Optimal. Leaf size=39 \[ -\frac {a \cos ^{m+1}(c+d x)}{d (m+1)}-\frac {b \cos ^m(c+d x)}{d m} \]
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Rubi [A] time = 0.07, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4377, 12, 2565, 30} \[ -\frac {a \cos ^{m+1}(c+d x)}{d (m+1)}-\frac {b \cos ^m(c+d x)}{d m} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2565
Rule 4377
Rubi steps
\begin {align*} \int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx &=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 \operatorname {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 \operatorname {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*}
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Mathematica [A] time = 0.09, size = 35, normalized size = 0.90 \[ -\frac {\cos ^m(c+d x) (a m \cos (c+d x)+b m+b)}{d m (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 35, normalized size = 0.90 \[ -\frac {{\left (a m \cos \left (d x + c\right ) + b m + b\right )} \cos \left (d x + c\right )^{m}}{d m^{2} + d m} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (d x + c\right ) + b \tan \left (d x + c\right )\right )} \cos \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 40, normalized size = 1.03 \[ -\frac {b \left (\cos ^{m}\left (d x +c \right )\right )}{d m}-\frac {a \left (\cos ^{1+m}\left (d x +c \right )\right )}{d \left (1+m \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 36, normalized size = 0.92 \[ -\frac {\frac {a \cos \left (d x + c\right )^{m + 1}}{m + 1} + \frac {b \cos \left (d x + c\right )^{m}}{m}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 35, normalized size = 0.90 \[ -\frac {{\cos \left (c+d\,x\right )}^m\,\left (b+b\,m+a\,m\,\cos \left (c+d\,x\right )\right )}{d\,m\,\left (m+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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