Integrand size = 8, antiderivative size = 54 \[ \int \cos ^7(a+b x) \, dx=\frac {\sin (a+b x)}{b}-\frac {\sin ^3(a+b x)}{b}+\frac {3 \sin ^5(a+b x)}{5 b}-\frac {\sin ^7(a+b x)}{7 b} \]
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Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2713} \[ \int \cos ^7(a+b x) \, dx=-\frac {\sin ^7(a+b x)}{7 b}+\frac {3 \sin ^5(a+b x)}{5 b}-\frac {\sin ^3(a+b x)}{b}+\frac {\sin (a+b x)}{b} \]
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Rule 2713
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (a+b x)\right )}{b} \\ & = \frac {\sin (a+b x)}{b}-\frac {\sin ^3(a+b x)}{b}+\frac {3 \sin ^5(a+b x)}{5 b}-\frac {\sin ^7(a+b x)}{7 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \cos ^7(a+b x) \, dx=\frac {\sin (a+b x)}{b}-\frac {\sin ^3(a+b x)}{b}+\frac {3 \sin ^5(a+b x)}{5 b}-\frac {\sin ^7(a+b x)}{7 b} \]
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Time = 1.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\left (\frac {16}{5}+\cos ^{6}\left (b x +a \right )+\frac {6 \left (\cos ^{4}\left (b x +a \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (b x +a \right )\right )}{5}\right ) \sin \left (b x +a \right )}{7 b}\) | \(42\) |
default | \(\frac {\left (\frac {16}{5}+\cos ^{6}\left (b x +a \right )+\frac {6 \left (\cos ^{4}\left (b x +a \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (b x +a \right )\right )}{5}\right ) \sin \left (b x +a \right )}{7 b}\) | \(42\) |
parallelrisch | \(\frac {1225 \sin \left (b x +a \right )+5 \sin \left (7 b x +7 a \right )+49 \sin \left (5 b x +5 a \right )+245 \sin \left (3 b x +3 a \right )}{2240 b}\) | \(48\) |
risch | \(\frac {35 \sin \left (b x +a \right )}{64 b}+\frac {\sin \left (7 b x +7 a \right )}{448 b}+\frac {7 \sin \left (5 b x +5 a \right )}{320 b}+\frac {7 \sin \left (3 b x +3 a \right )}{64 b}\) | \(55\) |
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \cos ^7(a+b x) \, dx=\frac {{\left (5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{35 \, b} \]
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Time = 0.51 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int \cos ^7(a+b x) \, dx=\begin {cases} \frac {16 \sin ^{7}{\left (a + b x \right )}}{35 b} + \frac {8 \sin ^{5}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{5 b} + \frac {2 \sin ^{3}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{b} + \frac {\sin {\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cos ^{7}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \cos ^7(a+b x) \, dx=-\frac {5 \, \sin \left (b x + a\right )^{7} - 21 \, \sin \left (b x + a\right )^{5} + 35 \, \sin \left (b x + a\right )^{3} - 35 \, \sin \left (b x + a\right )}{35 \, b} \]
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Time = 0.32 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \cos ^7(a+b x) \, dx=-\frac {5 \, \sin \left (b x + a\right )^{7} - 21 \, \sin \left (b x + a\right )^{5} + 35 \, \sin \left (b x + a\right )^{3} - 35 \, \sin \left (b x + a\right )}{35 \, b} \]
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Time = 13.99 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \cos ^7(a+b x) \, dx=-\frac {\sin \left (a+b\,x\right )\,\left (5\,{\sin \left (a+b\,x\right )}^6-21\,{\sin \left (a+b\,x\right )}^4+35\,{\sin \left (a+b\,x\right )}^2-35\right )}{35\,b} \]
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