Integrand size = 26, antiderivative size = 39 \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {b \cos ^m(c+d x)}{d m}-\frac {a \cos ^{1+m}(c+d x)}{d (1+m)} \]
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Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4462, 12, 2645, 30} \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {a \cos ^{m+1}(c+d x)}{d (m+1)}-\frac {b \cos ^m(c+d x)}{d m} \]
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Rule 12
Rule 30
Rule 2645
Rule 4462
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^m(c+d x) \sin (c+d x) \, dx+\int b \cos ^{-1+m}(c+d x) \sin (c+d x) \, dx \\ & = b \int \cos ^{-1+m}(c+d x) \sin (c+d x) \, dx-\frac {a \text {Subst}\left (\int x^m \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a \cos ^{1+m}(c+d x)}{d (1+m)}-\frac {b \text {Subst}\left (\int x^{-1+m} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {b \cos ^m(c+d x)}{d m}-\frac {a \cos ^{1+m}(c+d x)}{d (1+m)} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {\cos ^m(c+d x) (b+b m+a m \cos (c+d x))}{d m (1+m)} \]
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Time = 7.97 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03
method | result | size |
parts | \(-\frac {b \cos \left (d x +c \right )^{m}}{d m}-\frac {a \cos \left (d x +c \right )^{1+m}}{d \left (1+m \right )}\) | \(40\) |
default | \(-\frac {b \,{\mathrm e}^{m \ln \left (\cos \left (d x +c \right )\right )}}{d m}-\frac {a \cos \left (d x +c \right ) {\mathrm e}^{m \ln \left (\cos \left (d x +c \right )\right )}}{d \left (1+m \right )}\) | \(48\) |
risch | \(-\frac {a \left (\frac {1}{2}\right )^{m} \left ({\mathrm e}^{i \left (d x +c \right )}\right )^{-m} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{m} {\mathrm e}^{-\frac {i \left (m \pi \operatorname {csgn}\left (i \cos \left (d x +c \right )\right )^{3}-m \pi \operatorname {csgn}\left (i \cos \left (d x +c \right )\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-i \left (d x +c \right )}\right )-m \pi \operatorname {csgn}\left (i \cos \left (d x +c \right )\right )^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )\right )+m \pi \,\operatorname {csgn}\left (i \cos \left (d x +c \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i \left (d x +c \right )}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )\right )+2 d x +2 c \right )}{2}}}{2 d \left (1+m \right )}-\frac {\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{m} \left ({\mathrm e}^{i \left (d x +c \right )}\right )^{-m} \left (\frac {1}{2}\right )^{m} a \,{\mathrm e}^{\frac {i \left (-m \pi \operatorname {csgn}\left (i \cos \left (d x +c \right )\right )^{3}+m \pi \operatorname {csgn}\left (i \cos \left (d x +c \right )\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-i \left (d x +c \right )}\right )+m \pi \operatorname {csgn}\left (i \cos \left (d x +c \right )\right )^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )\right )-m \pi \,\operatorname {csgn}\left (i \cos \left (d x +c \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i \left (d x +c \right )}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )\right )+2 d x +2 c \right )}{2}}}{2 d \left (1+m \right )}-\frac {b \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{m} \left ({\mathrm e}^{i \left (d x +c \right )}\right )^{-m} \left (\frac {1}{2}\right )^{m} {\mathrm e}^{-\frac {i m \pi \,\operatorname {csgn}\left (i \cos \left (d x +c \right )\right ) \left (\operatorname {csgn}\left (i \cos \left (d x +c \right )\right )-\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )\right )\right ) \left (\operatorname {csgn}\left (i \cos \left (d x +c \right )\right )-\operatorname {csgn}\left (i {\mathrm e}^{-i \left (d x +c \right )}\right )\right )}{2}}}{m d}\) | \(446\) |
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {{\left (a m \cos \left (d x + c\right ) + b m + b\right )} \cos \left (d x + c\right )^{m}}{d m^{2} + d m} \]
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\[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=\int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {\frac {a \cos \left (d x + c\right )^{m + 1}}{m + 1} + \frac {b \cos \left (d x + c\right )^{m}}{m}}{d} \]
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\[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=\int { {\left (a \sin \left (d x + c\right ) + b \tan \left (d x + c\right )\right )} \cos \left (d x + c\right )^{m} \,d x } \]
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Time = 23.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {{\cos \left (c+d\,x\right )}^m\,\left (b+b\,m+a\,m\,\cos \left (c+d\,x\right )\right )}{d\,m\,\left (m+1\right )} \]
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