Optimal. Leaf size=167 \[ \frac {a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac {3 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {3 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac {3 a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {3 a \sin ^{n+6}(c+d x)}{d (n+6)}-\frac {a \sin ^{n+7}(c+d x)}{d (n+7)}-\frac {a \sin ^{n+8}(c+d x)}{d (n+8)} \]
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Rubi [A] time = 0.14, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2836, 88} \[ \frac {a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac {3 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {3 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac {3 a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {3 a \sin ^{n+6}(c+d x)}{d (n+6)}-\frac {a \sin ^{n+7}(c+d x)}{d (n+7)}-\frac {a \sin ^{n+8}(c+d x)}{d (n+8)} \]
Antiderivative was successfully verified.
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Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 \left (\frac {x}{a}\right )^n (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^7 \left (\frac {x}{a}\right )^n+a^7 \left (\frac {x}{a}\right )^{1+n}-3 a^7 \left (\frac {x}{a}\right )^{2+n}-3 a^7 \left (\frac {x}{a}\right )^{3+n}+3 a^7 \left (\frac {x}{a}\right )^{4+n}+3 a^7 \left (\frac {x}{a}\right )^{5+n}-a^7 \left (\frac {x}{a}\right )^{6+n}-a^7 \left (\frac {x}{a}\right )^{7+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {a \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {a \sin ^{2+n}(c+d x)}{d (2+n)}-\frac {3 a \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {3 a \sin ^{4+n}(c+d x)}{d (4+n)}+\frac {3 a \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {3 a \sin ^{6+n}(c+d x)}{d (6+n)}-\frac {a \sin ^{7+n}(c+d x)}{d (7+n)}-\frac {a \sin ^{8+n}(c+d x)}{d (8+n)}\\ \end {align*}
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Mathematica [B] time = 3.24, size = 659, normalized size = 3.95 \[ \frac {a \sin ^{n+1}(c+d x) \left (5 n^7 \sin (c+d x)+9 n^7 \sin (3 (c+d x))+5 n^7 \sin (5 (c+d x))+n^7 \sin (7 (c+d x))+2 n^7 \cos (6 (c+d x))+188 n^6 \sin (c+d x)+324 n^6 \sin (3 (c+d x))+164 n^6 \sin (5 (c+d x))+28 n^6 \sin (7 (c+d x))+58 n^6 \cos (6 (c+d x))+3050 n^5 \sin (c+d x)+4866 n^5 \sin (3 (c+d x))+2138 n^5 \sin (5 (c+d x))+322 n^5 \sin (7 (c+d x))+686 n^5 \cos (6 (c+d x))+28904 n^4 \sin (c+d x)+38232 n^4 \sin (3 (c+d x))+14360 n^4 \sin (5 (c+d x))+1960 n^4 \sin (7 (c+d x))+4270 n^4 \cos (6 (c+d x))+167669 n^3 \sin (c+d x)+165273 n^3 \sin (3 (c+d x))+53525 n^3 \sin (5 (c+d x))+6769 n^3 \sin (7 (c+d x))+15008 n^3 \cos (6 (c+d x))+552236 n^2 \sin (c+d x)+384948 n^2 \sin (3 (c+d x))+110036 n^2 \sin (5 (c+d x))+13132 n^2 \sin (7 (c+d x))+29512 n^2 \cos (6 (c+d x))+6 \left (5 n^7+177 n^6+2611 n^5+20499 n^4+90640 n^3+219828 n^2+262064 n+114816\right ) \cos (2 (c+d x))+12 \left (n^7+33 n^6+439 n^5+3027 n^4+11584 n^3+24372 n^2+25776 n+10368\right ) \cos (4 (c+d x))+879324 n \sin (c+d x)+439836 n \sin (3 (c+d x))+114252 n \sin (5 (c+d x))+13068 n \sin (7 (c+d x))+29664 n \cos (6 (c+d x))+468720 \sin (c+d x)+186480 \sin (3 (c+d x))+45360 \sin (5 (c+d x))+5040 \sin (7 (c+d x))+11520 \cos (6 (c+d x))+20 n^7+724 n^6+11084 n^5+94012 n^4+481280 n^3+1486096 n^2+2521536 n+1755648\right )}{64 d (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 445, normalized size = 2.66 \[ -\frac {{\left ({\left (a n^{7} + 28 \, a n^{6} + 322 \, a n^{5} + 1960 \, a n^{4} + 6769 \, a n^{3} + 13132 \, a n^{2} + 13068 \, a n + 5040 \, a\right )} \cos \left (d x + c\right )^{8} - {\left (a n^{7} + 22 \, a n^{6} + 190 \, a n^{5} + 820 \, a n^{4} + 1849 \, a n^{3} + 2038 \, a n^{2} + 840 \, a n\right )} \cos \left (d x + c\right )^{6} - 48 \, a n^{4} - 6 \, {\left (a n^{6} + 18 \, a n^{5} + 118 \, a n^{4} + 348 \, a n^{3} + 457 \, a n^{2} + 210 \, a n\right )} \cos \left (d x + c\right )^{4} - 768 \, a n^{3} - 4128 \, a n^{2} - 24 \, {\left (a n^{5} + 16 \, a n^{4} + 86 \, a n^{3} + 176 \, a n^{2} + 105 \, a n\right )} \cos \left (d x + c\right )^{2} - 8448 \, a n - {\left ({\left (a n^{7} + 29 \, a n^{6} + 343 \, a n^{5} + 2135 \, a n^{4} + 7504 \, a n^{3} + 14756 \, a n^{2} + 14832 \, a n + 5760 \, a\right )} \cos \left (d x + c\right )^{6} + 48 \, a n^{4} + 6 \, {\left (a n^{6} + 24 \, a n^{5} + 223 \, a n^{4} + 1020 \, a n^{3} + 2404 \, a n^{2} + 2736 \, a n + 1152 \, a\right )} \cos \left (d x + c\right )^{4} + 960 \, a n^{3} + 6720 \, a n^{2} + 24 \, {\left (a n^{5} + 21 \, a n^{4} + 160 \, a n^{3} + 540 \, a n^{2} + 784 \, a n + 384 \, a\right )} \cos \left (d x + c\right )^{2} + 19200 \, a n + 18432 \, a\right )} \sin \left (d x + c\right ) - 5040 \, a\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 36 \, d n^{7} + 546 \, d n^{6} + 4536 \, d n^{5} + 22449 \, d n^{4} + 67284 \, d n^{3} + 118124 \, d n^{2} + 109584 \, d n + 40320 \, d} \]
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 = 17.35, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 148, normalized size = 0.89 \[ -\frac {\frac {a \sin \left (d x + c\right )^{n + 8}}{n + 8} + \frac {a \sin \left (d x + c\right )^{n + 7}}{n + 7} - \frac {3 \, a \sin \left (d x + c\right )^{n + 6}}{n + 6} - \frac {3 \, a \sin \left (d x + c\right )^{n + 5}}{n + 5} + \frac {3 \, a \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {3 \, a \sin \left (d x + c\right )^{n + 3}}{n + 3} - \frac {a \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.90, size = 901, normalized size = 5.40 \[ \frac {a\,{\sin \left (c+d\,x\right )}^n\,\left (5\,n^7+188\,n^6+3050\,n^5+28904\,n^4+167669\,n^3+552236\,n^2+879324\,n+468720\right )}{128\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\cos \left (2\,c+2\,d\,x\right )\,\left (-n^7-34\,n^6-454\,n^5-2332\,n^4+599\,n^3+41822\,n^2+109872\,n+70560\right )}{32\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\sin \left (7\,c+7\,d\,x\right )\,\left (n^7\,1{}\mathrm {i}+n^6\,29{}\mathrm {i}+n^5\,343{}\mathrm {i}+n^4\,2135{}\mathrm {i}+n^3\,7504{}\mathrm {i}+n^2\,14756{}\mathrm {i}+n\,14832{}\mathrm {i}+5760{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\sin \left (5\,c+5\,d\,x\right )\,\left (n^7\,5{}\mathrm {i}+n^6\,169{}\mathrm {i}+n^5\,2291{}\mathrm {i}+n^4\,16027{}\mathrm {i}+n^3\,62000{}\mathrm {i}+n^2\,131476{}\mathrm {i}+n\,139824{}\mathrm {i}+56448{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\sin \left (3\,c+3\,d\,x\right )\,\left (n^7\,3{}\mathrm {i}+n^6\,111{}\mathrm {i}+n^5\,1733{}\mathrm {i}+n^4\,14445{}\mathrm {i}+n^3\,67472{}\mathrm {i}+n^2\,171084{}\mathrm {i}+n\,210512{}\mathrm {i}+94080{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\cos \left (8\,c+8\,d\,x\right )\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}{128\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\cos \left (6\,c+6\,d\,x\right )\,\left (n^7+34\,n^6+454\,n^5+3100\,n^4+11689\,n^3+24226\,n^2+25296\,n+10080\right )}{32\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\cos \left (4\,c+4\,d\,x\right )\,\left (n^7+40\,n^6+682\,n^5+5968\,n^4+27937\,n^3+68728\,n^2+81396\,n+35280\right )}{32\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {a\,\sin \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^n\,\left (n^7\,5{}\mathrm {i}+n^6\,193{}\mathrm {i}+n^5\,3251{}\mathrm {i}+n^4\,32515{}\mathrm {i}+n^3\,209360{}\mathrm {i}+n^2\,826612{}\mathrm {i}+n\,1735344{}\mathrm {i}+1411200{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,d\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\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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