\(\int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx\) [1310]

Optimal result

Integrand size = 29, antiderivative size = 212 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^8 d}+\frac {a^2 \left (a^2-b^2\right )^2 \sin (c+d x)}{b^7 d}-\frac {a \left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (a^2-b^2\right )^2 \sin ^3(c+d x)}{3 b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {a \sin ^6(c+d x)}{6 b^2 d}+\frac {\sin ^7(c+d x)}{7 b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2916, 12, 962} \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {a^2 \left (a^2-b^2\right )^2 \sin (c+d x)}{b^7 d}-\frac {a \left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (a^2-b^2\right )^2 \sin ^3(c+d x)}{3 b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^8 d}-\frac {a \sin ^6(c+d x)}{6 b^2 d}+\frac {\sin ^7(c+d x)}{7 b d} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 962

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))

Rule 2916

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*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3 \left (b^2-x^2\right )^2}{b^3 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {x^3 \left (b^2-x^2\right )^2}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^8 d} \\ & = \frac {\text {Subst}\left (\int \left (\left (a^3-a b^2\right )^2-a \left (a^2-b^2\right )^2 x+\left (a^2-b^2\right )^2 x^2-a \left (a^2-2 b^2\right ) x^3+\left (a^2-2 b^2\right ) x^4-a x^5+x^6-\frac {a^3 \left (a^2-b^2\right )^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^8 d} \\ & = -\frac {a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^8 d}+\frac {a^2 \left (a^2-b^2\right )^2 \sin (c+d x)}{b^7 d}-\frac {a \left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (a^2-b^2\right )^2 \sin ^3(c+d x)}{3 b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {a \sin ^6(c+d x)}{6 b^2 d}+\frac {\sin ^7(c+d x)}{7 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-420 a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))+420 b \left (a^3-a b^2\right )^2 \sin (c+d x)-210 a b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)+140 b^3 \left (a^2-b^2\right )^2 \sin ^3(c+d x)-105 a b^4 \left (a^2-2 b^2\right ) \sin ^4(c+d x)+84 b^5 \left (a^2-2 b^2\right ) \sin ^5(c+d x)-70 a b^6 \sin ^6(c+d x)+60 b^7 \sin ^7(c+d x)}{420 b^8 d} \]

[In]

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

[Out]

(-420*a^3*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]] + 420*b*(a^3 - a*b^2)^2*Sin[c + d*x] - 210*a*b^2*(a^2 - b^2)^2
*Sin[c + d*x]^2 + 140*b^3*(a^2 - b^2)^2*Sin[c + d*x]^3 - 105*a*b^4*(a^2 - 2*b^2)*Sin[c + d*x]^4 + 84*b^5*(a^2
- 2*b^2)*Sin[c + d*x]^5 - 70*a*b^6*Sin[c + d*x]^6 + 60*b^7*Sin[c + d*x]^7)/(420*b^8*d)

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\sin ^{7}\left (d x +c \right )\right ) b^{6}}{7}-\frac {a \,b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {a^{2} b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {2 b^{6} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {a^{3} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {a \,b^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{2}+\frac {a^{4} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {2 a^{2} b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {b^{6} \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {a^{5} b \left (\sin ^{2}\left (d x +c \right )\right )}{2}+a^{3} b^{3} \left (\sin ^{2}\left (d x +c \right )\right )-\frac {a \,b^{5} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+a^{6} \sin \left (d x +c \right )-2 a^{4} b^{2} \sin \left (d x +c \right )+a^{2} b^{4} \sin \left (d x +c \right )}{b^{7}}-\frac {a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{8}}}{d}\)	\(256\)
default	\(\frac {\frac {\frac {\left (\sin ^{7}\left (d x +c \right )\right ) b^{6}}{7}-\frac {a \,b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {a^{2} b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {2 b^{6} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {a^{3} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {a \,b^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{2}+\frac {a^{4} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {2 a^{2} b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {b^{6} \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {a^{5} b \left (\sin ^{2}\left (d x +c \right )\right )}{2}+a^{3} b^{3} \left (\sin ^{2}\left (d x +c \right )\right )-\frac {a \,b^{5} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+a^{6} \sin \left (d x +c \right )-2 a^{4} b^{2} \sin \left (d x +c \right )+a^{2} b^{4} \sin \left (d x +c \right )}{b^{7}}-\frac {a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{8}}}{d}\)	\(256\)
parallelrisch	\(\frac {-6720 a^{3} \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )+6720 a^{3} \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1680 a^{5} b^{2}-2520 a^{3} b^{4}+525 a \,b^{6}\right ) \cos \left (2 d x +2 c \right )+\left (-560 a^{4} b^{3}+700 a^{2} b^{5}-35 b^{7}\right ) \sin \left (3 d x +3 c \right )+\left (-210 a^{3} b^{4}+210 a \,b^{6}\right ) \cos \left (4 d x +4 c \right )+\left (84 a^{2} b^{5}-63 b^{7}\right ) \sin \left (5 d x +5 c \right )+35 a \,b^{6} \cos \left (6 d x +6 c \right )+6720 b \left (-\frac {b^{6} \sin \left (7 d x +7 c \right )}{448}+\left (a^{6}-\frac {7}{4} a^{4} b^{2}+\frac {5}{8} a^{2} b^{4}+\frac {5}{64} b^{6}\right ) \sin \left (d x +c \right )-\frac {a^{5} b}{4}+\frac {13 a^{3} b^{3}}{32}-\frac {11 a \,b^{5}}{96}\right )}{6720 b^{8} d}\)	\(282\)
risch	\(\frac {2 i a^{3} c}{b^{4} d}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )} a^{2}}{16 b^{3} d}-\frac {\sin \left (7 d x +7 c \right )}{448 b d}-\frac {3 \sin \left (5 d x +5 c \right )}{320 b d}-\frac {\sin \left (3 d x +3 c \right )}{192 b d}+\frac {a \cos \left (6 d x +6 c \right )}{192 b^{2} d}+\frac {2 i a^{7} c}{b^{8} d}-\frac {4 i a^{5} c}{b^{6} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{6}}{2 b^{7} d}+\frac {7 i {\mathrm e}^{i \left (d x +c \right )} a^{4}}{8 b^{5} d}-\frac {5 i {\mathrm e}^{i \left (d x +c \right )} a^{2}}{16 b^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{6}}{2 b^{7} d}-\frac {5 i {\mathrm e}^{i \left (d x +c \right )}}{128 b d}+\frac {5 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{128 d \,b^{2}}+\frac {a \cos \left (4 d x +4 c \right )}{32 b^{2} d}+\frac {\sin \left (5 d x +5 c \right ) a^{2}}{80 b^{3} d}-\frac {a^{3} \cos \left (4 d x +4 c \right )}{32 b^{4} d}-\frac {\sin \left (3 d x +3 c \right ) a^{4}}{12 b^{5} d}+\frac {5 \sin \left (3 d x +3 c \right ) a^{2}}{48 b^{3} d}-\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} a^{4}}{8 b^{5} d}-\frac {3 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{16 b^{4} d}+\frac {5 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{128 b^{2} d}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )}}{128 b d}+\frac {a^{5} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{6} d}-\frac {3 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 b^{4} d}+\frac {i a^{7} x}{b^{8}}+\frac {i a^{3} x}{b^{4}}-\frac {2 i a^{5} x}{b^{6}}+\frac {a^{5} {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{6} d}-\frac {a^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{8} d}+\frac {2 a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{6} d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{4} d}\)	\(633\)
norman	\(\frac {\frac {2 \left (315 a^{6}-530 a^{4} b^{2}+163 a^{2} b^{4}+4 b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 b^{7} d}+\frac {2 \left (315 a^{6}-530 a^{4} b^{2}+163 a^{2} b^{4}+4 b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 b^{7} d}+\frac {2 \left (3675 a^{6}-5950 a^{4} b^{2}+1883 a^{2} b^{4}+344 b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 b^{7} d}+\frac {2 \left (3675 a^{6}-5950 a^{4} b^{2}+1883 a^{2} b^{4}+344 b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 b^{7} d}-\frac {2 \left (60 a^{5}-84 a^{3} b^{2}+20 a \,b^{4}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,b^{6}}-\frac {2 \left (45 a^{5}-66 a^{3} b^{2}+13 a \,b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,b^{6}}-\frac {2 \left (45 a^{5}-66 a^{3} b^{2}+13 a \,b^{4}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,b^{6}}-\frac {4 \left (3 a^{5}-5 a^{3} b^{2}+a \,b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{6}}-\frac {4 \left (3 a^{5}-5 a^{3} b^{2}+a \,b^{4}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{6}}-\frac {2 \left (a^{5}-2 a^{3} b^{2}+a \,b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{6} d}-\frac {2 \left (a^{5}-2 a^{3} b^{2}+a \,b^{4}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{6} d}+\frac {2 a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{7} d}+\frac {2 a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{7} d}+\frac {2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (21 a^{2}+4 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b^{7} d}+\frac {2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (21 a^{2}+4 b^{2}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b^{7} d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{8} d}-\frac {a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{b^{8} d}\)	\(723\)

[In]

int(cos(d*x+c)^5*sin(d*x+c)^3/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {70 \, a b^{6} \cos \left (d x + c\right )^{6} - 105 \, a^{3} b^{4} \cos \left (d x + c\right )^{4} + 210 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} - 420 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 4 \, {\left (15 \, b^{7} \cos \left (d x + c\right )^{6} - 105 \, a^{6} b + 175 \, a^{4} b^{3} - 56 \, a^{2} b^{5} - 8 \, b^{7} - 3 \, {\left (7 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (35 \, a^{4} b^{3} - 28 \, a^{2} b^{5} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{420 \, b^{8} d} \]

[In]

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

[Out]

1/420*(70*a*b^6*cos(d*x + c)^6 - 105*a^3*b^4*cos(d*x + c)^4 + 210*(a^5*b^2 - a^3*b^4)*cos(d*x + c)^2 - 420*(a^
7 - 2*a^5*b^2 + a^3*b^4)*log(b*sin(d*x + c) + a) - 4*(15*b^7*cos(d*x + c)^6 - 105*a^6*b + 175*a^4*b^3 - 56*a^2
*b^5 - 8*b^7 - 3*(7*a^2*b^5 + b^7)*cos(d*x + c)^4 + (35*a^4*b^3 - 28*a^2*b^5 - 4*b^7)*cos(d*x + c)^2)*sin(d*x
+ c))/(b^8*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {60 \, b^{6} \sin \left (d x + c\right )^{7} - 70 \, a b^{5} \sin \left (d x + c\right )^{6} + 84 \, {\left (a^{2} b^{4} - 2 \, b^{6}\right )} \sin \left (d x + c\right )^{5} - 105 \, {\left (a^{3} b^{3} - 2 \, a b^{5}\right )} \sin \left (d x + c\right )^{4} + 140 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \sin \left (d x + c\right )^{3} - 210 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )^{2} + 420 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )}{b^{7}} - \frac {420 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{420 \, d} \]

[In]

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

[Out]

1/420*((60*b^6*sin(d*x + c)^7 - 70*a*b^5*sin(d*x + c)^6 + 84*(a^2*b^4 - 2*b^6)*sin(d*x + c)^5 - 105*(a^3*b^3 -
2*a*b^5)*sin(d*x + c)^4 + 140*(a^4*b^2 - 2*a^2*b^4 + b^6)*sin(d*x + c)^3 - 210*(a^5*b - 2*a^3*b^3 + a*b^5)*si
n(d*x + c)^2 + 420*(a^6 - 2*a^4*b^2 + a^2*b^4)*sin(d*x + c))/b^7 - 420*(a^7 - 2*a^5*b^2 + a^3*b^4)*log(b*sin(d
*x + c) + a)/b^8)/d

Giac [A] (verification not implemented)

none

Time = 0.70 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.23 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {60 \, b^{6} \sin \left (d x + c\right )^{7} - 70 \, a b^{5} \sin \left (d x + c\right )^{6} + 84 \, a^{2} b^{4} \sin \left (d x + c\right )^{5} - 168 \, b^{6} \sin \left (d x + c\right )^{5} - 105 \, a^{3} b^{3} \sin \left (d x + c\right )^{4} + 210 \, a b^{5} \sin \left (d x + c\right )^{4} + 140 \, a^{4} b^{2} \sin \left (d x + c\right )^{3} - 280 \, a^{2} b^{4} \sin \left (d x + c\right )^{3} + 140 \, b^{6} \sin \left (d x + c\right )^{3} - 210 \, a^{5} b \sin \left (d x + c\right )^{2} + 420 \, a^{3} b^{3} \sin \left (d x + c\right )^{2} - 210 \, a b^{5} \sin \left (d x + c\right )^{2} + 420 \, a^{6} \sin \left (d x + c\right ) - 840 \, a^{4} b^{2} \sin \left (d x + c\right ) + 420 \, a^{2} b^{4} \sin \left (d x + c\right )}{b^{7}} - \frac {420 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8}}}{420 \, d} \]

[In]

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

[Out]

1/420*((60*b^6*sin(d*x + c)^7 - 70*a*b^5*sin(d*x + c)^6 + 84*a^2*b^4*sin(d*x + c)^5 - 168*b^6*sin(d*x + c)^5 -
105*a^3*b^3*sin(d*x + c)^4 + 210*a*b^5*sin(d*x + c)^4 + 140*a^4*b^2*sin(d*x + c)^3 - 280*a^2*b^4*sin(d*x + c)
^3 + 140*b^6*sin(d*x + c)^3 - 210*a^5*b*sin(d*x + c)^2 + 420*a^3*b^3*sin(d*x + c)^2 - 210*a*b^5*sin(d*x + c)^2
+ 420*a^6*sin(d*x + c) - 840*a^4*b^2*sin(d*x + c) + 420*a^2*b^4*sin(d*x + c))/b^7 - 420*(a^7 - 2*a^5*b^2 + a^
3*b^4)*log(abs(b*sin(d*x + c) + a))/b^8)/d

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {{\sin \left (c+d\,x\right )}^5\,\left (\frac {2}{5\,b}-\frac {a^2}{5\,b^3}\right )-\frac {{\sin \left (c+d\,x\right )}^7}{7\,b}-{\sin \left (c+d\,x\right )}^3\,\left (\frac {1}{3\,b}-\frac {a^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{3\,b^2}\right )+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6\,b^2}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}{b^8}-\frac {a\,{\sin \left (c+d\,x\right )}^4\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{4\,b}+\frac {a\,{\sin \left (c+d\,x\right )}^2\,\left (\frac {1}{b}-\frac {a^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{b^2}\right )}{2\,b}-\frac {a^2\,\sin \left (c+d\,x\right )\,\left (\frac {1}{b}-\frac {a^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{b^2}\right )}{b^2}}{d} \]

[In]

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

[Out]

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