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

Optimal. Leaf size=86 \[ \frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^2(c+d x)}{2 d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{2 a \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0793945, antiderivative size = 86, 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} \[ \frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^2(c+d x)}{2 d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{2 a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.109221, size = 77, normalized size = 0.9 \[ \frac{a \sin ^3(c+d x)}{3 d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}-\frac{a \left (-\sin ^2(c+d x)+\csc ^2(c+d x)+4 \log (\sin (c+d x))\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.06, size = 139, normalized size = 1.6 \begin{align*} -{\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}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{2\,d}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{d}}-2\,{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15379, size = 92, normalized size = 1.07 \begin{align*} \frac{2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - 12 \, a \log \left (\sin \left (d x + c\right )\right ) - 12 \, a \sin \left (d x + c\right ) - \frac{3 \,{\left (2 \, a \sin \left (d x + c\right ) + a\right )}}{\sin \left (d x + c\right )^{2}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.15681, size = 258, normalized size = 3. \begin{align*} -\frac{6 \, a \cos \left (d x + c\right )^{4} - 9 \, a \cos \left (d x + c\right )^{2} + 24 \,{\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 4 \,{\left (a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} - 8 \, a\right )} \sin \left (d x + c\right ) - 3 \, a}{12 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*a*cos(d*x + c)^4 - 9*a*cos(d*x + c)^2 + 24*(a*cos(d*x + c)^2 - a)*log(1/2*sin(d*x + c)) + 4*(a*cos(d*
x + c)^4 + 4*a*cos(d*x + c)^2 - 8*a)*sin(d*x + c) - 3*a)/(d*cos(d*x + c)^2 - d)

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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)**3*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.29853, size = 111, normalized size = 1.29 \begin{align*} \frac{2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - 12 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 12 \, a \sin \left (d x + c\right ) + \frac{3 \,{\left (6 \, a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right ) - a\right )}}{\sin \left (d x + c\right )^{2}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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