Optimal. Leaf size=336 \[ -\frac{2 a^3 \left (A \left (32 n^3+224 n^2+478 n+301\right )+2 B \left (16 n^3+104 n^2+203 n+115\right )\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f (2 n+3) (2 n+5) (2 n+7) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^3 \left (A \left (8 n^2+50 n+77\right )+2 B \left (4 n^2+23 n+35\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) (2 n+5) (2 n+7) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 (A (2 n+7)+2 B (n+5)) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (d \sin (e+f x))^{n+1}}{d f (2 n+5) (2 n+7)}-\frac{2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (d \sin (e+f x))^{n+1}}{d f (2 n+7)} \]
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Rubi [A] time = 0.871863, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2976, 2981, 2776, 67, 65} \[ -\frac{2 a^3 \left (A \left (32 n^3+224 n^2+478 n+301\right )+2 B \left (16 n^3+104 n^2+203 n+115\right )\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f (2 n+3) (2 n+5) (2 n+7) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^3 \left (A \left (8 n^2+50 n+77\right )+2 B \left (4 n^2+23 n+35\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) (2 n+5) (2 n+7) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 (A (2 n+7)+2 B (n+5)) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (d \sin (e+f x))^{n+1}}{d f (2 n+5) (2 n+7)}-\frac{2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (d \sin (e+f x))^{n+1}}{d f (2 n+7)} \]
Antiderivative was successfully verified.
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Rule 2976
Rule 2981
Rule 2776
Rule 67
Rule 65
Rubi steps
\begin{align*} \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx &=-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}+\frac{2 \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} \left (\frac{1}{2} a d \left (2 B (1+n)+2 A \left (\frac{7}{2}+n\right )\right )+\frac{1}{2} a d (2 B (5+n)+A (7+2 n)) \sin (e+f x)\right ) \, dx}{d (7+2 n)}\\ &=-\frac{2 a^2 (2 B (5+n)+A (7+2 n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \sqrt{a+a \sin (e+f x)}}{d f (5+2 n) (7+2 n)}-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}+\frac{4 \int (d \sin (e+f x))^n \sqrt{a+a \sin (e+f x)} \left (\frac{1}{4} a^2 d^2 \left (2 B \left (15+19 n+4 n^2\right )+A \left (49+42 n+8 n^2\right )\right )+\frac{1}{4} a^2 d^2 \left (2 B \left (35+23 n+4 n^2\right )+A \left (77+50 n+8 n^2\right )\right ) \sin (e+f x)\right ) \, dx}{d^2 (5+2 n) (7+2 n)}\\ &=-\frac{2 a^3 \left (2 B \left (35+23 n+4 n^2\right )+A \left (77+50 n+8 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) (5+2 n) (7+2 n) \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 (2 B (5+n)+A (7+2 n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \sqrt{a+a \sin (e+f x)}}{d f (5+2 n) (7+2 n)}-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}+\frac{\left (a^2 \left (2 B \left (115+203 n+104 n^2+16 n^3\right )+A \left (301+478 n+224 n^2+32 n^3\right )\right )\right ) \int (d \sin (e+f x))^n \sqrt{a+a \sin (e+f x)} \, dx}{(3+2 n) (5+2 n) (7+2 n)}\\ &=-\frac{2 a^3 \left (2 B \left (35+23 n+4 n^2\right )+A \left (77+50 n+8 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) (5+2 n) (7+2 n) \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 (2 B (5+n)+A (7+2 n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \sqrt{a+a \sin (e+f x)}}{d f (5+2 n) (7+2 n)}-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}+\frac{\left (a^4 \left (2 B \left (115+203 n+104 n^2+16 n^3\right )+A \left (301+478 n+224 n^2+32 n^3\right )\right ) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(d x)^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{f (3+2 n) (5+2 n) (7+2 n) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{2 a^3 \left (2 B \left (35+23 n+4 n^2\right )+A \left (77+50 n+8 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) (5+2 n) (7+2 n) \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 (2 B (5+n)+A (7+2 n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \sqrt{a+a \sin (e+f x)}}{d f (5+2 n) (7+2 n)}-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}+\frac{\left (a^4 \left (2 B \left (115+203 n+104 n^2+16 n^3\right )+A \left (301+478 n+224 n^2+32 n^3\right )\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{x^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{f (3+2 n) (5+2 n) (7+2 n) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{2 a^3 \left (2 B \left (115+203 n+104 n^2+16 n^3\right )+A \left (301+478 n+224 n^2+32 n^3\right )\right ) \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f (3+2 n) (5+2 n) (7+2 n) \sqrt{a+a \sin (e+f x)}}-\frac{2 a^3 \left (2 B \left (35+23 n+4 n^2\right )+A \left (77+50 n+8 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) (5+2 n) (7+2 n) \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 (2 B (5+n)+A (7+2 n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \sqrt{a+a \sin (e+f x)}}{d f (5+2 n) (7+2 n)}-\frac{2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}\\ \end{align*}
Mathematica [A] time = 18.2937, size = 596, normalized size = 1.77 \[ \frac{2^{n+1} \tan \left (\frac{1}{2} (e+f x)\right ) \sec \left (\frac{1}{2} (e+f x)\right ) (a (\sin (e+f x)+1))^{5/2} \sin ^{-n}(e+f x) \left (\frac{\tan \left (\frac{1}{2} (e+f x)\right )}{\tan ^2\left (\frac{1}{2} (e+f x)\right )+1}\right )^n \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )^n (d \sin (e+f x))^n \left (\tan \left (\frac{1}{2} (e+f x)\right ) \left (\frac{(5 A+2 B) \, _2F_1\left (\frac{n+2}{2},n+\frac{9}{2};\frac{n+4}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n+2}+\tan \left (\frac{1}{2} (e+f x)\right ) \left (\frac{(11 A+10 B) \, _2F_1\left (\frac{n+3}{2},n+\frac{9}{2};\frac{n+5}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n+3}+\tan \left (\frac{1}{2} (e+f x)\right ) \left (\frac{5 (3 A+4 B) \, _2F_1\left (\frac{n+4}{2},n+\frac{9}{2};\frac{n+6}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n+4}+\tan \left (\frac{1}{2} (e+f x)\right ) \left (\frac{5 (3 A+4 B) \, _2F_1\left (n+\frac{9}{2},\frac{n+5}{2};\frac{n+7}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n+5}+\tan \left (\frac{1}{2} (e+f x)\right ) \left (\frac{(11 A+10 B) \, _2F_1\left (n+\frac{9}{2},\frac{n+6}{2};\frac{n+8}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n+6}+\frac{(5 A+2 B) \tan \left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (n+\frac{9}{2},\frac{n+7}{2};\frac{n+9}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n+7}\right )\right )\right )\right )\right )+\frac{A \tan ^7\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (\frac{n}{2}+4,n+\frac{9}{2};\frac{n}{2}+5;-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n+8}+\frac{A \, _2F_1\left (\frac{n+1}{2},n+\frac{9}{2};\frac{n+3}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n+1}\right )}{f \sqrt{\sec ^2\left (\frac{1}{2} (e+f x)\right )} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.499, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sin \left ( fx+e \right ) \right ) ^{n} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \left ( A+B\sin \left ( fx+e \right ) \right ) \, 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 (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}} \left (d \sin \left (f x + e\right )\right )^{n}\,{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 ({\left (A + 2 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2 \,{\left (A + B\right )} a^{2} +{\left (B a^{2} \cos \left (f x + e\right )^{2} - 2 \,{\left (A + B\right )} a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a} \left (d \sin \left (f x + e\right )\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(-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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