Optimal. Leaf size=92 \[ -\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+1}}{b^3 d (m+1)}+\frac {2 a (a+b \sin (c+d x))^{m+2}}{b^3 d (m+2)}-\frac {(a+b \sin (c+d x))^{m+3}}{b^3 d (m+3)} \]
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Rubi [A] time = 0.07, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2668, 697} \[ -\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+1}}{b^3 d (m+1)}+\frac {2 a (a+b \sin (c+d x))^{m+2}}{b^3 d (m+2)}-\frac {(a+b \sin (c+d x))^{m+3}}{b^3 d (m+3)} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+b \sin (c+d x))^m \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^m \left (b^2-x^2\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\left (-a^2+b^2\right ) (a+x)^m+2 a (a+x)^{1+m}-(a+x)^{2+m}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=-\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^{1+m}}{b^3 d (1+m)}+\frac {2 a (a+b \sin (c+d x))^{2+m}}{b^3 d (2+m)}-\frac {(a+b \sin (c+d x))^{3+m}}{b^3 d (3+m)}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 74, normalized size = 0.80 \[ \frac {(a+b \sin (c+d x))^{m+1} \left (\frac {b^2-a^2}{m+1}-\frac {(a+b \sin (c+d x))^2}{m+3}+\frac {2 a (a+b \sin (c+d x))}{m+2}\right )}{b^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 142, normalized size = 1.54 \[ \frac {{\left (4 \, a b^{2} m - 2 \, a^{3} + 6 \, a b^{2} + {\left (a b^{2} m^{2} + a b^{2} m\right )} \cos \left (d x + c\right )^{2} + {\left (4 \, b^{3} + {\left (b^{3} m^{2} + 3 \, b^{3} m + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{2} b + b^{3}\right )} m\right )} \sin \left (d x + c\right )\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{b^{3} d m^{3} + 6 \, b^{3} d m^{2} + 11 \, b^{3} d m + 6 \, b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 340, normalized size = 3.70 \[ -\frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{m} b^{3} m^{2} \sin \left (d x + c\right )^{3} + {\left (b \sin \left (d x + c\right ) + a\right )}^{m} a b^{2} m^{2} \sin \left (d x + c\right )^{2} + 3 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{m} b^{3} m \sin \left (d x + c\right )^{3} - {\left (b \sin \left (d x + c\right ) + a\right )}^{m} b^{3} m^{2} \sin \left (d x + c\right ) + {\left (b \sin \left (d x + c\right ) + a\right )}^{m} a b^{2} m \sin \left (d x + c\right )^{2} + 2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{m} b^{3} \sin \left (d x + c\right )^{3} - {\left (b \sin \left (d x + c\right ) + a\right )}^{m} a b^{2} m^{2} - 2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{m} a^{2} b m \sin \left (d x + c\right ) - 5 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{m} b^{3} m \sin \left (d x + c\right ) - 5 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{m} a b^{2} m - 6 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{m} b^{3} \sin \left (d x + c\right ) + 2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{m} a^{3} - 6 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{m} a b^{2}}{{\left (b^{3} m^{3} + 6 \, b^{3} m^{2} + 11 \, b^{3} m + 6 \, b^{3}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.49, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{3}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 117, normalized size = 1.27 \[ \frac {\frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{m + 1}}{b {\left (m + 1\right )}} - \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} \sin \left (d x + c\right )^{3} + {\left (m^{2} + m\right )} a b^{2} \sin \left (d x + c\right )^{2} - 2 \, a^{2} b m \sin \left (d x + c\right ) + 2 \, a^{3}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.51, size = 197, normalized size = 2.14 \[ \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m\,\left (24\,a\,b^2+18\,b^3\,\sin \left (c+d\,x\right )-8\,a^3+2\,b^3\,\sin \left (3\,c+3\,d\,x\right )+2\,a\,b^2\,m^2+3\,b^3\,m\,\sin \left (3\,c+3\,d\,x\right )+b^3\,m^2\,\sin \left (c+d\,x\right )+b^3\,m^2\,\sin \left (3\,c+3\,d\,x\right )+18\,a\,b^2\,m+11\,b^3\,m\,\sin \left (c+d\,x\right )+8\,a^2\,b\,m\,\sin \left (c+d\,x\right )-2\,a\,b^2\,m\,\left (2\,{\sin \left (c+d\,x\right )}^2-1\right )-2\,a\,b^2\,m^2\,\left (2\,{\sin \left (c+d\,x\right )}^2-1\right )\right )}{4\,b^3\,d\,\left (m^3+6\,m^2+11\,m+6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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