\(\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx\) [222]

Optimal result

Integrand size = 25, antiderivative size = 78 \[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {a^4 \csc (c+d x)}{d}+\frac {4 a^4 \log (\sin (c+d x))}{d}+\frac {6 a^4 \sin (c+d x)}{d}+\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {a^4 \sin ^3(c+d x)}{3 d} \]

[Out]

-a^4*csc(d*x+c)/d+4*a^4*ln(sin(d*x+c))/d+6*a^4*sin(d*x+c)/d+2*a^4*sin(d*x+c)^2/d+1/3*a^4*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2912, 12, 45} \[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 \sin ^3(c+d x)}{3 d}+\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {6 a^4 \sin (c+d x)}{d}-\frac {a^4 \csc (c+d x)}{d}+\frac {4 a^4 \log (\sin (c+d x))}{d} \]

[In]

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

[Out]

-((a^4*Csc[c + d*x])/d) + (4*a^4*Log[Sin[c + d*x]])/d + (6*a^4*Sin[c + d*x])/d + (2*a^4*Sin[c + d*x]^2)/d + (a
^4*Sin[c + d*x]^3)/(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 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 {a^2 (a+x)^4}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a \text {Subst}\left (\int \frac {(a+x)^4}{x^2} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a \text {Subst}\left (\int \left (6 a^2+\frac {a^4}{x^2}+\frac {4 a^3}{x}+4 a x+x^2\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {a^4 \csc (c+d x)}{d}+\frac {4 a^4 \log (\sin (c+d x))}{d}+\frac {6 a^4 \sin (c+d x)}{d}+\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {a^4 \sin ^3(c+d x)}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {a^4 \csc (c+d x)}{d}+\frac {4 a^4 \log (\sin (c+d x))}{d}+\frac {6 a^4 \sin (c+d x)}{d}+\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {a^4 \sin ^3(c+d x)}{3 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(-\frac {a^{4} \left (\csc \left (d x +c \right )+4 \ln \left (\csc \left (d x +c \right )\right )-\frac {6}{\csc \left (d x +c \right )}-\frac {2}{\csc \left (d x +c \right )^{2}}-\frac {1}{3 \csc \left (d x +c \right )^{3}}\right )}{d}\)	\(55\)
default	\(-\frac {a^{4} \left (\csc \left (d x +c \right )+4 \ln \left (\csc \left (d x +c \right )\right )-\frac {6}{\csc \left (d x +c \right )}-\frac {2}{\csc \left (d x +c \right )^{2}}-\frac {1}{3 \csc \left (d x +c \right )^{3}}\right )}{d}\)	\(55\)
parallelrisch	\(-\frac {\left (-2+8 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (-37+\frac {73 \cos \left (d x +c \right )}{2}+\cos \left (2 d x +2 c \right )-\frac {\cos \left (3 d x +3 c \right )}{2}\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{3}+\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \cos \left (2 d x +2 c \right )\right ) a^{4}}{2 d}\)	\(108\)
risch	\(-4 i a^{4} x -\frac {a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{2 d}-\frac {25 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {25 i a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{2 d}-\frac {8 i a^{4} c}{d}-\frac {2 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {4 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {a^{4} \sin \left (3 d x +3 c \right )}{12 d}\)	\(157\)
norman	\(\frac {-\frac {a^{4}}{2 d}+\frac {19 a^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {101 a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {101 a^{4} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {19 a^{4} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a^{4} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {8 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {4 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{4} \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(230\)

[In]

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

[Out]

-1/d*a^4*(csc(d*x+c)+4*ln(csc(d*x+c))-6/csc(d*x+c)-2/csc(d*x+c)^2-1/3/csc(d*x+c)^3)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.17 \[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \cos \left (d x + c\right )^{4} - 20 \, a^{4} \cos \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 16 \, a^{4} - 3 \, {\left (2 \, a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \sin \left (d x + c\right )}{3 \, d \sin \left (d x + c\right )} \]

[In]

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

[Out]

1/3*(a^4*cos(d*x + c)^4 - 20*a^4*cos(d*x + c)^2 + 12*a^4*log(1/2*sin(d*x + c))*sin(d*x + c) + 16*a^4 - 3*(2*a^
4*cos(d*x + c)^2 - a^4)*sin(d*x + c))/(d*sin(d*x + c))

Sympy [F]

\[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx=a^{4} \left (\int \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int 4 \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

a**4*(Integral(cos(c + d*x)*csc(c + d*x)**2, x) + Integral(4*sin(c + d*x)*cos(c + d*x)*csc(c + d*x)**2, x) + I
ntegral(6*sin(c + d*x)**2*cos(c + d*x)*csc(c + d*x)**2, x) + Integral(4*sin(c + d*x)**3*cos(c + d*x)*csc(c + d
*x)**2, x) + Integral(sin(c + d*x)**4*cos(c + d*x)*csc(c + d*x)**2, x))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.86 \[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \sin \left (d x + c\right )^{3} + 6 \, a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) + 18 \, a^{4} \sin \left (d x + c\right ) - \frac {3 \, a^{4}}{\sin \left (d x + c\right )}}{3 \, d} \]

[In]

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

[Out]

1/3*(a^4*sin(d*x + c)^3 + 6*a^4*sin(d*x + c)^2 + 12*a^4*log(sin(d*x + c)) + 18*a^4*sin(d*x + c) - 3*a^4/sin(d*
x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.87 \[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \sin \left (d x + c\right )^{3} + 6 \, a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 18 \, a^{4} \sin \left (d x + c\right ) - \frac {3 \, a^{4}}{\sin \left (d x + c\right )}}{3 \, d} \]

[In]

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

[Out]

1/3*(a^4*sin(d*x + c)^3 + 6*a^4*sin(d*x + c)^2 + 12*a^4*log(abs(sin(d*x + c))) + 18*a^4*sin(d*x + c) - 3*a^4/s
in(d*x + c))/d

Mupad [B] (verification not implemented)

Time = 9.22 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.01 \[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {8\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d}-\frac {8\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}-\frac {4\,a^4\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {4\,a^4\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {28\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {16\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {8\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {23\,a^4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {a^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]

[In]

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

[Out]

(8*a^4*cos(c/2 + (d*x)/2)^2)/d - (8*a^4*cos(c/2 + (d*x)/2)^4)/d - (4*a^4*log(1/cos(c/2 + (d*x)/2)^2))/d + (4*a
^4*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d - (28*a^4*cos(c/2 + (d*x)/2)^3)/(3*d*sin(c/2 + (d*x)/2)) - (1
6*a^4*cos(c/2 + (d*x)/2)^5)/(3*d*sin(c/2 + (d*x)/2)) + (8*a^4*cos(c/2 + (d*x)/2)^7)/(3*d*sin(c/2 + (d*x)/2)) +
(23*a^4*cos(c/2 + (d*x)/2))/(2*d*sin(c/2 + (d*x)/2)) - (a^4*sin(c/2 + (d*x)/2))/(2*d*cos(c/2 + (d*x)/2))