3.623 \(\int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=178 \[ \frac {4 a^4 \cos ^5(c+d x)}{5 d}-\frac {4 a^4 \cos (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {4 a^4 \cot (c+d x)}{d}+\frac {a^4 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac {23 a^4 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {89 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {6 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {135 a^4 x}{16} \]

[Out]

-135/16*a^4*x+6*a^4*arctanh(cos(d*x+c))/d-4*a^4*cos(d*x+c)/d+4/5*a^4*cos(d*x+c)^5/d-4*a^4*cot(d*x+c)/d-1/3*a^4
*cot(d*x+c)^3/d-2*a^4*cot(d*x+c)*csc(d*x+c)/d-89/16*a^4*cos(d*x+c)*sin(d*x+c)/d+23/24*a^4*cos(d*x+c)*sin(d*x+c
)^3/d+1/6*a^4*cos(d*x+c)*sin(d*x+c)^5/d

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Rubi [A]  time = 0.28, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2872, 3770, 3767, 8, 3768, 2635, 2633} \[ \frac {4 a^4 \cos ^5(c+d x)}{5 d}-\frac {4 a^4 \cos (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {4 a^4 \cot (c+d x)}{d}+\frac {a^4 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac {23 a^4 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {89 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {6 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {135 a^4 x}{16} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]

[Out]

(-135*a^4*x)/16 + (6*a^4*ArcTanh[Cos[c + d*x]])/d - (4*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x]^5)/(5*d) - (4
*a^4*Cot[c + d*x])/d - (a^4*Cot[c + d*x]^3)/(3*d) - (2*a^4*Cot[c + d*x]*Csc[c + d*x])/d - (89*a^4*Cos[c + d*x]
*Sin[c + d*x])/(16*d) + (23*a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^4*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2872

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac {\int \left (-14 a^{10}-8 a^{10} \csc (c+d x)+3 a^{10} \csc ^2(c+d x)+4 a^{10} \csc ^3(c+d x)+a^{10} \csc ^4(c+d x)+14 a^{10} \sin ^2(c+d x)+8 a^{10} \sin ^3(c+d x)-3 a^{10} \sin ^4(c+d x)-4 a^{10} \sin ^5(c+d x)-a^{10} \sin ^6(c+d x)\right ) \, dx}{a^6}\\ &=-14 a^4 x+a^4 \int \csc ^4(c+d x) \, dx-a^4 \int \sin ^6(c+d x) \, dx+\left (3 a^4\right ) \int \csc ^2(c+d x) \, dx-\left (3 a^4\right ) \int \sin ^4(c+d x) \, dx+\left (4 a^4\right ) \int \csc ^3(c+d x) \, dx-\left (4 a^4\right ) \int \sin ^5(c+d x) \, dx-\left (8 a^4\right ) \int \csc (c+d x) \, dx+\left (8 a^4\right ) \int \sin ^3(c+d x) \, dx+\left (14 a^4\right ) \int \sin ^2(c+d x) \, dx\\ &=-14 a^4 x+\frac {8 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {7 a^4 \cos (c+d x) \sin (c+d x)}{d}+\frac {3 a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {a^4 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {1}{6} \left (5 a^4\right ) \int \sin ^4(c+d x) \, dx+\left (2 a^4\right ) \int \csc (c+d x) \, dx-\frac {1}{4} \left (9 a^4\right ) \int \sin ^2(c+d x) \, dx+\left (7 a^4\right ) \int 1 \, dx-\frac {a^4 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (3 a^4\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac {\left (4 a^4\right ) \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (8 a^4\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-7 a^4 x+\frac {6 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^5(c+d x)}{5 d}-\frac {4 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {47 a^4 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {23 a^4 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^4 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {1}{8} \left (5 a^4\right ) \int \sin ^2(c+d x) \, dx-\frac {1}{8} \left (9 a^4\right ) \int 1 \, dx\\ &=-\frac {65 a^4 x}{8}+\frac {6 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^5(c+d x)}{5 d}-\frac {4 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {89 a^4 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {23 a^4 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^4 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {1}{16} \left (5 a^4\right ) \int 1 \, dx\\ &=-\frac {135 a^4 x}{16}+\frac {6 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^5(c+d x)}{5 d}-\frac {4 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {89 a^4 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {23 a^4 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^4 \cos (c+d x) \sin ^5(c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]  time = 1.66, size = 229, normalized size = 1.29 \[ \frac {a^4 (\sin (c+d x)+1)^4 \left (-8100 (c+d x)-2415 \sin (2 (c+d x))-135 \sin (4 (c+d x))+5 \sin (6 (c+d x))-3360 \cos (c+d x)+240 \cos (3 (c+d x))+48 \cos (5 (c+d x))+1760 \tan \left (\frac {1}{2} (c+d x)\right )-1760 \cot \left (\frac {1}{2} (c+d x)\right )-480 \csc ^2\left (\frac {1}{2} (c+d x)\right )+480 \sec ^2\left (\frac {1}{2} (c+d x)\right )-5760 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+5760 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-20 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+320 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)\right )}{960 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]

[Out]

(a^4*(1 + Sin[c + d*x])^4*(-8100*(c + d*x) - 3360*Cos[c + d*x] + 240*Cos[3*(c + d*x)] + 48*Cos[5*(c + d*x)] -
1760*Cot[(c + d*x)/2] - 480*Csc[(c + d*x)/2]^2 + 5760*Log[Cos[(c + d*x)/2]] - 5760*Log[Sin[(c + d*x)/2]] + 480
*Sec[(c + d*x)/2]^2 + 320*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 20*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 2415*Sin[2*
(c + d*x)] - 135*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)] + 1760*Tan[(c + d*x)/2]))/(960*d*(Cos[(c + d*x)/2] + Si
n[(c + d*x)/2])^8)

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fricas [A]  time = 0.89, size = 245, normalized size = 1.38 \[ -\frac {40 \, a^{4} \cos \left (d x + c\right )^{9} - 390 \, a^{4} \cos \left (d x + c\right )^{7} - 405 \, a^{4} \cos \left (d x + c\right )^{5} + 2700 \, a^{4} \cos \left (d x + c\right )^{3} - 2025 \, a^{4} \cos \left (d x + c\right ) - 720 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 720 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left (64 \, a^{4} \cos \left (d x + c\right )^{7} - 64 \, a^{4} \cos \left (d x + c\right )^{5} - 675 \, a^{4} d x \cos \left (d x + c\right )^{2} - 320 \, a^{4} \cos \left (d x + c\right )^{3} + 675 \, a^{4} d x + 480 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/240*(40*a^4*cos(d*x + c)^9 - 390*a^4*cos(d*x + c)^7 - 405*a^4*cos(d*x + c)^5 + 2700*a^4*cos(d*x + c)^3 - 20
25*a^4*cos(d*x + c) - 720*(a^4*cos(d*x + c)^2 - a^4)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 720*(a^4*cos(d
*x + c)^2 - a^4)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3*(64*a^4*cos(d*x + c)^7 - 64*a^4*cos(d*x + c)^5
- 675*a^4*d*x*cos(d*x + c)^2 - 320*a^4*cos(d*x + c)^3 + 675*a^4*d*x + 480*a^4*cos(d*x + c))*sin(d*x + c))/((d*
cos(d*x + c)^2 - d)*sin(d*x + c))

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giac [A]  time = 0.49, size = 324, normalized size = 1.82 \[ \frac {10 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2025 \, {\left (d x + c\right )} a^{4} - 1440 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 450 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {10 \, {\left (264 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{4}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {2 \, {\left (1335 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3085 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3840 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1110 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7680 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1110 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 7680 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3085 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4608 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1335 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 768 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/240*(10*a^4*tan(1/2*d*x + 1/2*c)^3 + 120*a^4*tan(1/2*d*x + 1/2*c)^2 - 2025*(d*x + c)*a^4 - 1440*a^4*log(abs(
tan(1/2*d*x + 1/2*c))) + 450*a^4*tan(1/2*d*x + 1/2*c) + 10*(264*a^4*tan(1/2*d*x + 1/2*c)^3 - 45*a^4*tan(1/2*d*
x + 1/2*c)^2 - 12*a^4*tan(1/2*d*x + 1/2*c) - a^4)/tan(1/2*d*x + 1/2*c)^3 + 2*(1335*a^4*tan(1/2*d*x + 1/2*c)^11
+ 3085*a^4*tan(1/2*d*x + 1/2*c)^9 - 3840*a^4*tan(1/2*d*x + 1/2*c)^8 + 1110*a^4*tan(1/2*d*x + 1/2*c)^7 - 7680*
a^4*tan(1/2*d*x + 1/2*c)^6 - 1110*a^4*tan(1/2*d*x + 1/2*c)^5 - 7680*a^4*tan(1/2*d*x + 1/2*c)^4 - 3085*a^4*tan(
1/2*d*x + 1/2*c)^3 - 4608*a^4*tan(1/2*d*x + 1/2*c)^2 - 1335*a^4*tan(1/2*d*x + 1/2*c) - 768*a^4)/(tan(1/2*d*x +
1/2*c)^2 + 1)^6)/d

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maple [A]  time = 0.44, size = 223, normalized size = 1.25 \[ -\frac {9 a^{4} \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{2 d}-\frac {6 a^{4} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d}-\frac {a^{4} \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}-\frac {14 a^{4} \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )}-\frac {2 a^{4} \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )^{2}}-\frac {135 a^{4} x}{16}-\frac {45 a^{4} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8 d}-\frac {135 a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{16 d}-\frac {135 a^{4} c}{16 d}-\frac {2 a^{4} \left (\cos ^{3}\left (d x +c \right )\right )}{d}-\frac {6 a^{4} \cos \left (d x +c \right )}{d}-\frac {6 a^{4} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x)

[Out]

-9/2*a^4*cos(d*x+c)^5*sin(d*x+c)/d-6/5*a^4*cos(d*x+c)^5/d-1/3/d*a^4/sin(d*x+c)^3*cos(d*x+c)^7-14/3/d*a^4/sin(d
*x+c)*cos(d*x+c)^7-2/d*a^4/sin(d*x+c)^2*cos(d*x+c)^7-135/16*a^4*x-45/8*a^4*cos(d*x+c)^3*sin(d*x+c)/d-135/16*a^
4*cos(d*x+c)*sin(d*x+c)/d-135/16/d*a^4*c-2*a^4*cos(d*x+c)^3/d-6*a^4*cos(d*x+c)/d-6/d*a^4*ln(csc(d*x+c)-cot(d*x
+c))

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maxima [A]  time = 0.41, size = 294, normalized size = 1.65 \[ \frac {128 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} - 320 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 720 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{4} + 160 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{4}}{960 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/960*(128*(6*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 30*cos(d*x + c) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x
+ c) - 1))*a^4 - 320*(4*cos(d*x + c)^3 - 6*cos(d*x + c)/(cos(d*x + c)^2 - 1) + 24*cos(d*x + c) - 15*log(cos(d
*x + c) + 1) + 15*log(cos(d*x + c) - 1))*a^4 - 5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) -
48*sin(2*d*x + 2*c))*a^4 - 720*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 25*tan(d*x + c)^2 + 8)/(tan(d*x + c)^5 +
2*tan(d*x + c)^3 + tan(d*x + c)))*a^4 + 160*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 - 2)/(tan(
d*x + c)^5 + tan(d*x + c)^3))*a^4)/d

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mupad [B]  time = 8.98, size = 474, normalized size = 2.66 \[ \frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d}+\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {6\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {135\,a^4\,\mathrm {atan}\left (\frac {18225\,a^8}{64\,\left (\frac {405\,a^8}{2}-\frac {18225\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}+\frac {405\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {405\,a^8}{2}-\frac {18225\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}\right )}{8\,d}-\frac {-74\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-\frac {346\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+280\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+153\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+572\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+379\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+592\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {1312\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {1836\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+184\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {376\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+17\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+120\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+160\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+120\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {15\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^4)/sin(c + d*x)^4,x)

[Out]

(a^4*tan(c/2 + (d*x)/2)^2)/(2*d) + (a^4*tan(c/2 + (d*x)/2)^3)/(24*d) - (6*a^4*log(tan(c/2 + (d*x)/2)))/d - (13
5*a^4*atan((18225*a^8)/(64*((405*a^8)/2 - (18225*a^8*tan(c/2 + (d*x)/2))/64)) + (405*a^8*tan(c/2 + (d*x)/2))/(
2*((405*a^8)/2 - (18225*a^8*tan(c/2 + (d*x)/2))/64))))/(8*d) - (17*a^4*tan(c/2 + (d*x)/2)^2 + (376*a^4*tan(c/2
+ (d*x)/2)^3)/5 + 184*a^4*tan(c/2 + (d*x)/2)^4 + (1836*a^4*tan(c/2 + (d*x)/2)^5)/5 + (1312*a^4*tan(c/2 + (d*x
)/2)^6)/3 + 592*a^4*tan(c/2 + (d*x)/2)^7 + 379*a^4*tan(c/2 + (d*x)/2)^8 + 572*a^4*tan(c/2 + (d*x)/2)^9 + 153*a
^4*tan(c/2 + (d*x)/2)^10 + 280*a^4*tan(c/2 + (d*x)/2)^11 - (346*a^4*tan(c/2 + (d*x)/2)^12)/3 + 4*a^4*tan(c/2 +
(d*x)/2)^13 - 74*a^4*tan(c/2 + (d*x)/2)^14 + a^4/3 + 4*a^4*tan(c/2 + (d*x)/2))/(d*(8*tan(c/2 + (d*x)/2)^3 + 4
8*tan(c/2 + (d*x)/2)^5 + 120*tan(c/2 + (d*x)/2)^7 + 160*tan(c/2 + (d*x)/2)^9 + 120*tan(c/2 + (d*x)/2)^11 + 48*
tan(c/2 + (d*x)/2)^13 + 8*tan(c/2 + (d*x)/2)^15)) + (15*a^4*tan(c/2 + (d*x)/2))/(8*d)

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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)**6*csc(d*x+c)**4*(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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