Optimal. Leaf size=109 \[ \frac {8 (a+a \sin (c+d x))^{4+m}}{a^4 d (4+m)}-\frac {12 (a+a \sin (c+d x))^{5+m}}{a^5 d (5+m)}+\frac {6 (a+a \sin (c+d x))^{6+m}}{a^6 d (6+m)}-\frac {(a+a \sin (c+d x))^{7+m}}{a^7 d (7+m)} \]
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Rubi [A]
time = 0.06, 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 = {2746, 45}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {\text {Subst}\left (\int (a-x)^3 (a+x)^{3+m} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\text {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 [C] Result contains complex when optimal does not.
time = 6.14, size = 796, normalized size = 7.30 \begin {gather*} \frac {(a (1+\sin (c+d x)))^m \left (\frac {6144+1084 m+117 m^2+5 m^3}{16 (4+m) (5+m) (6+m) (7+m)}+\frac {\left (29400+2578 m+171 m^2+5 m^3\right ) \left (-\frac {1}{128} i \cos (c+d x)+\frac {1}{128} \sin (c+d x)\right )}{(4+m) (5+m) (6+m) (7+m)}+\frac {\left (29400+2578 m+171 m^2+5 m^3\right ) \left (\frac {1}{128} i \cos (c+d x)+\frac {1}{128} \sin (c+d x)\right )}{(4+m) (5+m) (6+m) (7+m)}+\frac {\left (804 m+109 m^2+5 m^3\right ) \left (\frac {3}{64} \cos (2 (c+d x))-\frac {3}{64} i \sin (2 (c+d x))\right )}{(4+m) (5+m) (6+m) (7+m)}+\frac {\left (804 m+109 m^2+5 m^3\right ) \left (\frac {3}{64} \cos (2 (c+d x))+\frac {3}{64} i \sin (2 (c+d x))\right )}{(4+m) (5+m) (6+m) (7+m)}+\frac {\left (1960+1070 m+93 m^2+3 m^3\right ) \left (-\frac {3}{128} i \cos (3 (c+d x))+\frac {3}{128} \sin (3 (c+d x))\right )}{(4+m) (5+m) (6+m) (7+m)}+\frac {\left (1960+1070 m+93 m^2+3 m^3\right ) \left (\frac {3}{128} i \cos (3 (c+d x))+\frac {3}{128} \sin (3 (c+d x))\right )}{(4+m) (5+m) (6+m) (7+m)}+\frac {\left (44 m+17 m^2+m^3\right ) \left (\frac {3}{32} \cos (4 (c+d x))-\frac {3}{32} i \sin (4 (c+d x))\right )}{(4+m) (5+m) (6+m) (7+m)}+\frac {\left (44 m+17 m^2+m^3\right ) \left (\frac {3}{32} \cos (4 (c+d x))+\frac {3}{32} i \sin (4 (c+d x))\right )}{(4+m) (5+m) (6+m) (7+m)}+\frac {\left (294+103 m+5 m^2\right ) \left (-\frac {1}{128} i \cos (5 (c+d x))+\frac {1}{128} \sin (5 (c+d x))\right )}{(5+m) (6+m) (7+m)}+\frac {\left (294+103 m+5 m^2\right ) \left (\frac {1}{128} i \cos (5 (c+d x))+\frac {1}{128} \sin (5 (c+d x))\right )}{(5+m) (6+m) (7+m)}+\frac {\frac {1}{64} m \cos (6 (c+d x))-\frac {1}{64} i m \sin (6 (c+d x))}{(6+m) (7+m)}+\frac {\frac {1}{64} m \cos (6 (c+d x))+\frac {1}{64} i m \sin (6 (c+d x))}{(6+m) (7+m)}+\frac {-\frac {1}{128} i \cos (7 (c+d x))+\frac {1}{128} \sin (7 (c+d x))}{7+m}+\frac {\frac {1}{128} i \cos (7 (c+d x))+\frac {1}{128} \sin (7 (c+d x))}{7+m}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.30, 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] Leaf count of result is larger than twice the leaf count of optimal. 520 vs.
\(2 (109) = 218\).
time = 0.31, size = 520, normalized size = 4.77 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 153, normalized size = 1.40 \begin {gather*} \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} \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. 517 vs.
\(2 (109) = 218\).
time = 5.11, size = 517, normalized size = 4.74 \begin {gather*} -\frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{3} - 6 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m^{3} + 12 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m^{3} - 8 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m^{3} + 15 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{2} - 96 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m^{2} + 204 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m^{2} - 144 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m^{2} + 74 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m - 498 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m + 1128 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m - 856 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m + 120 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} - 840 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a + 2016 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} - 1680 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3}}{{\left (a^{6} m^{4} + 22 \, a^{6} m^{3} + 179 \, a^{6} m^{2} + 638 \, a^{6} m + 840 \, a^{6}\right )} a d} \end {gather*}
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 \begin {gather*} {\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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