Optimal. Leaf size=298 \[ \frac {a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac {a^2 d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))}{f (1-n)}+\frac {a d^3 \left (3 b^2 (1-n)+a^2 (2-n)\right ) \cos (e+f x) (d \csc (e+f x))^{-3+n} \, _2F_1\left (\frac {1}{2},\frac {3-n}{2};\frac {5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt {\cos ^2(e+f x)}}+\frac {b d^4 \left (b^2 (2-n)+3 a^2 (3-n)\right ) \cos (e+f x) (d \csc (e+f x))^{-4+n} \, _2F_1\left (\frac {1}{2},\frac {4-n}{2};\frac {6-n}{2};\sin ^2(e+f x)\right )}{f (2-n) (4-n) \sqrt {\cos ^2(e+f x)}} \]
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Rubi [A]
time = 0.38, 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 = {3317, 3927,
4132, 3857, 2722, 4131} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3317
Rule 3857
Rule 3927
Rule 4131
Rule 4132
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 = 0.38, size = 167, normalized size = 0.56 \begin {gather*} -\frac {d \cos (e+f x) (d \csc (e+f x))^{-1+n} \sin ^2(e+f x)^{\frac {1}{2} (-1+n)} \left (3 a b^2 \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1+n);\frac {3}{2};\cos ^2(e+f x)\right )+a^3 \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3}{2};\cos ^2(e+f x)\right )+b \csc (e+f x) \left (b^2 \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2+n);\frac {3}{2};\cos ^2(e+f x)\right )+3 a^2 \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {3}{2};\cos ^2(e+f x)\right )\right ) \sqrt {\sin ^2(e+f x)}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.56, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (d \csc {\left (e + f x \right )}\right )^{n} \left (a + b \sin {\left (e + f x \right )}\right )^{3}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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