\(\int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx\) [1204]

Optimal result

Integrand size = 27, antiderivative size = 83 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \csc (c+d x)}{d}+\frac {b \log (\sin (c+d x))}{d}-\frac {2 a \sin (c+d x)}{d}-\frac {b \sin ^2(c+d x)}{d}+\frac {a \sin ^3(c+d x)}{3 d}+\frac {b \sin ^4(c+d x)}{4 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 83, 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 = {2916, 12, 780} \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a \sin ^3(c+d x)}{3 d}-\frac {2 a \sin (c+d x)}{d}-\frac {a \csc (c+d x)}{d}+\frac {b \sin ^4(c+d x)}{4 d}-\frac {b \sin ^2(c+d x)}{d}+\frac {b \log (\sin (c+d x))}{d} \]

[In]

Int[Cos[c + d*x]^3*Cot[c + d*x]^2*(a + b*Sin[c + d*x]),x]

[Out]

-((a*Csc[c + d*x])/d) + (b*Log[Sin[c + d*x]])/d - (2*a*Sin[c + d*x])/d - (b*Sin[c + d*x]^2)/d + (a*Sin[c + d*x
]^3)/(3*d) + (b*Sin[c + d*x]^4)/(4*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 780

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 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 {b^2 (a+x) \left (b^2-x^2\right )^2}{x^2} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+x) \left (b^2-x^2\right )^2}{x^2} \, dx,x,b \sin (c+d x)\right )}{b^3 d} \\ & = \frac {\text {Subst}\left (\int \left (-2 a b^2+\frac {a b^4}{x^2}+\frac {b^4}{x}-2 b^2 x+a x^2+x^3\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d} \\ & = -\frac {a \csc (c+d x)}{d}+\frac {b \log (\sin (c+d x))}{d}-\frac {2 a \sin (c+d x)}{d}-\frac {b \sin ^2(c+d x)}{d}+\frac {a \sin ^3(c+d x)}{3 d}+\frac {b \sin ^4(c+d x)}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \csc (c+d x)}{d}+\frac {b \log (\sin (c+d x))}{d}-\frac {2 a \sin (c+d x)}{d}-\frac {b \sin ^2(c+d x)}{d}+\frac {a \sin ^3(c+d x)}{3 d}+\frac {b \sin ^4(c+d x)}{4 d} \]

[In]

Integrate[Cos[c + d*x]^3*Cot[c + d*x]^2*(a + b*Sin[c + d*x]),x]

[Out]

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

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.02

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+b \left (\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(85\)
default	\(\frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+b \left (\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(85\)
parallelrisch	\(\frac {-24 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +10 \left (\cos \left (2 d x +2 c \right )+\frac {\cos \left (4 d x +4 c \right )}{20}-\frac {9}{4}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) a \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+9 b \left (\cos \left (2 d x +2 c \right )+\frac {\cos \left (4 d x +4 c \right )}{12}-\frac {13}{12}\right )}{24 d}\)	\(103\)
risch	\(-i x b +\frac {3 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{16 d}+\frac {7 i a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {7 i a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}-\frac {2 i b c}{d}-\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {b \cos \left (4 d x +4 c \right )}{32 d}-\frac {a \sin \left (3 d x +3 c \right )}{12 d}\)	\(153\)
norman	\(\frac {-\frac {a}{2 d}-\frac {13 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {43 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {43 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {13 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {4 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {b \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(207\)

[In]

int(cos(d*x+c)^5*csc(d*x+c)^2*(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.10 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {32 \, a \cos \left (d x + c\right )^{4} + 128 \, a \cos \left (d x + c\right )^{2} + 96 \, b \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 3 \, {\left (8 \, b \cos \left (d x + c\right )^{4} + 16 \, b \cos \left (d x + c\right )^{2} - 11 \, b\right )} \sin \left (d x + c\right ) - 256 \, a}{96 \, d \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^2*(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/96*(32*a*cos(d*x + c)^4 + 128*a*cos(d*x + c)^2 + 96*b*log(1/2*sin(d*x + c))*sin(d*x + c) + 3*(8*b*cos(d*x +
c)^4 + 16*b*cos(d*x + c)^2 - 11*b)*sin(d*x + c) - 256*a)/(d*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**2*(a+b*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {3 \, b \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 12 \, b \sin \left (d x + c\right )^{2} + 12 \, b \log \left (\sin \left (d x + c\right )\right ) - 24 \, a \sin \left (d x + c\right ) - \frac {12 \, a}{\sin \left (d x + c\right )}}{12 \, d} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^2*(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/12*(3*b*sin(d*x + c)^4 + 4*a*sin(d*x + c)^3 - 12*b*sin(d*x + c)^2 + 12*b*log(sin(d*x + c)) - 24*a*sin(d*x +
c) - 12*a/sin(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {3 \, b \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 12 \, b \sin \left (d x + c\right )^{2} + 12 \, b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 24 \, a \sin \left (d x + c\right ) - \frac {12 \, {\left (b \sin \left (d x + c\right ) + a\right )}}{\sin \left (d x + c\right )}}{12 \, d} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^2*(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/12*(3*b*sin(d*x + c)^4 + 4*a*sin(d*x + c)^3 - 12*b*sin(d*x + c)^2 + 12*b*log(abs(sin(d*x + c))) - 24*a*sin(d
*x + c) - 12*(b*sin(d*x + c) + a)/sin(d*x + c))/d

Mupad [B] (verification not implemented)

Time = 11.52 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.01 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {b\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}-\frac {4\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d}+\frac {8\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}-\frac {8\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{d}+\frac {4\,b\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d}-\frac {9\,a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {20\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {16\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {8\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]

[In]

int((cos(c + d*x)^5*(a + b*sin(c + d*x)))/sin(c + d*x)^2,x)

[Out]

(b*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d - (b*log(1/cos(c/2 + (d*x)/2)^2))/d - (4*b*cos(c/2 + (d*x)/2)
^2)/d + (8*b*cos(c/2 + (d*x)/2)^4)/d - (8*b*cos(c/2 + (d*x)/2)^6)/d + (4*b*cos(c/2 + (d*x)/2)^8)/d - (9*a*cos(
c/2 + (d*x)/2))/(2*d*sin(c/2 + (d*x)/2)) - (a*sin(c/2 + (d*x)/2))/(2*d*cos(c/2 + (d*x)/2)) + (20*a*cos(c/2 + (
d*x)/2)^3)/(3*d*sin(c/2 + (d*x)/2)) - (16*a*cos(c/2 + (d*x)/2)^5)/(3*d*sin(c/2 + (d*x)/2)) + (8*a*cos(c/2 + (d
*x)/2)^7)/(3*d*sin(c/2 + (d*x)/2))