Optimal. Leaf size=109 \[ -\frac {(a \sin (c+d x)+a)^{m+7}}{a^7 d (m+7)}+\frac {6 (a \sin (c+d x)+a)^{m+6}}{a^6 d (m+6)}-\frac {12 (a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac {8 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)} \]
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Rubi [A] time = 0.09, antiderivative size = 109, 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 = {2667, 43} \[ \frac {8 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}-\frac {12 (a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac {6 (a \sin (c+d x)+a)^{m+6}}{a^6 d (m+6)}-\frac {(a \sin (c+d x)+a)^{m+7}}{a^7 d (m+7)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^{3+m} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (8 a^3 (a+x)^{3+m}-12 a^2 (a+x)^{4+m}+6 a (a+x)^{5+m}-(a+x)^{6+m}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {8 (a+a \sin (c+d x))^{4+m}}{a^4 d (4+m)}-\frac {12 (a+a \sin (c+d x))^{5+m}}{a^5 d (5+m)}+\frac {6 (a+a \sin (c+d x))^{6+m}}{a^6 d (6+m)}-\frac {(a+a \sin (c+d x))^{7+m}}{a^7 d (7+m)}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 89, normalized size = 0.82 \[ \frac {(a (\sin (c+d x)+1))^{m+4} \left (\frac {6 a^3 (\sin (c+d x)+1)^2}{m+6}-\frac {12 a^3 (\sin (c+d x)+1)}{m+5}+\frac {8 a^3}{m+4}-\frac {(a \sin (c+d x)+a)^3}{m+7}\right )}{a^7 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 153, normalized size = 1.40 \[ \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 20 \, m\right )} \cos \left (d x + c\right )^{6} + 12 \, {\left (m^{2} + 3 \, m\right )} \cos \left (d x + c\right )^{4} + 96 \, m \cos \left (d x + c\right )^{2} + {\left ({\left (m^{3} + 15 \, m^{2} + 74 \, m + 120\right )} \cos \left (d x + c\right )^{6} + 12 \, {\left (m^{2} + 7 \, m + 12\right )} \cos \left (d x + c\right )^{4} + 96 \, {\left (m + 2\right )} \cos \left (d x + c\right )^{2} + 384\right )} \sin \left (d x + c\right ) + 384\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{4} + 22 \, d m^{3} + 179 \, d m^{2} + 638 \, d m + 840 \, d} \]
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 = 6.77, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{7}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 520, normalized size = 4.77 \[ -\frac {\frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} a^{m} \sin \left (d x + c\right )^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} a^{m} \sin \left (d x + c\right )^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} a^{m} \sin \left (d x + c\right )^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (d x + c\right )^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (d x + c\right )^{3} + 360 \, {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 720 \, a^{m} m \sin \left (d x + c\right ) + 720 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} - \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} a^{m} \sin \left (d x + c\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (d x + c\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (d x + c\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 24 \, a^{m} m \sin \left (d x + c\right ) + 24 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} a^{m} \sin \left (d x + c\right )^{3} + {\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 2 \, a^{m} m \sin \left (d x + c\right ) + 2 \, a^{m}\right )} {\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} - \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m + 1}}{a {\left (m + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.48, size = 555, normalized size = 5.09 \[ {\mathrm {e}}^{-c\,7{}\mathrm {i}-d\,x\,7{}\mathrm {i}}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m\,\left (\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\left (m^3\,40{}\mathrm {i}+m^2\,936{}\mathrm {i}+m\,8672{}\mathrm {i}+49152{}\mathrm {i}\right )}{128\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\cos \left (2\,c+2\,d\,x\right )\,\left (m^3\,30{}\mathrm {i}+m^2\,654{}\mathrm {i}+m\,4824{}\mathrm {i}\right )}{64\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\sin \left (5\,c+5\,d\,x\right )\,\left (5\,m^3+123\,m^2+706\,m+1176\right )\,1{}\mathrm {i}}{64\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\sin \left (3\,c+3\,d\,x\right )\,\left (9\,m^3+279\,m^2+3210\,m+5880\right )\,1{}\mathrm {i}}{64\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\sin \left (7\,c+7\,d\,x\right )\,\left (m^3+15\,m^2+74\,m+120\right )\,1{}\mathrm {i}}{64\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\sin \left (c+d\,x\right )\,\left (5\,m^3+171\,m^2+2578\,m+29400\right )\,1{}\mathrm {i}}{64\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {m\,{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\cos \left (6\,c+6\,d\,x\right )\,\left (m^2\,1{}\mathrm {i}+m\,9{}\mathrm {i}+20{}\mathrm {i}\right )}{32\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}+\frac {3\,m\,{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\cos \left (4\,c+4\,d\,x\right )\,\left (m^2\,1{}\mathrm {i}+m\,17{}\mathrm {i}+44{}\mathrm {i}\right )}{16\,d\,\left (m^4\,1{}\mathrm {i}+m^3\,22{}\mathrm {i}+m^2\,179{}\mathrm {i}+m\,638{}\mathrm {i}+840{}\mathrm {i}\right )}\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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