3.223 \(\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=80 \[ \frac{a^4 \sin ^2(c+d x)}{2 d}+\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^2(c+d x)}{2 d}-\frac{4 a^4 \csc (c+d x)}{d}+\frac{6 a^4 \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0761459, antiderivative size = 80, 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 = {2833, 12, 43} \[ \frac{a^4 \sin ^2(c+d x)}{2 d}+\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^2(c+d x)}{2 d}-\frac{4 a^4 \csc (c+d x)}{d}+\frac{6 a^4 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]

[Out]

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

Rule 2833

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*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 12

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

Rule 43

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])

Rubi steps

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

Mathematica [A]  time = 0.0722376, size = 54, normalized size = 0.68 \[ -\frac{a^4 \left (-\sin ^2(c+d x)-8 \sin (c+d x)+\csc ^2(c+d x)+8 \csc (c+d x)-12 \log (\sin (c+d x))\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]

[Out]

-(a^4*(8*Csc[c + d*x] + Csc[c + d*x]^2 - 12*Log[Sin[c + d*x]] - 8*Sin[c + d*x] - Sin[c + d*x]^2))/(2*d)

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Maple [A]  time = 0.044, size = 79, normalized size = 1. \begin{align*}{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+4\,{\frac{{a}^{4}\sin \left ( dx+c \right ) }{d}}-4\,{\frac{{a}^{4}}{d\sin \left ( dx+c \right ) }}+6\,{\frac{{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{4}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^4,x)

[Out]

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

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Maxima [A]  time = 1.06831, size = 89, normalized size = 1.11 \begin{align*} \frac{a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) + 8 \, a^{4} \sin \left (d x + c\right ) - \frac{8 \, a^{4} \sin \left (d x + c\right ) + a^{4}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/2*(a^4*sin(d*x + c)^2 + 12*a^4*log(sin(d*x + c)) + 8*a^4*sin(d*x + c) - (8*a^4*sin(d*x + c) + a^4)/sin(d*x +
c)^2)/d

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Fricas [A]  time = 1.41718, size = 232, normalized size = 2.9 \begin{align*} -\frac{2 \, a^{4} \cos \left (d x + c\right )^{4} - 16 \, a^{4} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 3 \, a^{4} \cos \left (d x + c\right )^{2} - a^{4} - 24 \,{\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right )}{4 \,{\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)*csc(d*x+c)^3*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/4*(2*a^4*cos(d*x + c)^4 - 16*a^4*cos(d*x + c)^2*sin(d*x + c) - 3*a^4*cos(d*x + c)^2 - a^4 - 24*(a^4*cos(d*x
+ c)^2 - a^4)*log(1/2*sin(d*x + c)))/(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)*csc(d*x+c)**3*(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.17421, size = 90, normalized size = 1.12 \begin{align*} \frac{a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 8 \, a^{4} \sin \left (d x + c\right ) - \frac{8 \, a^{4} \sin \left (d x + c\right ) + a^{4}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/2*(a^4*sin(d*x + c)^2 + 12*a^4*log(abs(sin(d*x + c))) + 8*a^4*sin(d*x + c) - (8*a^4*sin(d*x + c) + a^4)/sin(
d*x + c)^2)/d