Optimal. Leaf size=600 \[ \frac {2^{-3-n} e^{i (3 a-c n)+i (3 b-d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (\frac {1}{2} \left (\frac {3 b}{d}-n\right ),-n;\frac {1}{2} \left (2+\frac {3 b}{d}-n\right );e^{2 i (c+d x)}\right )}{3 b-d n}-\frac {3\ 2^{-3-n} e^{i (a-c n)+i (b-d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (-n,\frac {b-d n}{2 d};\frac {1}{2} \left (2+\frac {b}{d}-n\right );e^{2 i (c+d x)}\right )}{b-d n}-\frac {3\ 2^{-3-n} e^{-i (a+c n)-i (b+d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (-n,-\frac {b+d n}{2 d};1-\frac {b+d n}{2 d};e^{2 i (c+d x)}\right )}{b+d n}+\frac {2^{-3-n} e^{-i (3 a+c n)-i (3 b+d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (-n,-\frac {3 b+d n}{2 d};\frac {1}{2} \left (2-\frac {3 b}{d}-n\right );e^{2 i (c+d x)}\right )}{3 b+d n} \]
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Rubi [A]
time = 1.19, antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {4649, 2323,
2285, 2284, 2283} \begin {gather*} \frac {2^{-n-3} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac {1}{2} \left (\frac {3 b}{d}-n\right ),-n;\frac {1}{2} \left (\frac {3 b}{d}-n+2\right );e^{2 i (c+d x)}\right ) \exp (i (3 a-c n)+i x (3 b-d n)+i n (c+d x))}{3 b-d n}-\frac {3\ 2^{-n-3} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (-n,\frac {b-d n}{2 d};\frac {1}{2} \left (\frac {b}{d}-n+2\right );e^{2 i (c+d x)}\right ) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac {3\ 2^{-n-3} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (-n,-\frac {b+d n}{2 d};1-\frac {b+d n}{2 d};e^{2 i (c+d x)}\right ) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n}+\frac {2^{-n-3} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (-n,-\frac {3 b+d n}{2 d};\frac {1}{2} \left (-\frac {3 b}{d}-n+2\right );e^{2 i (c+d x)}\right ) \exp (-i (3 a+c n)-i x (3 b+d n)+i n (c+d x))}{3 b+d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2283
Rule 2284
Rule 2285
Rule 2323
Rule 4649
Rubi steps
\begin {align*} \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx &=2^{-3-n} \int \left (3 i e^{-i a-i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n-3 i e^{i a+i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n-i e^{-3 i a-3 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n+i e^{3 i a+3 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \, dx\\ &=-\left (\left (i 2^{-3-n}\right ) \int e^{-3 i a-3 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, dx\right )+\left (i 2^{-3-n}\right ) \int e^{3 i a+3 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, dx+\left (3 i 2^{-3-n}\right ) \int e^{-i a-i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, dx-\left (3 i 2^{-3-n}\right ) \int e^{i a+i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, dx\\ &=-\left (\left (i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-3 i a-3 i b x-i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx\right )+\left (i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{3 i a+3 i b x-i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx+\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i a-i b x-i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx-\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i a+i b x-i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx\\ &=\left (i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i (3 a-c n)+i (3 b-d n) x} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx-\left (i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i (3 a+c n)-i (3 b+d n) x} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx-\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i (a-c n)+i (b-d n) x} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx+\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i (a+c n)-i (b+d n) x} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx\\ &=\left (i 2^{-3-n} e^{i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i (3 a-c n)+i (3 b-d n) x} \left (1-e^{2 i c+2 i d x}\right )^n \, dx-\left (i 2^{-3-n} e^{i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i (3 a+c n)-i (3 b+d n) x} \left (1-e^{2 i c+2 i d x}\right )^n \, dx-\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i (a-c n)+i (b-d n) x} \left (1-e^{2 i c+2 i d x}\right )^n \, dx+\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i (a+c n)-i (b+d n) x} \left (1-e^{2 i c+2 i d x}\right )^n \, dx\\ &=\frac {2^{-3-n} \exp (i (3 a-c n)+i (3 b-d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (\frac {1}{2} \left (\frac {3 b}{d}-n\right ),-n;\frac {1}{2} \left (2+\frac {3 b}{d}-n\right );e^{2 i (c+d x)}\right )}{3 b-d n}-\frac {3\ 2^{-3-n} \exp (i (a-c n)+i (b-d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (-n,\frac {b-d n}{2 d};\frac {1}{2} \left (2+\frac {b}{d}-n\right );e^{2 i (c+d x)}\right )}{b-d n}-\frac {3\ 2^{-3-n} \exp (-i (a+c n)-i (b+d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (-n,-\frac {b+d n}{2 d};1-\frac {b+d n}{2 d};e^{2 i (c+d x)}\right )}{b+d n}+\frac {2^{-3-n} \exp (-i (3 a+c n)-i (3 b+d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (-n,-\frac {3 b+d n}{2 d};\frac {1}{2} \left (2-\frac {3 b}{d}-n\right );e^{2 i (c+d x)}\right )}{3 b+d n}\\ \end {align*}
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Mathematica [F]
time = 0.60, size = 0, normalized size = 0.00 \begin {gather*} \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (\sin ^{3}\left (b x +a \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\sin \left (a+b\,x\right )}^3\,{\sin \left (c+d\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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