Optimal. Leaf size=127 \[ \frac {F_1\left (1+m;-\frac {1}{2},-\frac {1}{2};2+m;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) \cos (c+d x) (a+b \sin (c+d x))^{1+m}}{b d (1+m) \sqrt {1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt {1-\frac {a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A]
time = 0.06, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2783, 143}
\begin {gather*} \frac {\cos (c+d x) (a+b \sin (c+d x))^{m+1} F_1\left (m+1;-\frac {1}{2},-\frac {1}{2};m+2;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) \sqrt {1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt {1-\frac {a+b \sin (c+d x)}{a+b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 2783
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \sin (c+d x))^m \, dx &=\frac {\cos (c+d x) \text {Subst}\left (\int (a+b x)^m \sqrt {-\frac {b}{a-b}-\frac {b x}{a-b}} \sqrt {\frac {b}{a+b}-\frac {b x}{a+b}} \, dx,x,\sin (c+d x)\right )}{d \sqrt {1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt {1-\frac {a+b \sin (c+d x)}{a+b}}}\\ &=\frac {F_1\left (1+m;-\frac {1}{2},-\frac {1}{2};2+m;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) \cos (c+d x) (a+b \sin (c+d x))^{1+m}}{b d (1+m) \sqrt {1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt {1-\frac {a+b \sin (c+d x)}{a+b}}}\\ \end {align*}
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Mathematica [F]
time = 2.76, size = 0, normalized size = 0.00 \begin {gather*} \int \cos ^2(c+d x) (a+b \sin (c+d x))^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (\cos ^{2}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{m}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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