Optimal. Leaf size=298 \[ \frac {b d^4 \left (3 a^2 (3-n)+b^2 (2-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left (\frac {1}{2},\frac {4-n}{2};\frac {6-n}{2};\sin ^2(e+f x)\right )}{f (2-n) (4-n) \sqrt {\cos ^2(e+f x)}}+\frac {a d^3 \left (a^2 (2-n)+3 b^2 (1-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac {1}{2},\frac {3-n}{2};\frac {5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt {\cos ^2(e+f x)}}+\frac {a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac {a^2 d^3 \cot (e+f x) (a \csc (e+f x)+b) (d \csc (e+f x))^{n-3}}{f (1-n)} \]
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Rubi [A] time = 0.57, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3238, 3842, 4047, 3772, 2643, 4046} \[ \frac {b d^4 \left (3 a^2 (3-n)+b^2 (2-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left (\frac {1}{2},\frac {4-n}{2};\frac {6-n}{2};\sin ^2(e+f x)\right )}{f (2-n) (4-n) \sqrt {\cos ^2(e+f x)}}+\frac {a d^3 \left (a^2 (2-n)+3 b^2 (1-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac {1}{2},\frac {3-n}{2};\frac {5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt {\cos ^2(e+f x)}}+\frac {a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac {a^2 d^3 \cot (e+f x) (a \csc (e+f x)+b) (d \csc (e+f x))^{n-3}}{f (1-n)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3238
Rule 3772
Rule 3842
Rule 4046
Rule 4047
Rubi steps
\begin {align*} \int (d \csc (e+f x))^n (a+b \sin (e+f x))^3 \, dx &=d^3 \int (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))^3 \, dx\\ &=\frac {a^2 d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))}{f (1-n)}-\frac {d^2 \int (d \csc (e+f x))^{-3+n} \left (-b d \left (b^2 (1-n)+a^2 (3-n)\right )-a d \left (3 b^2 (1-n)+a^2 (2-n)\right ) \csc (e+f x)-a^2 b d (1-2 n) \csc ^2(e+f x)\right ) \, dx}{1-n}\\ &=\frac {a^2 d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))}{f (1-n)}-\frac {d^2 \int (d \csc (e+f x))^{-3+n} \left (-b d \left (b^2 (1-n)+a^2 (3-n)\right )-a^2 b d (1-2 n) \csc ^2(e+f x)\right ) \, dx}{1-n}+\frac {\left (a d^2 \left (3 b^2 (1-n)+a^2 (2-n)\right )\right ) \int (d \csc (e+f x))^{-2+n} \, dx}{1-n}\\ &=\frac {a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac {a^2 d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))}{f (1-n)}+\frac {\left (b d^3 \left (b^2 (2-n)+3 a^2 (3-n)\right )\right ) \int (d \csc (e+f x))^{-3+n} \, dx}{2-n}+\frac {\left (a d^2 \left (3 b^2 (1-n)+a^2 (2-n)\right ) (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{2-n} \, dx}{1-n}\\ &=\frac {a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac {a^2 d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))}{f (1-n)}+\frac {a \left (3 b^2 (1-n)+a^2 (2-n)\right ) \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {3-n}{2};\frac {5-n}{2};\sin ^2(e+f x)\right ) \sin ^3(e+f x)}{f (1-n) (3-n) \sqrt {\cos ^2(e+f x)}}+\frac {\left (b d^3 \left (b^2 (2-n)+3 a^2 (3-n)\right ) (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{3-n} \, dx}{2-n}\\ &=\frac {a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac {a^2 d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))}{f (1-n)}+\frac {a \left (3 b^2 (1-n)+a^2 (2-n)\right ) \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {3-n}{2};\frac {5-n}{2};\sin ^2(e+f x)\right ) \sin ^3(e+f x)}{f (1-n) (3-n) \sqrt {\cos ^2(e+f x)}}+\frac {b \left (b^2 (2-n)+3 a^2 (3-n)\right ) \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {4-n}{2};\frac {6-n}{2};\sin ^2(e+f x)\right ) \sin ^4(e+f x)}{f (2-n) (4-n) \sqrt {\cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 167, normalized size = 0.56 \[ -\frac {d \cos (e+f x) \sin ^2(e+f x)^{\frac {n-1}{2}} (d \csc (e+f x))^{n-1} \left (a^3 \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {3}{2};\cos ^2(e+f x)\right )+b \sqrt {\sin ^2(e+f x)} \csc (e+f x) \left (3 a^2 \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {3}{2};\cos ^2(e+f x)\right )+b^2 \, _2F_1\left (\frac {1}{2},\frac {n-2}{2};\frac {3}{2};\cos ^2(e+f x)\right )\right )+3 a b^2 \, _2F_1\left (\frac {1}{2},\frac {n-1}{2};\frac {3}{2};\cos ^2(e+f x)\right )\right )}{f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a b^{2} \cos \left (f x + e\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (f x + e\right )\right )} \left (d \csc \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )}^{3} \left (d \csc \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.32, size = 0, normalized size = 0.00 \[ \int \left (d \csc \left (f x +e \right )\right )^{n} \left (a +b \sin \left (f x +e \right )\right )^{3}\, dx \]
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 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )}^{3} \left (d \csc \left (f x + e\right )\right )^{n}\,{d x} \]
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 \[ \int {\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \csc {\left (e + f x \right )}\right )^{n} \left (a + b \sin {\left (e + f x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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