3.531 \(\int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=148 \[ \frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{2 a^4 \sin ^2(c+d x)}{d}-\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^4(c+d x)}{4 d}-\frac{4 a^4 \csc ^3(c+d x)}{3 d}-\frac{2 a^4 \csc ^2(c+d x)}{d}+\frac{4 a^4 \csc (c+d x)}{d}-\frac{10 a^4 \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0788179, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 88} \[ \frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{2 a^4 \sin ^2(c+d x)}{d}-\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^4(c+d x)}{4 d}-\frac{4 a^4 \csc ^3(c+d x)}{3 d}-\frac{2 a^4 \csc ^2(c+d x)}{d}+\frac{4 a^4 \csc (c+d x)}{d}-\frac{10 a^4 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^4,x]

[Out]

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

Rule 2707

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

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 \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)^6}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-4 a^3+\frac{a^8}{x^5}+\frac{4 a^7}{x^4}+\frac{4 a^6}{x^3}-\frac{4 a^5}{x^2}-\frac{10 a^4}{x}+4 a^2 x+4 a x^2+x^3\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{4 a^4 \csc (c+d x)}{d}-\frac{2 a^4 \csc ^2(c+d x)}{d}-\frac{4 a^4 \csc ^3(c+d x)}{3 d}-\frac{a^4 \csc ^4(c+d x)}{4 d}-\frac{10 a^4 \log (\sin (c+d x))}{d}-\frac{4 a^4 \sin (c+d x)}{d}+\frac{2 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin ^4(c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.148453, size = 96, normalized size = 0.65 \[ \frac{a^4 \left (3 \sin ^4(c+d x)+16 \sin ^3(c+d x)+24 \sin ^2(c+d x)-48 \sin (c+d x)-3 \csc ^4(c+d x)-16 \csc ^3(c+d x)-24 \csc ^2(c+d x)+48 \csc (c+d x)-120 \log (\sin (c+d x))\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^4,x]

[Out]

(a^4*(48*Csc[c + d*x] - 24*Csc[c + d*x]^2 - 16*Csc[c + d*x]^3 - 3*Csc[c + d*x]^4 - 120*Log[Sin[c + d*x]] - 48*
Sin[c + d*x] + 24*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3 + 3*Sin[c + d*x]^4))/(12*d)

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Maple [A]  time = 0.095, size = 129, normalized size = 0.9 \begin{align*} -{\frac{11\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{11\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-10\,{\frac{{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{4\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^5*(a+a*sin(d*x+c))^4,x)

[Out]

-11/4/d*a^4*cos(d*x+c)^4-11/2/d*a^4*cos(d*x+c)^2-10*a^4*ln(sin(d*x+c))/d-3/d*a^4/sin(d*x+c)^2*cos(d*x+c)^6-4/3
/d*a^4/sin(d*x+c)^3*cos(d*x+c)^6-1/4/d*a^4*cot(d*x+c)^4+1/2/d*a^4*cot(d*x+c)^2

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Maxima [A]  time = 1.12739, size = 162, normalized size = 1.09 \begin{align*} \frac{3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 24 \, a^{4} \sin \left (d x + c\right )^{2} - 120 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) - 48 \, a^{4} \sin \left (d x + c\right ) + \frac{48 \, a^{4} \sin \left (d x + c\right )^{3} - 24 \, a^{4} \sin \left (d x + c\right )^{2} - 16 \, a^{4} \sin \left (d x + c\right ) - 3 \, a^{4}}{\sin \left (d x + c\right )^{4}}}{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)^5*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 24*a^4*sin(d*x + c)^2 - 120*a^4*log(sin(d*x + c)) - 48*a^
4*sin(d*x + c) + (48*a^4*sin(d*x + c)^3 - 24*a^4*sin(d*x + c)^2 - 16*a^4*sin(d*x + c) - 3*a^4)/sin(d*x + c)^4)
/d

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Fricas [A]  time = 1.59987, size = 371, normalized size = 2.51 \begin{align*} \frac{24 \, a^{4} \cos \left (d x + c\right )^{8} - 128 \, a^{4} \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - 288 \, a^{4} \cos \left (d x + c\right )^{6} + 615 \, a^{4} \cos \left (d x + c\right )^{4} - 270 \, a^{4} \cos \left (d x + c\right )^{2} - 105 \, a^{4} - 960 \,{\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right )}{96 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, 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)^5*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/96*(24*a^4*cos(d*x + c)^8 - 128*a^4*cos(d*x + c)^6*sin(d*x + c) - 288*a^4*cos(d*x + c)^6 + 615*a^4*cos(d*x +
c)^4 - 270*a^4*cos(d*x + c)^2 - 105*a^4 - 960*(a^4*cos(d*x + c)^4 - 2*a^4*cos(d*x + c)^2 + a^4)*log(1/2*sin(d
*x + c)))/(d*cos(d*x + c)^4 - 2*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)**5*(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.38203, size = 181, normalized size = 1.22 \begin{align*} \frac{3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 24 \, a^{4} \sin \left (d x + c\right )^{2} - 120 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 48 \, a^{4} \sin \left (d x + c\right ) + \frac{250 \, a^{4} \sin \left (d x + c\right )^{4} + 48 \, a^{4} \sin \left (d x + c\right )^{3} - 24 \, a^{4} \sin \left (d x + c\right )^{2} - 16 \, a^{4} \sin \left (d x + c\right ) - 3 \, a^{4}}{\sin \left (d x + c\right )^{4}}}{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)^5*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 24*a^4*sin(d*x + c)^2 - 120*a^4*log(abs(sin(d*x + c))) -
48*a^4*sin(d*x + c) + (250*a^4*sin(d*x + c)^4 + 48*a^4*sin(d*x + c)^3 - 24*a^4*sin(d*x + c)^2 - 16*a^4*sin(d*x
+ c) - 3*a^4)/sin(d*x + c)^4)/d