\(\int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [557]

Optimal result

Integrand size = 27, antiderivative size = 45 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\log (\sin (c+d x))}{a^3 d}-\frac {4 \log (1+\sin (c+d x))}{a^3 d}+\frac {\sin (c+d x)}{a^3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 45, 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 = {2915, 12, 84} \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\sin (c+d x)}{a^3 d}+\frac {\log (\sin (c+d x))}{a^3 d}-\frac {4 \log (\sin (c+d x)+1)}{a^3 d} \]

[In]

Int[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]

[Out]

Log[Sin[c + d*x]]/(a^3*d) - (4*Log[1 + Sin[c + d*x]])/(a^3*d) + Sin[c + d*x]/(a^3*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 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 2915

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.71 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\log (\sin (c+d x))-4 \log (1+\sin (c+d x))+\sin (c+d x)}{a^3 d} \]

[In]

Integrate[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {\sin \left (d x +c \right )+\ln \left (\sin \left (d x +c \right )\right )-4 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\)	\(33\)
default	\(\frac {\sin \left (d x +c \right )+\ln \left (\sin \left (d x +c \right )\right )-4 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\)	\(33\)
parallelrisch	\(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+3 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin \left (d x +c \right )}{d \,a^{3}}\)	\(53\)
risch	\(\frac {3 i x}{a^{3}}-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{3}}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{3}}+\frac {6 i c}{d \,a^{3}}-\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{3}}\)	\(95\)
norman	\(\frac {\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {2 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {50 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {50 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {76 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {76 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {10 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {10 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {26 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {26 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {92 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {92 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}-\frac {8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{3}}+\frac {3 \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\)	\(320\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + \sin \left (d x + c\right )}{a^{3} d} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {4 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} - \frac {\log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac {\sin \left (d x + c\right )}{a^{3}}}{d} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (45) = 90\).

Time = 0.34 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.29 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {3 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}{a^{3}} - \frac {8 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}}}{d} \]

[In]

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

[Out]

(3*log(tan(1/2*d*x + 1/2*c)^2 + 1)/a^3 - 8*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^3 + log(abs(tan(1/2*d*x + 1/2*
c)))/a^3 - (3*tan(1/2*d*x + 1/2*c)^2 - 2*tan(1/2*d*x + 1/2*c) + 3)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^3))/d

Mupad [B] (verification not implemented)

Time = 10.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.11 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {8\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^3\,d}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^3\,d} \]

[In]

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

[Out]

log(tan(c/2 + (d*x)/2))/(a^3*d) - (8*log(tan(c/2 + (d*x)/2) + 1))/(a^3*d) + (2*tan(c/2 + (d*x)/2))/(d*(a^3*tan
(c/2 + (d*x)/2)^2 + a^3)) + (3*log(tan(c/2 + (d*x)/2)^2 + 1))/(a^3*d)