Optimal. Leaf size=487 \[ \frac {3 \left (a^2 (n+6)+b^2 (n+1)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(c+d x)\right )}{d (n+1) (n+2) (n+4) (n+6) \sqrt {\cos ^2(c+d x)}}-\frac {2 a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+13 n+39\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{b d (n+3) (n+4) (n+5) (n+6)}-\frac {\left (a^2 (n+2) (n+3)-b^2 (n+5) (n+7)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (n+4) (n+5) (n+6)}-\frac {\left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+13 n+40\right )+3 b^4 (n+5)\right ) \cos (c+d x) \sin ^{n+1}(c+d x)}{b^2 d (n+2) (n+4) (n+5) (n+6)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (n+5) (n+6)}+\frac {6 a b \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\sin ^2(c+d x)\right )}{d (n+2) (n+3) (n+5) \sqrt {\cos ^2(c+d x)}}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^3}{b d (n+6)} \]
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Rubi [A] time = 1.12, antiderivative size = 487, normalized size of antiderivative = 1.00, 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 2643
Rule 2748
Rule 2895
Rule 3023
Rule 3033
Rule 3049
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*}
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Mathematica [A] time = 0.34, 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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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b^{2} \cos \left (d x + c\right )^{6} - 2 \, a b \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )^{n}, x\right ) \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{2} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 17.91, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{2}\, dx \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{2} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^n\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]
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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