Optimal. Leaf size=136 \[ -\frac {\sin \left (-\tan ^{-1}(a,b)+c+d x\right ) (a \cos (c+d x)+b \sin (c+d x))^n \left (\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}\right )^{-n} \cos ^{n+1}\left (-\tan ^{-1}(a,b)+c+d x\right ) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2\left (c+d x-\tan ^{-1}(a,b)\right )\right )}{d (n+1) \sqrt {\sin ^2\left (-\tan ^{-1}(a,b)+c+d x\right )}} \]
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Rubi [A] time = 0.06, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3078, 2643} \[ -\frac {\sin \left (-\tan ^{-1}(a,b)+c+d x\right ) (a \cos (c+d x)+b \sin (c+d x))^n \left (\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}\right )^{-n} \cos ^{n+1}\left (-\tan ^{-1}(a,b)+c+d x\right ) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2\left (c+d x-\tan ^{-1}(a,b)\right )\right )}{d (n+1) \sqrt {\sin ^2\left (-\tan ^{-1}(a,b)+c+d x\right )}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3078
Rubi steps
\begin {align*} \int (a \cos (c+d x)+b \sin (c+d x))^n \, dx &=\left ((a \cos (c+d x)+b \sin (c+d x))^n \left (\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}\right )^{-n}\right ) \int \cos ^n\left (c+d x-\tan ^{-1}(a,b)\right ) \, dx\\ &=-\frac {\cos ^{1+n}\left (c+d x-\tan ^{-1}(a,b)\right ) \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2\left (c+d x-\tan ^{-1}(a,b)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^n \left (\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}\right )^{-n} \sin \left (c+d x-\tan ^{-1}(a,b)\right )}{d (1+n) \sqrt {\sin ^2\left (c+d x-\tan ^{-1}(a,b)\right )}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 94, normalized size = 0.69 \[ -\frac {\sin \left (2 \left (\tan ^{-1}\left (\frac {a}{b}\right )+c+d x\right )\right ) \sin ^2\left (\tan ^{-1}\left (\frac {a}{b}\right )+c+d x\right )^{-\frac {n}{2}-\frac {1}{2}} (a \cos (c+d x)+b \sin (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3}{2};\cos ^2\left (c+d x+\tan ^{-1}\left (\frac {a}{b}\right )\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )}^{n}, x\right ) \]
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 \[ \int {\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.00, size = 0, normalized size = 0.00 \[ \int \left (a \cos \left (d x +c \right )+b \sin \left (d x +c \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,\cos \left (c+d\,x\right )+b\,\sin \left (c+d\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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