Optimal. Leaf size=167 \[ \frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{1+m}}{b^5 d (1+m)}-\frac {4 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{2+m}}{b^5 d (2+m)}+\frac {2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{3+m}}{b^5 d (3+m)}-\frac {4 a (a+b \sin (c+d x))^{4+m}}{b^5 d (4+m)}+\frac {(a+b \sin (c+d x))^{5+m}}{b^5 d (5+m)} \]
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Rubi [A]
time = 0.08, 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 = {2747, 711}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rule 2747
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+b \sin (c+d x))^m \, dx &=\frac {\text {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 {\text {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 = 0.95, size = 330, normalized size = 1.98 \begin {gather*} \frac {(a+b \sin (c+d x))^{1+m} \left (192 a^4-544 a^2 b^2+712 b^4-144 a^2 b^2 m+758 b^4 m+16 a^2 b^2 m^2+281 b^4 m^2+46 b^4 m^3+3 b^4 m^4+4 b^2 \left (2+3 m+m^2\right ) \left (-12 a^2+b^2 \left (28+11 m+m^2\right )\right ) \cos (2 (c+d x))+b^4 \left (24+50 m+35 m^2+10 m^3+m^4\right ) \cos (4 (c+d x))-192 a^3 b \sin (c+d x)+496 a b^3 \sin (c+d x)-192 a^3 b m \sin (c+d x)+664 a b^3 m \sin (c+d x)+176 a b^3 m^2 \sin (c+d x)+8 a b^3 m^3 \sin (c+d x)+48 a b^3 \sin (3 (c+d x))+88 a b^3 m \sin (3 (c+d x))+48 a b^3 m^2 \sin (3 (c+d x))+8 a b^3 m^3 \sin (3 (c+d x))\right )}{8 b^5 d (1+m) (2+m) (3+m) (4+m) (5+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, 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 = 0.30, size = 286, normalized size = 1.71 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 381 vs.
\(2 (167) = 334\).
time = 0.41, size = 381, normalized size = 2.28 \begin {gather*} \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} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1410 vs.
\(2 (167) = 334\).
time = 4.55, size = 1410, normalized size = 8.44 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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