3.500 \(\int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=83 \[ \frac{a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.078624, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ \frac{a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2836

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*x)/b
)^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]

Rule 12

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^2 (a-x)^2 (a+x)^3}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)^3}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2 a^3+\frac{a^5}{x^2}+\frac{a^4}{x}-2 a^2 x+a x^2+x^3\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \sin ^2(c+d x)}{d}+\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^4(c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.0419195, size = 83, normalized size = 1. \[ \frac{a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.05, size = 116, normalized size = 1.4 \begin{align*}{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d\sin \left ( dx+c \right ) }}-{\frac{8\,a\sin \left ( dx+c \right ) }{3\,d}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}a}{d}}-{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{3\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09919, size = 93, normalized size = 1.12 \begin{align*} \frac{3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 12 \, a \sin \left (d x + c\right )^{2} + 12 \, a \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.15737, size = 250, normalized size = 3.01 \begin{align*} \frac{32 \, a \cos \left (d x + c\right )^{4} + 128 \, a \cos \left (d x + c\right )^{2} + 96 \, a \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 3 \,{\left (8 \, a \cos \left (d x + c\right )^{4} + 16 \, a \cos \left (d x + c\right )^{2} - 11 \, a\right )} \sin \left (d x + c\right ) - 256 \, a}{96 \, d \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^2*(a+a*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*a*log(1/2*sin(d*x + c))*sin(d*x + c) + 3*(8*a*cos(d*x +
c)^4 + 16*a*cos(d*x + c)^2 - 11*a)*sin(d*x + c) - 256*a)/(d*sin(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.28746, size = 107, normalized size = 1.29 \begin{align*} \frac{3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 12 \, a \sin \left (d x + c\right )^{2} + 12 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 24 \, a \sin \left (d x + c\right ) - \frac{12 \,{\left (a \sin \left (d x + c\right ) + a\right )}}{\sin \left (d x + c\right )}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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