Optimal. Leaf size=167 \[ \frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{m+1}}{b^5 d (m+1)}-\frac {4 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+2}}{b^5 d (m+2)}+\frac {2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{m+3}}{b^5 d (m+3)}-\frac {4 a (a+b \sin (c+d x))^{m+4}}{b^5 d (m+4)}+\frac {(a+b \sin (c+d x))^{m+5}}{b^5 d (m+5)} \]
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Rubi [A] time = 0.11, antiderivative size = 167, 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} \[ \frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{m+1}}{b^5 d (m+1)}-\frac {4 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+2}}{b^5 d (m+2)}+\frac {2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{m+3}}{b^5 d (m+3)}-\frac {4 a (a+b \sin (c+d x))^{m+4}}{b^5 d (m+4)}+\frac {(a+b \sin (c+d x))^{m+5}}{b^5 d (m+5)} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \cos ^5(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 )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\left (a^2-b^2\right )^2 (a+x)^m-4 \left (a^3-a b^2\right ) (a+x)^{1+m}+2 \left (3 a^2-b^2\right ) (a+x)^{2+m}-4 a (a+x)^{3+m}+(a+x)^{4+m}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{1+m}}{b^5 d (1+m)}-\frac {4 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{2+m}}{b^5 d (2+m)}+\frac {2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{3+m}}{b^5 d (3+m)}-\frac {4 a (a+b \sin (c+d x))^{4+m}}{b^5 d (4+m)}+\frac {(a+b \sin (c+d x))^{5+m}}{b^5 d (5+m)}\\ \end {align*}
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Mathematica [A] time = 0.90, size = 169, normalized size = 1.01 \[ \frac {(a+b \sin (c+d x))^{m+1} \left (4 \left (b^2-a^2\right ) \left (\frac {b^2-a^2}{m+1}-\frac {(a+b \sin (c+d x))^2}{m+3}+\frac {2 a (a+b \sin (c+d x))}{m+2}\right )+4 a (a+b \sin (c+d x)) \left (\frac {b^2-a^2}{m+2}-\frac {(a+b \sin (c+d x))^2}{m+4}+\frac {2 a (a+b \sin (c+d x))}{m+3}\right )+b^4 \cos ^4(c+d x)\right )}{b^5 d (m+5)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 381, normalized size = 2.28 \[ \frac {{\left (24 \, a^{5} - 80 \, a^{3} b^{2} + 120 \, a b^{4} + {\left (a b^{4} m^{4} + 6 \, a b^{4} m^{3} + 11 \, a b^{4} m^{2} + 6 \, a b^{4} m\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (a^{3} b^{2} + 3 \, a b^{4}\right )} m^{2} + 4 \, {\left (2 \, a b^{4} m^{3} - 3 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} m^{2} - {\left (3 \, a^{3} b^{2} - 7 \, a b^{4}\right )} m\right )} \cos \left (d x + c\right )^{2} - 24 \, {\left (a^{3} b^{2} - 5 \, a b^{4}\right )} m + {\left (64 \, b^{5} + {\left (b^{5} m^{4} + 10 \, b^{5} m^{3} + 35 \, b^{5} m^{2} + 50 \, b^{5} m + 24 \, b^{5}\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (3 \, a^{2} b^{3} + b^{5}\right )} m^{2} + 4 \, {\left (8 \, b^{5} + {\left (a^{2} b^{3} + b^{5}\right )} m^{3} + {\left (3 \, a^{2} b^{3} + 7 \, b^{5}\right )} m^{2} + 2 \, {\left (a^{2} b^{3} + 7 \, b^{5}\right )} m\right )} \cos \left (d x + c\right )^{2} - 24 \, {\left (a^{4} b - 3 \, a^{2} b^{3} - 2 \, b^{5}\right )} m\right )} \sin \left (d x + c\right )\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{b^{5} d m^{5} + 15 \, b^{5} d m^{4} + 85 \, b^{5} d m^{3} + 225 \, b^{5} d m^{2} + 274 \, b^{5} d m + 120 \, b^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{5}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 286, normalized size = 1.71 \[ \frac {\frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{m + 1}}{b {\left (m + 1\right )}} - \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} \sin \left (d x + c\right )^{3} + {\left (m^{2} + m\right )} a b^{2} \sin \left (d x + c\right )^{2} - 2 \, a^{2} b m \sin \left (d x + c\right ) + 2 \, a^{3}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{5} \sin \left (d x + c\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a b^{4} \sin \left (d x + c\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{2} b^{3} \sin \left (d x + c\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{3} b^{2} \sin \left (d x + c\right )^{2} - 24 \, a^{4} b m \sin \left (d x + c\right ) + 24 \, a^{5}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} b^{5}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.62, size = 641, normalized size = 3.84 \[ \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m\,\left (1920\,a\,b^4+1200\,b^5\,\sin \left (c+d\,x\right )+384\,a^5-1280\,a^3\,b^2+200\,b^5\,\sin \left (3\,c+3\,d\,x\right )+24\,b^5\,\sin \left (5\,c+5\,d\,x\right )-480\,a^3\,b^2\,m+738\,a\,b^4\,m^2+100\,a\,b^4\,m^3+6\,a\,b^4\,m^4+374\,b^5\,m\,\sin \left (3\,c+3\,d\,x\right )+50\,b^5\,m\,\sin \left (5\,c+5\,d\,x\right )+310\,b^5\,m^2\,\sin \left (c+d\,x\right )+36\,b^5\,m^3\,\sin \left (c+d\,x\right )+2\,b^5\,m^4\,\sin \left (c+d\,x\right )+32\,a^3\,b^2\,m^2+217\,b^5\,m^2\,\sin \left (3\,c+3\,d\,x\right )+46\,b^5\,m^3\,\sin \left (3\,c+3\,d\,x\right )+3\,b^5\,m^4\,\sin \left (3\,c+3\,d\,x\right )+35\,b^5\,m^2\,\sin \left (5\,c+5\,d\,x\right )+10\,b^5\,m^3\,\sin \left (5\,c+5\,d\,x\right )+b^5\,m^4\,\sin \left (5\,c+5\,d\,x\right )+2180\,a\,b^4\,m+1092\,b^5\,m\,\sin \left (c+d\,x\right )-96\,a^3\,b^2\,m\,\cos \left (2\,c+2\,d\,x\right )+376\,a\,b^4\,m^2\,\cos \left (2\,c+2\,d\,x\right )+112\,a\,b^4\,m^3\,\cos \left (2\,c+2\,d\,x\right )+8\,a\,b^4\,m^4\,\cos \left (2\,c+2\,d\,x\right )+22\,a\,b^4\,m^2\,\cos \left (4\,c+4\,d\,x\right )+12\,a\,b^4\,m^3\,\cos \left (4\,c+4\,d\,x\right )+2\,a\,b^4\,m^4\,\cos \left (4\,c+4\,d\,x\right )+32\,a^2\,b^3\,m\,\sin \left (3\,c+3\,d\,x\right )+432\,a^2\,b^3\,m^2\,\sin \left (c+d\,x\right )+16\,a^2\,b^3\,m^3\,\sin \left (c+d\,x\right )-384\,a^4\,b\,m\,\sin \left (c+d\,x\right )-96\,a^3\,b^2\,m^2\,\cos \left (2\,c+2\,d\,x\right )+48\,a^2\,b^3\,m^2\,\sin \left (3\,c+3\,d\,x\right )+16\,a^2\,b^3\,m^3\,\sin \left (3\,c+3\,d\,x\right )+272\,a\,b^4\,m\,\cos \left (2\,c+2\,d\,x\right )+12\,a\,b^4\,m\,\cos \left (4\,c+4\,d\,x\right )+1184\,a^2\,b^3\,m\,\sin \left (c+d\,x\right )\right )}{16\,b^5\,d\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )} \]
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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