Optimal. Leaf size=162 \[ \frac {i 2^{-m-3} e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 i b (c+d x)}{d}\right )}{b}-\frac {i 2^{-m-3} e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 i b (c+d x)}{d}\right )}{b}+\frac {(c+d x)^{m+1}}{2 d (m+1)} \]
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Rubi [A] time = 0.22, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3312, 3307, 2181} \[ \frac {i 2^{-m-3} e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {2 i b (c+d x)}{d}\right )}{b}-\frac {i 2^{-m-3} e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {2 i b (c+d x)}{d}\right )}{b}+\frac {(c+d x)^{m+1}}{2 d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rule 3312
Rubi steps
\begin {align*} \int (c+d x)^m \sin ^2(a+b x) \, dx &=\int \left (\frac {1}{2} (c+d x)^m-\frac {1}{2} (c+d x)^m \cos (2 a+2 b x)\right ) \, dx\\ &=\frac {(c+d x)^{1+m}}{2 d (1+m)}-\frac {1}{2} \int (c+d x)^m \cos (2 a+2 b x) \, dx\\ &=\frac {(c+d x)^{1+m}}{2 d (1+m)}-\frac {1}{4} \int e^{-i (2 a+2 b x)} (c+d x)^m \, dx-\frac {1}{4} \int e^{i (2 a+2 b x)} (c+d x)^m \, dx\\ &=\frac {(c+d x)^{1+m}}{2 d (1+m)}+\frac {i 2^{-3-m} e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 i b (c+d x)}{d}\right )}{b}-\frac {i 2^{-3-m} e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i b (c+d x)}{d}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 211, normalized size = 1.30 \[ \frac {2^{-m-3} (c+d x)^m \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (-i d (m+1) \left (-\frac {i b (c+d x)}{d}\right )^m \left (\cos \left (2 a-\frac {2 b c}{d}\right )-i \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \Gamma \left (m+1,\frac {2 i b (c+d x)}{d}\right )+i d (m+1) \left (\frac {i b (c+d x)}{d}\right )^m \left (\cos \left (2 a-\frac {2 b c}{d}\right )+i \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \Gamma \left (m+1,-\frac {2 i b (c+d x)}{d}\right )+b 2^{m+2} (c+d x) \left (\frac {b^2 (c+d x)^2}{d^2}\right )^m\right )}{b d (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 134, normalized size = 0.83 \[ \frac {{\left (-i \, d m - i \, d\right )} e^{\left (-\frac {d m \log \left (\frac {2 i \, b}{d}\right ) - 2 i \, b c + 2 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {2 i \, b d x + 2 i \, b c}{d}\right ) + {\left (i \, d m + i \, d\right )} e^{\left (-\frac {d m \log \left (-\frac {2 i \, b}{d}\right ) + 2 i \, b c - 2 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-2 i \, b d x - 2 i \, b c}{d}\right ) + 4 \, {\left (b d x + b c\right )} {\left (d x + c\right )}^{m}}{8 \, {\left (b d m + b d\right )}} \]
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 \[ \int {\left (d x + c\right )}^{m} \sin \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (\sin ^{2}\left (b x +a \right )\right )\, dx \]
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 \[ -\frac {{\left (d m + d\right )} \int {\left (d x + c\right )}^{m} \cos \left (2 \, b x + 2 \, a\right )\,{d x} - e^{\left (m \log \left (d x + c\right ) + \log \left (d x + c\right )\right )}}{2 \, {\left (d m + d\right )}} \]
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 \[ \int {\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^m \,d x \]
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 \[ \int \left (c + d x\right )^{m} \sin ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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