3.565 \(\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=181 \[ \frac {a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac {a^3 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {5 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac {5 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac {3 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}+\frac {a^3 \sin ^{n+8}(c+d x)}{d (n+8)} \]

[Out]

a^3*sin(d*x+c)^(1+n)/d/(1+n)+3*a^3*sin(d*x+c)^(2+n)/d/(2+n)+a^3*sin(d*x+c)^(3+n)/d/(3+n)-5*a^3*sin(d*x+c)^(4+n
)/d/(4+n)-5*a^3*sin(d*x+c)^(5+n)/d/(5+n)+a^3*sin(d*x+c)^(6+n)/d/(6+n)+3*a^3*sin(d*x+c)^(7+n)/d/(7+n)+a^3*sin(d
*x+c)^(8+n)/d/(8+n)

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Rubi [A]  time = 0.18, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2836, 88} \[ \frac {a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac {a^3 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {5 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac {5 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac {3 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}+\frac {a^3 \sin ^{n+8}(c+d x)}{d (n+8)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (a^3*Sin[c + d*x]^(3 + n))
/(d*(3 + n)) - (5*a^3*Sin[c + d*x]^(4 + n))/(d*(4 + n)) - (5*a^3*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (a^3*Sin[
c + d*x]^(6 + n))/(d*(6 + n)) + (3*a^3*Sin[c + d*x]^(7 + n))/(d*(7 + n)) + (a^3*Sin[c + d*x]^(8 + n))/(d*(8 +
n))

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2836

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rubi steps

\begin {align*} \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 \left (\frac {x}{a}\right )^n (a+x)^5 \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^7 \left (\frac {x}{a}\right )^n+3 a^7 \left (\frac {x}{a}\right )^{1+n}+a^7 \left (\frac {x}{a}\right )^{2+n}-5 a^7 \left (\frac {x}{a}\right )^{3+n}-5 a^7 \left (\frac {x}{a}\right )^{4+n}+a^7 \left (\frac {x}{a}\right )^{5+n}+3 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^5 d}\\ &=\frac {a^3 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {3 a^3 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac {a^3 \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {5 a^3 \sin ^{4+n}(c+d x)}{d (4+n)}-\frac {5 a^3 \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {a^3 \sin ^{6+n}(c+d x)}{d (6+n)}+\frac {3 a^3 \sin ^{7+n}(c+d x)}{d (7+n)}+\frac {a^3 \sin ^{8+n}(c+d x)}{d (8+n)}\\ \end {align*}

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Mathematica [A]  time = 0.59, size = 123, normalized size = 0.68 \[ \frac {a^3 \sin ^{n+1}(c+d x) \left (\frac {\sin ^7(c+d x)}{n+8}+\frac {3 \sin ^6(c+d x)}{n+7}+\frac {\sin ^5(c+d x)}{n+6}-\frac {5 \sin ^4(c+d x)}{n+5}-\frac {5 \sin ^3(c+d x)}{n+4}+\frac {\sin ^2(c+d x)}{n+3}+\frac {3 \sin (c+d x)}{n+2}+\frac {1}{n+1}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*Sin[c + d*x]^(1 + n)*((1 + n)^(-1) + (3*Sin[c + d*x])/(2 + n) + Sin[c + d*x]^2/(3 + n) - (5*Sin[c + d*x]^
3)/(4 + n) - (5*Sin[c + d*x]^4)/(5 + n) + Sin[c + d*x]^5/(6 + n) + (3*Sin[c + d*x]^6)/(7 + n) + Sin[c + d*x]^7
/(8 + n)))/d

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fricas [B]  time = 0.77, size = 616, normalized size = 3.40 \[ \frac {{\left ({\left (a^{3} n^{7} + 28 \, a^{3} n^{6} + 322 \, a^{3} n^{5} + 1960 \, a^{3} n^{4} + 6769 \, a^{3} n^{3} + 13132 \, a^{3} n^{2} + 13068 \, a^{3} n + 5040 \, a^{3}\right )} \cos \left (d x + c\right )^{8} + 32 \, a^{3} n^{5} + 720 \, a^{3} n^{4} - {\left (5 \, a^{3} n^{7} + 142 \, a^{3} n^{6} + 1654 \, a^{3} n^{5} + 10180 \, a^{3} n^{4} + 35485 \, a^{3} n^{3} + 69358 \, a^{3} n^{2} + 69416 \, a^{3} n + 26880 \, a^{3}\right )} \cos \left (d x + c\right )^{6} + 6080 \, a^{3} n^{3} + 23520 \, a^{3} n^{2} + 2 \, {\left (2 \, a^{3} n^{7} + 49 \, a^{3} n^{6} + 470 \, a^{3} n^{5} + 2230 \, a^{3} n^{4} + 5438 \, a^{3} n^{3} + 6361 \, a^{3} n^{2} + 2730 \, a^{3} n\right )} \cos \left (d x + c\right )^{4} + 39968 \, a^{3} n + 21840 \, a^{3} + 8 \, {\left (2 \, a^{3} n^{6} + 45 \, a^{3} n^{5} + 380 \, a^{3} n^{4} + 1470 \, a^{3} n^{3} + 2498 \, a^{3} n^{2} + 1365 \, a^{3} n\right )} \cos \left (d x + c\right )^{2} + {\left (32 \, a^{3} n^{5} + 720 \, a^{3} n^{4} - 3 \, {\left (a^{3} n^{7} + 29 \, a^{3} n^{6} + 343 \, a^{3} n^{5} + 2135 \, a^{3} n^{4} + 7504 \, a^{3} n^{3} + 14756 \, a^{3} n^{2} + 14832 \, a^{3} n + 5760 \, a^{3}\right )} \cos \left (d x + c\right )^{6} + 6080 \, a^{3} n^{3} + 24000 \, a^{3} n^{2} + 2 \, {\left (2 \, a^{3} n^{7} + 53 \, a^{3} n^{6} + 566 \, a^{3} n^{5} + 3155 \, a^{3} n^{4} + 9908 \, a^{3} n^{3} + 17492 \, a^{3} n^{2} + 15984 \, a^{3} n + 5760 \, a^{3}\right )} \cos \left (d x + c\right )^{4} + 44288 \, a^{3} n + 30720 \, a^{3} + 8 \, {\left (2 \, a^{3} n^{6} + 47 \, a^{3} n^{5} + 425 \, a^{3} n^{4} + 1880 \, a^{3} n^{3} + 4268 \, a^{3} n^{2} + 4688 \, a^{3} n + 1920 \, a^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 36 \, d n^{7} + 546 \, d n^{6} + 4536 \, d n^{5} + 22449 \, d n^{4} + 67284 \, d n^{3} + 118124 \, d n^{2} + 109584 \, d n + 40320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

((a^3*n^7 + 28*a^3*n^6 + 322*a^3*n^5 + 1960*a^3*n^4 + 6769*a^3*n^3 + 13132*a^3*n^2 + 13068*a^3*n + 5040*a^3)*c
os(d*x + c)^8 + 32*a^3*n^5 + 720*a^3*n^4 - (5*a^3*n^7 + 142*a^3*n^6 + 1654*a^3*n^5 + 10180*a^3*n^4 + 35485*a^3
*n^3 + 69358*a^3*n^2 + 69416*a^3*n + 26880*a^3)*cos(d*x + c)^6 + 6080*a^3*n^3 + 23520*a^3*n^2 + 2*(2*a^3*n^7 +
49*a^3*n^6 + 470*a^3*n^5 + 2230*a^3*n^4 + 5438*a^3*n^3 + 6361*a^3*n^2 + 2730*a^3*n)*cos(d*x + c)^4 + 39968*a^
3*n + 21840*a^3 + 8*(2*a^3*n^6 + 45*a^3*n^5 + 380*a^3*n^4 + 1470*a^3*n^3 + 2498*a^3*n^2 + 1365*a^3*n)*cos(d*x
+ c)^2 + (32*a^3*n^5 + 720*a^3*n^4 - 3*(a^3*n^7 + 29*a^3*n^6 + 343*a^3*n^5 + 2135*a^3*n^4 + 7504*a^3*n^3 + 147
56*a^3*n^2 + 14832*a^3*n + 5760*a^3)*cos(d*x + c)^6 + 6080*a^3*n^3 + 24000*a^3*n^2 + 2*(2*a^3*n^7 + 53*a^3*n^6
+ 566*a^3*n^5 + 3155*a^3*n^4 + 9908*a^3*n^3 + 17492*a^3*n^2 + 15984*a^3*n + 5760*a^3)*cos(d*x + c)^4 + 44288*
a^3*n + 30720*a^3 + 8*(2*a^3*n^6 + 47*a^3*n^5 + 425*a^3*n^4 + 1880*a^3*n^3 + 4268*a^3*n^2 + 4688*a^3*n + 1920*
a^3)*cos(d*x + c)^2)*sin(d*x + c))*sin(d*x + c)^n/(d*n^8 + 36*d*n^7 + 546*d*n^6 + 4536*d*n^5 + 22449*d*n^4 + 6
7284*d*n^3 + 118124*d*n^2 + 109584*d*n + 40320*d)

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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.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 21.58, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x)

[Out]

int(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x)

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maxima [A]  time = 0.93, size = 161, normalized size = 0.89 \[ \frac {\frac {a^{3} \sin \left (d x + c\right )^{n + 8}}{n + 8} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 7}}{n + 7} + \frac {a^{3} \sin \left (d x + c\right )^{n + 6}}{n + 6} - \frac {5 \, a^{3} \sin \left (d x + c\right )^{n + 5}}{n + 5} - \frac {5 \, a^{3} \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {a^{3} \sin \left (d x + c\right )^{n + 3}}{n + 3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a^{3} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

(a^3*sin(d*x + c)^(n + 8)/(n + 8) + 3*a^3*sin(d*x + c)^(n + 7)/(n + 7) + a^3*sin(d*x + c)^(n + 6)/(n + 6) - 5*
a^3*sin(d*x + c)^(n + 5)/(n + 5) - 5*a^3*sin(d*x + c)^(n + 4)/(n + 4) + a^3*sin(d*x + c)^(n + 3)/(n + 3) + 3*a
^3*sin(d*x + c)^(n + 2)/(n + 2) + a^3*sin(d*x + c)^(n + 1)/(n + 1))/d

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mupad [B]  time = 15.45, size = 923, normalized size = 5.10 \[ \frac {a^3\,{\sin \left (c+d\,x\right )}^n\,\left (27\,n^7+1028\,n^6+17366\,n^5+162200\,n^4+870443\,n^3+2585492\,n^2+3757604\,n+1896720\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^3\,{\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^3\,\sin \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^n\,\left (n^7\,17{}\mathrm {i}+n^6\,669{}\mathrm {i}+n^5\,11975{}\mathrm {i}+n^4\,118935{}\mathrm {i}+n^3\,675728{}\mathrm {i}+n^2\,2140836{}\mathrm {i}+n\,3467760{}\mathrm {i}+2217600{}\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^3\,{\sin \left (c+d\,x\right )}^n\,\cos \left (6\,c+6\,d\,x\right )\,\left (3\,n^7+86\,n^6+1010\,n^5+6260\,n^4+21947\,n^3+43094\,n^2+43280\,n+16800\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^3\,{\sin \left (c+d\,x\right )}^n\,\cos \left (4\,c+4\,d\,x\right )\,\left (7\,n^7+264\,n^6+3910\,n^5+29520\,n^4+122023\,n^3+273336\,n^2+303180\,n+126000\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^3\,{\sin \left (c+d\,x\right )}^n\,\cos \left (2\,c+2\,d\,x\right )\,\left (-3\,n^7-86\,n^6-498\,n^5+5260\,n^4+75333\,n^3+333226\,n^2+596208\,n+332640\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^3\,{\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 )\,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^3\,{\sin \left (c+d\,x\right )}^n\,\sin \left (5\,c+5\,d\,x\right )\,\left (-n^7\,1{}\mathrm {i}+n^6\,11{}\mathrm {i}+n^5\,617{}\mathrm {i}+n^4\,6785{}\mathrm {i}+n^3\,33296{}\mathrm {i}+n^2\,81404{}\mathrm {i}+n\,94608{}\mathrm {i}+40320{}\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^3\,{\sin \left (c+d\,x\right )}^n\,\sin \left (3\,c+3\,d\,x\right )\,\left (n^7\,21{}\mathrm {i}+n^6\,745{}\mathrm {i}+n^5\,10339{}\mathrm {i}+n^4\,72475{}\mathrm {i}+n^3\,275824{}\mathrm {i}+n^2\,567700{}\mathrm {i}+n\,583216{}\mathrm {i}+228480{}\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.

[In]

int(cos(c + d*x)^5*sin(c + d*x)^n*(a + a*sin(c + d*x))^3,x)

[Out]

(a^3*sin(c + d*x)^n*(3757604*n + 2585492*n^2 + 870443*n^3 + 162200*n^4 + 17366*n^5 + 1028*n^6 + 27*n^7 + 18967
20))/(128*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (a^
3*sin(c + d*x)^n*cos(8*c + 8*d*x)*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040))
/(128*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (a^3*si
n(c + d*x)*sin(c + d*x)^n*(n*3467760i + n^2*2140836i + n^3*675728i + n^4*118935i + n^5*11975i + n^6*669i + n^7
*17i + 2217600i)*1i)/(64*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8
+ 40320)) - (a^3*sin(c + d*x)^n*cos(6*c + 6*d*x)*(43280*n + 43094*n^2 + 21947*n^3 + 6260*n^4 + 1010*n^5 + 86*n
^6 + 3*n^7 + 16800))/(32*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8
+ 40320)) - (a^3*sin(c + d*x)^n*cos(4*c + 4*d*x)*(303180*n + 273336*n^2 + 122023*n^3 + 29520*n^4 + 3910*n^5 +
264*n^6 + 7*n^7 + 126000))/(32*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7
+ n^8 + 40320)) - (a^3*sin(c + d*x)^n*cos(2*c + 2*d*x)*(596208*n + 333226*n^2 + 75333*n^3 + 5260*n^4 - 498*n^5
- 86*n^6 - 3*n^7 + 332640))/(32*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^
7 + n^8 + 40320)) + (a^3*sin(c + d*x)^n*sin(7*c + 7*d*x)*(n*14832i + n^2*14756i + n^3*7504i + n^4*2135i + n^5*
343i + n^6*29i + n^7*1i + 5760i)*3i)/(64*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6
+ 36*n^7 + n^8 + 40320)) + (a^3*sin(c + d*x)^n*sin(5*c + 5*d*x)*(n*94608i + n^2*81404i + n^3*33296i + n^4*678
5i + n^5*617i + n^6*11i - n^7*1i + 40320i)*1i)/(64*d*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5
+ 546*n^6 + 36*n^7 + n^8 + 40320)) - (a^3*sin(c + d*x)^n*sin(3*c + 3*d*x)*(n*583216i + n^2*567700i + n^3*2758
24i + n^4*72475i + n^5*10339i + n^6*745i + n^7*21i + 228480i)*1i)/(64*d*(109584*n + 118124*n^2 + 67284*n^3 + 2
2449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320))

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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.

[In]

integrate(cos(d*x+c)**5*sin(d*x+c)**n*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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