\(\int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [232]

Optimal result

Integrand size = 27, antiderivative size = 87 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {4 \log (1+\sin (c+d x))}{a^2 d}+\frac {3 \sin (c+d x)}{a^2 d}-\frac {\sin ^2(c+d x)}{a^2 d}+\frac {\sin ^3(c+d x)}{3 a^2 d}-\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )} \]

[Out]

-4*ln(1+sin(d*x+c))/a^2/d+3*sin(d*x+c)/a^2/d-sin(d*x+c)^2/a^2/d+1/3*sin(d*x+c)^3/a^2/d-1/d/(a^2+a^2*sin(d*x+c)
)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2912, 12, 45} \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\sin ^3(c+d x)}{3 a^2 d}-\frac {\sin ^2(c+d x)}{a^2 d}+\frac {3 \sin (c+d x)}{a^2 d}-\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )}-\frac {4 \log (\sin (c+d x)+1)}{a^2 d} \]

[In]

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

[Out]

(-4*Log[1 + Sin[c + d*x]])/(a^2*d) + (3*Sin[c + d*x])/(a^2*d) - Sin[c + d*x]^2/(a^2*d) + Sin[c + d*x]^3/(3*a^2
*d) - 1/(d*(a^2 + a^2*Sin[c + d*x]))

Rule 12

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2912

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{a^4 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \frac {x^4}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \left (3 a^2-2 a x+x^2+\frac {a^4}{(a+x)^2}-\frac {4 a^3}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = -\frac {4 \log (1+\sin (c+d x))}{a^2 d}+\frac {3 \sin (c+d x)}{a^2 d}-\frac {\sin ^2(c+d x)}{a^2 d}+\frac {\sin ^3(c+d x)}{3 a^2 d}-\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {-3 (1+4 \log (1+\sin (c+d x)))+(9-12 \log (1+\sin (c+d x))) \sin (c+d x)+6 \sin ^2(c+d x)-2 \sin ^3(c+d x)+\sin ^4(c+d x)}{3 a^2 d (1+\sin (c+d x))} \]

[In]

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

[Out]

(-3*(1 + 4*Log[1 + Sin[c + d*x]]) + (9 - 12*Log[1 + Sin[c + d*x]])*Sin[c + d*x] + 6*Sin[c + d*x]^2 - 2*Sin[c +
d*x]^3 + Sin[c + d*x]^4)/(3*a^2*d*(1 + Sin[c + d*x]))

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\left (\sin ^{2}\left (d x +c \right )\right )+3 \sin \left (d x +c \right )-4 \ln \left (1+\sin \left (d x +c \right )\right )-\frac {1}{1+\sin \left (d x +c \right )}}{d \,a^{2}}\)	\(60\)
default	\(\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\left (\sin ^{2}\left (d x +c \right )\right )+3 \sin \left (d x +c \right )-4 \ln \left (1+\sin \left (d x +c \right )\right )-\frac {1}{1+\sin \left (d x +c \right )}}{d \,a^{2}}\)	\(60\)
parallelrisch	\(\frac {\left (96 \sin \left (d x +c \right )+96\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-192 \sin \left (d x +c \right )-192\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-28 \cos \left (2 d x +2 c \right )+\cos \left (4 d x +4 c \right )+84 \sin \left (d x +c \right )+4 \sin \left (3 d x +3 c \right )+27}{24 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )}\)	\(106\)
risch	\(\frac {4 i x}{a^{2}}-\frac {13 i {\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{2}}+\frac {13 i {\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{2}}+\frac {8 i c}{d \,a^{2}}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )}}{d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2}}-\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{2}}-\frac {\sin \left (3 d x +3 c \right )}{12 d \,a^{2}}+\frac {\cos \left (2 d x +2 c \right )}{2 d \,a^{2}}\)	\(142\)
norman	\(\frac {\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {8 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {72 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {72 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {16 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {128 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {128 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {368 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {368 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {304 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {304 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{2}}+\frac {4 \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\)	\(303\)

[In]

int(cos(d*x+c)*sin(d*x+c)^4/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d/a^2*(1/3*sin(d*x+c)^3-sin(d*x+c)^2+3*sin(d*x+c)-4*ln(1+sin(d*x+c))-1/(1+sin(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2} - 24 \, {\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (4 \, \cos \left (d x + c\right )^{2} + 17\right )} \sin \left (d x + c\right ) + 11}{6 \, {\left (a^{2} d \sin \left (d x + c\right ) + a^{2} d\right )}} \]

[In]

integrate(cos(d*x+c)*sin(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/6*(2*cos(d*x + c)^4 - 16*cos(d*x + c)^2 - 24*(sin(d*x + c) + 1)*log(sin(d*x + c) + 1) + (4*cos(d*x + c)^2 +
17)*sin(d*x + c) + 11)/(a^2*d*sin(d*x + c) + a^2*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (75) = 150\).

Time = 0.79 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.31 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\begin {cases} - \frac {12 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} - \frac {12 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} + \frac {\sin ^{4}{\left (c + d x \right )}}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} - \frac {2 \sin ^{3}{\left (c + d x \right )}}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} + \frac {6 \sin ^{2}{\left (c + d x \right )}}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} - \frac {12}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{4}{\left (c \right )} \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)*sin(d*x+c)**4/(a+a*sin(d*x+c))**2,x)

[Out]

Piecewise((-12*log(sin(c + d*x) + 1)*sin(c + d*x)/(3*a**2*d*sin(c + d*x) + 3*a**2*d) - 12*log(sin(c + d*x) + 1
)/(3*a**2*d*sin(c + d*x) + 3*a**2*d) + sin(c + d*x)**4/(3*a**2*d*sin(c + d*x) + 3*a**2*d) - 2*sin(c + d*x)**3/
(3*a**2*d*sin(c + d*x) + 3*a**2*d) + 6*sin(c + d*x)**2/(3*a**2*d*sin(c + d*x) + 3*a**2*d) - 12/(3*a**2*d*sin(c
+ d*x) + 3*a**2*d), Ne(d, 0)), (x*sin(c)**4*cos(c)/(a*sin(c) + a)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {3}{a^{2} \sin \left (d x + c\right ) + a^{2}} - \frac {\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 9 \, \sin \left (d x + c\right )}{a^{2}} + \frac {12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}}}{3 \, d} \]

[In]

integrate(cos(d*x+c)*sin(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/3*(3/(a^2*sin(d*x + c) + a^2) - (sin(d*x + c)^3 - 3*sin(d*x + c)^2 + 9*sin(d*x + c))/a^2 + 12*log(sin(d*x +
c) + 1)/a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.23 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3} {\left (\frac {6 \, a}{a \sin \left (d x + c\right ) + a} - \frac {18 \, a^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}} - 1\right )}}{a^{5}} - \frac {12 \, \log \left (\frac {{\left | a \sin \left (d x + c\right ) + a \right |}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2} {\left | a \right |}}\right )}{a^{2}} + \frac {3}{{\left (a \sin \left (d x + c\right ) + a\right )} a}}{3 \, d} \]

[In]

integrate(cos(d*x+c)*sin(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/3*((a*sin(d*x + c) + a)^3*(6*a/(a*sin(d*x + c) + a) - 18*a^2/(a*sin(d*x + c) + a)^2 - 1)/a^5 - 12*log(abs(a
*sin(d*x + c) + a)/((a*sin(d*x + c) + a)^2*abs(a)))/a^2 + 3/((a*sin(d*x + c) + a)*a))/d

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.83 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {1}{a^2\,\sin \left (c+d\,x\right )+a^2}+\frac {4\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^2}-\frac {3\,\sin \left (c+d\,x\right )}{a^2}+\frac {{\sin \left (c+d\,x\right )}^2}{a^2}-\frac {{\sin \left (c+d\,x\right )}^3}{3\,a^2}}{d} \]

[In]

int((cos(c + d*x)*sin(c + d*x)^4)/(a + a*sin(c + d*x))^2,x)

[Out]

-(1/(a^2*sin(c + d*x) + a^2) + (4*log(sin(c + d*x) + 1))/a^2 - (3*sin(c + d*x))/a^2 + sin(c + d*x)^2/a^2 - sin
(c + d*x)^3/(3*a^2))/d