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

Optimal. Leaf size=81 \[ \frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{3 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \log (\sin (c+d x))}{d} \]

[Out]

(a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (3*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 = 0.0472288, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2707, 43} \[ \frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{3 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (3*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 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) (a+a \sin (c+d x))^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+x)^4}{x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^3+\frac{a^4}{x}+6 a^2 x+4 a x^2+x^3\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^4 \log (\sin (c+d x))}{d}+\frac{4 a^4 \sin (c+d x)}{d}+\frac{3 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 = 0.0365896, size = 81, normalized size = 1. \[ \frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{3 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (3*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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Maple [A]  time = 0.032, size = 78, normalized size = 1. \begin{align*}{\frac{{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+4\,{\frac{{a}^{4}\sin \left ( dx+c \right ) }{d}}+3\,{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{4\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14212, size = 92, normalized size = 1.14 \begin{align*} \frac{3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 36 \, a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) + 48 \, a^{4} \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)*csc(d*x+c)*(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 + 36*a^4*sin(d*x + c)^2 + 12*a^4*log(sin(d*x + c)) + 48*a^4
*sin(d*x + c))/d

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Fricas [A]  time = 1.45256, size = 180, normalized size = 2.22 \begin{align*} \frac{3 \, a^{4} \cos \left (d x + c\right )^{4} - 42 \, a^{4} \cos \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 16 \,{\left (a^{4} \cos \left (d x + c\right )^{2} - 4 \, a^{4}\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)*csc(d*x+c)*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/12*(3*a^4*cos(d*x + c)^4 - 42*a^4*cos(d*x + c)^2 + 12*a^4*log(1/2*sin(d*x + c)) - 16*(a^4*cos(d*x + c)^2 - 4
*a^4)*sin(d*x + c))/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)*(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.19535, size = 93, normalized size = 1.15 \begin{align*} \frac{3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 36 \, a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 48 \, a^{4} \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)*csc(d*x+c)*(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 + 36*a^4*sin(d*x + c)^2 + 12*a^4*log(abs(sin(d*x + c))) + 4
8*a^4*sin(d*x + c))/d