Optimal. Leaf size=127 \[ \frac {\cos (c+d x) (a+b \sin (c+d x))^{m+1} F_1\left (m+1;-\frac {1}{2},-\frac {1}{2};m+2;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) \sqrt {1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt {1-\frac {a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.09, 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 = {2704, 138} \[ \frac {\cos (c+d x) (a+b \sin (c+d x))^{m+1} F_1\left (m+1;-\frac {1}{2},-\frac {1}{2};m+2;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) \sqrt {1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt {1-\frac {a+b \sin (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 138
Rule 2704
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \sin (c+d x))^m \, dx &=\frac {\cos (c+d x) \operatorname {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 = 5.88, size = 0, normalized size = 0.00 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^m \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{2}, x\right ) \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, 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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m \,d x \]
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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