3.410 \(\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx\)
Optimal. Leaf size=117 \[ \frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos ^3(c+d x)}{3 a d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos (c+d x)}{16 a d}-\frac{x}{16 a} \]
[Out]
-x/(16*a) - Cos[c + d*x]^3/(3*a*d) + Cos[c + d*x]^5/(5*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (Cos[c +
d*x]^3*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a*d)
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Rubi [A] time = 0.191169, antiderivative size = 117, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used =
{2839, 2565, 14, 2568, 2635, 8} \[ \frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos ^3(c+d x)}{3 a d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos (c+d x)}{16 a d}-\frac{x}{16 a} \]
Antiderivative was successfully verified.
[In]
Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
[Out]
-x/(16*a) - Cos[c + d*x]^3/(3*a*d) + Cos[c + d*x]^5/(5*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (Cos[c +
d*x]^3*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a*d)
Rule 2839
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]
Rule 2565
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&& !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rule 2568
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]
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 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \cos ^2(c+d x) \sin ^3(c+d x) \, dx}{a}-\frac{\int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{a}\\ &=\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{6 a d}-\frac{\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{2 a}-\frac{\operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac{\cos ^3(c+d x) \sin (c+d x)}{8 a d}+\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{6 a d}-\frac{\int \cos ^2(c+d x) \, dx}{8 a}-\frac{\operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac{\cos ^3(c+d x)}{3 a d}+\frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos (c+d x) \sin (c+d x)}{16 a d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{8 a d}+\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{6 a d}-\frac{\int 1 \, dx}{16 a}\\ &=-\frac{x}{16 a}-\frac{\cos ^3(c+d x)}{3 a d}+\frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos (c+d x) \sin (c+d x)}{16 a d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{8 a d}+\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{6 a d}\\ \end{align*}
Mathematica [B] time = 4.84269, size = 377, normalized size = 3.22 \[ \frac{-120 d x \sin \left (\frac{c}{2}\right )+120 \sin \left (\frac{c}{2}+d x\right )-120 \sin \left (\frac{3 c}{2}+d x\right )+15 \sin \left (\frac{3 c}{2}+2 d x\right )+15 \sin \left (\frac{5 c}{2}+2 d x\right )+20 \sin \left (\frac{5 c}{2}+3 d x\right )-20 \sin \left (\frac{7 c}{2}+3 d x\right )+15 \sin \left (\frac{7 c}{2}+4 d x\right )+15 \sin \left (\frac{9 c}{2}+4 d x\right )-12 \sin \left (\frac{9 c}{2}+5 d x\right )+12 \sin \left (\frac{11 c}{2}+5 d x\right )-5 \sin \left (\frac{11 c}{2}+6 d x\right )-5 \sin \left (\frac{13 c}{2}+6 d x\right )+30 \cos \left (\frac{c}{2}\right ) (3 c-4 d x)-120 \cos \left (\frac{c}{2}+d x\right )-120 \cos \left (\frac{3 c}{2}+d x\right )+15 \cos \left (\frac{3 c}{2}+2 d x\right )-15 \cos \left (\frac{5 c}{2}+2 d x\right )-20 \cos \left (\frac{5 c}{2}+3 d x\right )-20 \cos \left (\frac{7 c}{2}+3 d x\right )+15 \cos \left (\frac{7 c}{2}+4 d x\right )-15 \cos \left (\frac{9 c}{2}+4 d x\right )+12 \cos \left (\frac{9 c}{2}+5 d x\right )+12 \cos \left (\frac{11 c}{2}+5 d x\right )-5 \cos \left (\frac{11 c}{2}+6 d x\right )+5 \cos \left (\frac{13 c}{2}+6 d x\right )+90 c \sin \left (\frac{c}{2}\right )-180 \sin \left (\frac{c}{2}\right )}{1920 a d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
[Out]
(30*(3*c - 4*d*x)*Cos[c/2] - 120*Cos[c/2 + d*x] - 120*Cos[(3*c)/2 + d*x] + 15*Cos[(3*c)/2 + 2*d*x] - 15*Cos[(5
*c)/2 + 2*d*x] - 20*Cos[(5*c)/2 + 3*d*x] - 20*Cos[(7*c)/2 + 3*d*x] + 15*Cos[(7*c)/2 + 4*d*x] - 15*Cos[(9*c)/2
+ 4*d*x] + 12*Cos[(9*c)/2 + 5*d*x] + 12*Cos[(11*c)/2 + 5*d*x] - 5*Cos[(11*c)/2 + 6*d*x] + 5*Cos[(13*c)/2 + 6*d
*x] - 180*Sin[c/2] + 90*c*Sin[c/2] - 120*d*x*Sin[c/2] + 120*Sin[c/2 + d*x] - 120*Sin[(3*c)/2 + d*x] + 15*Sin[(
3*c)/2 + 2*d*x] + 15*Sin[(5*c)/2 + 2*d*x] + 20*Sin[(5*c)/2 + 3*d*x] - 20*Sin[(7*c)/2 + 3*d*x] + 15*Sin[(7*c)/2
+ 4*d*x] + 15*Sin[(9*c)/2 + 4*d*x] - 12*Sin[(9*c)/2 + 5*d*x] + 12*Sin[(11*c)/2 + 5*d*x] - 5*Sin[(11*c)/2 + 6*
d*x] - 5*Sin[(13*c)/2 + 6*d*x])/(1920*a*d*(Cos[c/2] + Sin[c/2]))
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Maple [B] time = 0.074, size = 347, normalized size = 3. \begin{align*} -{\frac{1}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{11} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-{\frac{17}{24\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-4\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{6}}}+{\frac{19}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-{\frac{8}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-{\frac{19}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}+{\frac{17}{24\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-{\frac{8}{5\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}+{\frac{1}{8\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-{\frac{4}{15\,da} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-{\frac{1}{8\,da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^4*sin(d*x+c)^3/(a+a*sin(d*x+c)),x)
[Out]
-1/8/d/a/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11-17/24/d/a/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2
*c)^9-4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8+19/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/
2*c)^7-8/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^6-19/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x
+1/2*c)^5+17/24/d/a/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3-8/5/d/a/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2
*d*x+1/2*c)^2+1/8/d/a/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)-4/15/d/a/(1+tan(1/2*d*x+1/2*c)^2)^6-1/8/a/
d*arctan(tan(1/2*d*x+1/2*c))
________________________________________________________________________________________
Maxima [B] time = 1.70372, size = 458, normalized size = 3.91 \begin{align*} \frac{\frac{\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{192 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{85 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{570 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{320 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{570 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{480 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{85 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{15 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 32}{a + \frac{6 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{15 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{20 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{15 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{6 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac{15 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
1/120*((15*sin(d*x + c)/(cos(d*x + c) + 1) - 192*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 85*sin(d*x + c)^3/(cos(
d*x + c) + 1)^3 - 570*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 320*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 570*sin(
d*x + c)^7/(cos(d*x + c) + 1)^7 - 480*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 85*sin(d*x + c)^9/(cos(d*x + c) +
1)^9 - 15*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 32)/(a + 6*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a*sin(
d*x + c)^4/(cos(d*x + c) + 1)^4 + 20*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a*sin(d*x + c)^8/(cos(d*x + c)
+ 1)^8 + 6*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 15*arctan(sin
(d*x + c)/(cos(d*x + c) + 1))/a)/d
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Fricas [A] time = 1.15258, size = 182, normalized size = 1.56 \begin{align*} \frac{48 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 15 \, d x - 5 \,{\left (8 \, \cos \left (d x + c\right )^{5} - 14 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
1/240*(48*cos(d*x + c)^5 - 80*cos(d*x + c)^3 - 15*d*x - 5*(8*cos(d*x + c)^5 - 14*cos(d*x + c)^3 + 3*cos(d*x +
c))*sin(d*x + c))/(a*d)
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Sympy [A] time = 75.9437, size = 2429, normalized size = 20.76 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**4*sin(d*x+c)**3/(a+a*sin(d*x+c)),x)
[Out]
Piecewise((-105*d*x*tan(c/2 + d*x/2)**12/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 252
00*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2
+ d*x/2)**2 + 1680*a*d) - 630*d*x*tan(c/2 + d*x/2)**10/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d
*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10
080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 1575*d*x*tan(c/2 + d*x/2)**8/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*
a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 +
d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 2100*d*x*tan(c/2 + d*x/2)**6/(1680*a*d*tan(c/2 + d*x/
2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d*x/2)**6 + 2520
0*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 1575*d*x*tan(c/2 + d*x/2)**4/(1680*a*d
*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d
*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 630*d*x*tan(c/2 + d*x/2
)**2/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a
*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 105*d*x/(
1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan
(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 30*tan(c/2 + d*
x/2)**12/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 336
00*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 210*t
an(c/2 + d*x/2)**11/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/
2)**8 + 33600*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a
*d) + 180*tan(c/2 + d*x/2)**10/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan
(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)*
*2 + 1680*a*d) - 1190*tan(c/2 + d*x/2)**9/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25
200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/
2 + d*x/2)**2 + 1680*a*d) - 6270*tan(c/2 + d*x/2)**8/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/
2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080
*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 7980*tan(c/2 + d*x/2)**7/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan
(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)
**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 3880*tan(c/2 + d*x/2)**6/(1680*a*d*tan(c/2 + d*x/2)**12 + 10
080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c
/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 7980*tan(c/2 + d*x/2)**5/(1680*a*d*tan(c/2 + d*x/
2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d*x/2)**6 + 2520
0*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 450*tan(c/2 + d*x/2)**4/(1680*a*d*tan(
c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c/2 + d*x/2)
**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 1190*tan(c/2 + d*x/2)**3/(16
80*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a*d*tan(c
/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) - 2508*tan(c/2 + d*
x/2)**2/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 3360
0*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 210*ta
n(c/2 + d*x/2)/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8
+ 33600*a*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) -
418/(1680*a*d*tan(c/2 + d*x/2)**12 + 10080*a*d*tan(c/2 + d*x/2)**10 + 25200*a*d*tan(c/2 + d*x/2)**8 + 33600*a
*d*tan(c/2 + d*x/2)**6 + 25200*a*d*tan(c/2 + d*x/2)**4 + 10080*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d), Ne(d, 0)),
(x*sin(c)**3*cos(c)**4/(a*sin(c) + a), True))
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Giac [A] time = 1.3679, size = 207, normalized size = 1.77 \begin{align*} -\frac{\frac{15 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 85 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 480 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 570 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 320 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 570 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 85 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 192 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 32\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{6} a}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
-1/240*(15*(d*x + c)/a + 2*(15*tan(1/2*d*x + 1/2*c)^11 + 85*tan(1/2*d*x + 1/2*c)^9 + 480*tan(1/2*d*x + 1/2*c)^
8 - 570*tan(1/2*d*x + 1/2*c)^7 + 320*tan(1/2*d*x + 1/2*c)^6 + 570*tan(1/2*d*x + 1/2*c)^5 - 85*tan(1/2*d*x + 1/
2*c)^3 + 192*tan(1/2*d*x + 1/2*c)^2 - 15*tan(1/2*d*x + 1/2*c) + 32)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a))/d