Optimal. Leaf size=173 \[ \frac {(2 n+3) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(c+d x)\right )}{a^2 d (n+1) (n+2) \sqrt {\cos ^2(c+d x)}}-\frac {2 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\sin ^2(c+d x)\right )}{a^2 d (n+2) \sqrt {\cos ^2(c+d x)}}-\frac {\cos (c+d x) \sin ^{n+1}(c+d x)}{a^2 d (n+2)} \]
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Rubi [A] time = 0.24, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2869, 2763, 2748, 2643} \[ \frac {(2 n+3) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(c+d x)\right )}{a^2 d (n+1) (n+2) \sqrt {\cos ^2(c+d x)}}-\frac {2 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\sin ^2(c+d x)\right )}{a^2 d (n+2) \sqrt {\cos ^2(c+d x)}}-\frac {\cos (c+d x) \sin ^{n+1}(c+d x)}{a^2 d (n+2)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 2763
Rule 2869
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \sin ^n(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=-\frac {\cos (c+d x) \sin ^{1+n}(c+d x)}{a^2 d (2+n)}+\frac {\int \sin ^n(c+d x) \left (a^2 (3+2 n)-2 a^2 (2+n) \sin (c+d x)\right ) \, dx}{a^4 (2+n)}\\ &=-\frac {\cos (c+d x) \sin ^{1+n}(c+d x)}{a^2 d (2+n)}-\frac {2 \int \sin ^{1+n}(c+d x) \, dx}{a^2}+\frac {(3+2 n) \int \sin ^n(c+d x) \, dx}{a^2 (2+n)}\\ &=-\frac {\cos (c+d x) \sin ^{1+n}(c+d x)}{a^2 d (2+n)}+\frac {(3+2 n) \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)}{a^2 d (1+n) (2+n) \sqrt {\cos ^2(c+d x)}}-\frac {2 \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)}{a^2 d (2+n) \sqrt {\cos ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 4.92, size = 312, normalized size = 1.80 \[ \frac {2 \tan \left (\frac {1}{2} (c+d x)\right ) \sin ^n(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )^n \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (\frac {\, _2F_1\left (\frac {n+1}{2},n+3;\frac {n+3}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{n+1}+\tan \left (\frac {1}{2} (c+d x)\right ) \left (\tan \left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan ^2\left (\frac {1}{2} (c+d x)\right ) \, _2F_1\left (n+3,\frac {n+5}{2};\frac {n+7}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{n+5}-\frac {4 \tan \left (\frac {1}{2} (c+d x)\right ) \, _2F_1\left (n+3,\frac {n+4}{2};\frac {n+6}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{n+4}+\frac {6 \, _2F_1\left (\frac {n+3}{2},n+3;\frac {n+5}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{n+3}\right )-\frac {4 \, _2F_1\left (\frac {n+2}{2},n+3;\frac {n+4}{2};-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{n+2}\right )\right )}{d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}}{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}}, x\right ) \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.95, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{2}}\, dx \]
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 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^n}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]
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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