Optimal. Leaf size=250 \[ -\frac{a \left (a^2 (m+2)+3 b^2 (m+1)\right ) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{b \left (3 a^2 (m+3)+b^2 (m+2)\right ) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{a b^2 (2 m+7) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3)}+\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))}{d (m+3)} \]
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Rubi [A] time = 0.316659, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2793, 3023, 2748, 2643} \[ -\frac{a \left (a^2 (m+2)+3 b^2 (m+1)\right ) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{b \left (3 a^2 (m+3)+b^2 (m+2)\right ) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{a b^2 (2 m+7) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3)}+\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))}{d (m+3)} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \cos ^m(c+d x) (a+b \cos (c+d x))^3 \, dx &=\frac{b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{d (3+m)}+\frac{\int \cos ^m(c+d x) \left (a \left (b^2 (1+m)+a^2 (3+m)\right )+b \left (b^2 (2+m)+3 a^2 (3+m)\right ) \cos (c+d x)+a b^2 (7+2 m) \cos ^2(c+d x)\right ) \, dx}{3+m}\\ &=\frac{a b^2 (7+2 m) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (3+m)}+\frac{b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{d (3+m)}+\frac{\int \cos ^m(c+d x) \left (a (3+m) \left (3 b^2 (1+m)+a^2 (2+m)\right )+b (2+m) \left (b^2 (2+m)+3 a^2 (3+m)\right ) \cos (c+d x)\right ) \, dx}{6+5 m+m^2}\\ &=\frac{a b^2 (7+2 m) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (3+m)}+\frac{b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{d (3+m)}+\left (a \left (a^2+\frac{3 b^2 (1+m)}{2+m}\right )\right ) \int \cos ^m(c+d x) \, dx+\left (b \left (3 a^2+\frac{b^2 (2+m)}{3+m}\right )\right ) \int \cos ^{1+m}(c+d x) \, dx\\ &=\frac{a b^2 (7+2 m) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (3+m)}+\frac{b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{d (3+m)}-\frac{a \left (a^2+\frac{3 b^2 (1+m)}{2+m}\right ) \cos ^{1+m}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+m) \sqrt{\sin ^2(c+d x)}}-\frac{b \left (3 a^2+\frac{b^2 (2+m)}{3+m}\right ) \cos ^{2+m}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2+m) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.807916, size = 197, normalized size = 0.79 \[ \frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left (b \cos (c+d x) \left (b \cos (c+d x) \left (-\frac{3 a \, _2F_1\left (\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right )}{m+3}-\frac{b \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(c+d x)\right )}{m+4}\right )-\frac{3 a^2 \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right )}{m+2}\right )-\frac{a^3 \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right )}{m+1}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.426, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{m} \left ( a+b\cos \left ( dx+c \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \cos \left (d x + c\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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