Optimal. Leaf size=487 \[ \frac{3 \left (a^2 (n+6)+b^2 (n+1)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right )}{d (n+1) (n+2) (n+4) (n+6) \sqrt{\cos ^2(c+d x)}}-\frac{\left (-2 a^2 b^2 \left (n^2+13 n+40\right )+2 a^4 \left (n^2+5 n+6\right )+3 b^4 (n+5)\right ) \cos (c+d x) \sin ^{n+1}(c+d x)}{b^2 d (n+2) (n+4) (n+5) (n+6)}-\frac{2 a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+13 n+39\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{b d (n+3) (n+4) (n+5) (n+6)}-\frac{\left (a^2 (n+2) (n+3)-b^2 (n+5) (n+7)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (n+4) (n+5) (n+6)}+\frac{a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (n+5) (n+6)}+\frac{6 a b \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right )}{d (n+2) (n+3) (n+5) \sqrt{\cos ^2(c+d x)}}-\frac{\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^3}{b d (n+6)} \]
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Rubi [A] time = 1.12029, antiderivative size = 487, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2895, 3049, 3033, 3023, 2748, 2643} \[ \frac{3 \left (a^2 (n+6)+b^2 (n+1)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right )}{d (n+1) (n+2) (n+4) (n+6) \sqrt{\cos ^2(c+d x)}}-\frac{\left (-2 a^2 b^2 \left (n^2+13 n+40\right )+2 a^4 \left (n^2+5 n+6\right )+3 b^4 (n+5)\right ) \cos (c+d x) \sin ^{n+1}(c+d x)}{b^2 d (n+2) (n+4) (n+5) (n+6)}-\frac{2 a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+13 n+39\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{b d (n+3) (n+4) (n+5) (n+6)}-\frac{\left (a^2 (n+2) (n+3)-b^2 (n+5) (n+7)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (n+4) (n+5) (n+6)}+\frac{a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (n+5) (n+6)}+\frac{6 a b \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right )}{d (n+2) (n+3) (n+5) \sqrt{\cos ^2(c+d x)}}-\frac{\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^3}{b d (n+6)} \]
Antiderivative was successfully verified.
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Rule 2895
Rule 3049
Rule 3033
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n)}-\frac{\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^3}{b d (6+n)}-\frac{\int \sin ^n(c+d x) (a+b \sin (c+d x))^2 \left (a^2 (1+n) (3+n)-b^2 (5+n) (6+n)+2 a b \sin (c+d x)-\left (a^2 (2+n) (3+n)-b^2 (5+n) (7+n)\right ) \sin ^2(c+d x)\right ) \, dx}{b^2 (5+n) (6+n)}\\ &=-\frac{\left (a^2 (2+n) (3+n)-b^2 (5+n) (7+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (4+n) (5+n) (6+n)}+\frac{a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n)}-\frac{\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^3}{b d (6+n)}-\frac{\int \sin ^n(c+d x) (a+b \sin (c+d x)) \left (a \left (2 a^2 \left (3+4 n+n^2\right )-b^2 \left (85+27 n+2 n^2\right )\right )+b \left (2 a^2-3 b^2 (5+n)\right ) \sin (c+d x)-2 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (39+13 n+n^2\right )\right ) \sin ^2(c+d x)\right ) \, dx}{b^2 (4+n) (5+n) (6+n)}\\ &=-\frac{2 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (39+13 n+n^2\right )\right ) \cos (c+d x) \sin ^{2+n}(c+d x)}{b d (3+n) (4+n) (5+n) (6+n)}-\frac{\left (a^2 (2+n) (3+n)-b^2 (5+n) (7+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (4+n) (5+n) (6+n)}+\frac{a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n)}-\frac{\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^3}{b d (6+n)}-\frac{\int \sin ^n(c+d x) \left (a^2 (3+n) \left (2 a^2 \left (3+4 n+n^2\right )-b^2 \left (85+27 n+2 n^2\right )\right )-6 a b^3 (4+n) (6+n) \sin (c+d x)-(3+n) \left (3 b^4 (5+n)+2 a^4 \left (6+5 n+n^2\right )-2 a^2 b^2 \left (40+13 n+n^2\right )\right ) \sin ^2(c+d x)\right ) \, dx}{b^2 (3+n) (4+n) (5+n) (6+n)}\\ &=-\frac{\left (3 b^4 (5+n)+2 a^4 \left (6+5 n+n^2\right )-2 a^2 b^2 \left (40+13 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x)}{b^2 d (2+n) (4+n) (5+n) (6+n)}-\frac{2 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (39+13 n+n^2\right )\right ) \cos (c+d x) \sin ^{2+n}(c+d x)}{b d (3+n) (4+n) (5+n) (6+n)}-\frac{\left (a^2 (2+n) (3+n)-b^2 (5+n) (7+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (4+n) (5+n) (6+n)}+\frac{a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n)}-\frac{\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^3}{b d (6+n)}-\frac{\int \sin ^n(c+d x) \left (-3 b^2 \left (15+8 n+n^2\right ) \left (b^2 (1+n)+a^2 (6+n)\right )-6 a b^3 (2+n) (4+n) (6+n) \sin (c+d x)\right ) \, dx}{b^2 (2+n) (3+n) (4+n) (5+n) (6+n)}\\ &=-\frac{\left (3 b^4 (5+n)+2 a^4 \left (6+5 n+n^2\right )-2 a^2 b^2 \left (40+13 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x)}{b^2 d (2+n) (4+n) (5+n) (6+n)}-\frac{2 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (39+13 n+n^2\right )\right ) \cos (c+d x) \sin ^{2+n}(c+d x)}{b d (3+n) (4+n) (5+n) (6+n)}-\frac{\left (a^2 (2+n) (3+n)-b^2 (5+n) (7+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (4+n) (5+n) (6+n)}+\frac{a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n)}-\frac{\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^3}{b d (6+n)}+\frac{(6 a b) \int \sin ^{1+n}(c+d x) \, dx}{15+8 n+n^2}+\frac{\left (3 \left (b^2 (1+n)+a^2 (6+n)\right )\right ) \int \sin ^n(c+d x) \, dx}{(2+n) (4+n) (6+n)}\\ &=-\frac{\left (3 b^4 (5+n)+2 a^4 \left (6+5 n+n^2\right )-2 a^2 b^2 \left (40+13 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x)}{b^2 d (2+n) (4+n) (5+n) (6+n)}+\frac{3 \left (b^2 (1+n)+a^2 (6+n)\right ) \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1+n}{2};\frac{3+n}{2};\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{d (1+n) (2+n) (4+n) (6+n) \sqrt{\cos ^2(c+d x)}}-\frac{2 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (39+13 n+n^2\right )\right ) \cos (c+d x) \sin ^{2+n}(c+d x)}{b d (3+n) (4+n) (5+n) (6+n)}+\frac{6 a b \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{2+n}{2};\frac{4+n}{2};\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{d (2+n) \left (15+8 n+n^2\right ) \sqrt{\cos ^2(c+d x)}}-\frac{\left (a^2 (2+n) (3+n)-b^2 (5+n) (7+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (4+n) (5+n) (6+n)}+\frac{a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n)}-\frac{\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^3}{b d (6+n)}\\ \end{align*}
Mathematica [A] time = 0.315392, size = 167, normalized size = 0.34 \[ \frac{\sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{n+1}(c+d x) \left (a^2 \left (n^2+5 n+6\right ) \, _2F_1\left (-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right )+b (n+1) \sin (c+d x) \left (2 a (n+3) \, _2F_1\left (-\frac{3}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right )+b (n+2) \sin (c+d x) \, _2F_1\left (-\frac{3}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right )\right )\right )}{d (n+1) (n+2) (n+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 7.588, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{n} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}\, 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 \sin \left (d x + c\right ) + a\right )}^{2} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}\,{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^{2} \cos \left (d x + c\right )^{6} - 2 \, a b \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) -{\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )^{n}, 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 \sin \left (d x + c\right ) + a\right )}^{2} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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