frac {6\,a^2\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{35\,d}+\frac {a^2\,{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )}{7\,d}+\frac {4\,b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{105\,d}+\frac {b^2\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{35\,d}-\frac {b^2\,{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )}{7\,d}-\frac {2\,a\,b\,{\cos \left (c+d\,x\right )}^7}{7\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.32, size = 187, normalized size = 1.36 \[ \begin {cases} \frac {16 a^{2} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {2 a b \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac {8 b^{2} \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {4 b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{2} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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