3.37 \(\int \cot ^3(c+d x) (a+a \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=102 \[ -\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{5 a^4 \sin ^2(c+d x)}{2 d}-\frac{a^4 \csc ^2(c+d x)}{2 d}-\frac{4 a^4 \csc (c+d x)}{d}+\frac{5 a^4 \log (\sin (c+d x))}{d} \]

[Out]

(-4*a^4*Csc[c + d*x])/d - (a^4*Csc[c + d*x]^2)/(2*d) + (5*a^4*Log[Sin[c + d*x]])/d - (5*a^4*Sin[c + d*x]^2)/(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 = 0.0667044, antiderivative size = 102, 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, 75} \[ -\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{5 a^4 \sin ^2(c+d x)}{2 d}-\frac{a^4 \csc ^2(c+d x)}{2 d}-\frac{4 a^4 \csc (c+d x)}{d}+\frac{5 a^4 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^3*(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) + (5*a^4*Log[Sin[c + d*x]])/d - (5*a^4*Sin[c + d*x]^2)/(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 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

\begin{align*} \int \cot ^3(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x) (a+x)^5}{x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^6}{x^3}+\frac{4 a^5}{x^2}+\frac{5 a^4}{x}-5 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{a^4 \csc ^2(c+d x)}{2 d}+\frac{5 a^4 \log (\sin (c+d x))}{d}-\frac{5 a^4 \sin ^2(c+d x)}{2 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 = 0.137296, size = 78, normalized size = 0.76 \[ -\frac{a^4 \sin ^4(c+d x) \left (6 \csc ^6(c+d x)+48 \csc ^5(c+d x)+30 \csc ^2(c+d x)+16 \csc (c+d x)+\csc ^4(c+d x) (90-60 \log (\sin (c+d x)))+3\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^3*(a + a*Sin[c + d*x])^4,x]

[Out]

-(a^4*(3 + 16*Csc[c + d*x] + 30*Csc[c + d*x]^2 + 48*Csc[c + d*x]^5 + 6*Csc[c + d*x]^6 + Csc[c + d*x]^4*(90 - 6
0*Log[Sin[c + d*x]]))*Sin[c + d*x]^4)/(12*d)

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Maple [A]  time = 0.054, size = 125, normalized size = 1.2 \begin{align*} -{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{8\,{a}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,d}}-{\frac{16\,{a}^{4}\sin \left ( dx+c \right ) }{3\,d}}+3\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{d}}+5\,{\frac{{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-4\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d\sin \left ( dx+c \right ) }}-{\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(cot(d*x+c)^3*(a+a*sin(d*x+c))^4,x)

[Out]

-1/4/d*a^4*cos(d*x+c)^4-8/3/d*a^4*sin(d*x+c)*cos(d*x+c)^2-16/3*a^4*sin(d*x+c)/d+3/d*a^4*cos(d*x+c)^2+5*a^4*ln(
sin(d*x+c))/d-4/d*a^4/sin(d*x+c)*cos(d*x+c)^4-1/2/d*a^4*cot(d*x+c)^2

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Maxima [A]  time = 1.11802, size = 111, 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} + 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) + \frac{6 \,{\left (8 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )}}{\sin \left (d x + c\right )^{2}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(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 + 30*a^4*sin(d*x + c)^2 - 60*a^4*log(sin(d*x + c)) + 6*(8*
a^4*sin(d*x + c) + a^4)/sin(d*x + c)^2)/d

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Fricas [A]  time = 1.73429, size = 324, normalized size = 3.18 \begin{align*} -\frac{24 \, a^{4} \cos \left (d x + c\right )^{6} - 312 \, a^{4} \cos \left (d x + c\right )^{4} + 423 \, a^{4} \cos \left (d x + c\right )^{2} - 183 \, a^{4} - 480 \,{\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 ) - 128 \,{\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + 4 \, a^{4}\right )} \sin \left (d x + c\right )}{96 \,{\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(cot(d*x+c)^3*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/96*(24*a^4*cos(d*x + c)^6 - 312*a^4*cos(d*x + c)^4 + 423*a^4*cos(d*x + c)^2 - 183*a^4 - 480*(a^4*cos(d*x +
c)^2 - a^4)*log(1/2*sin(d*x + c)) - 128*(a^4*cos(d*x + c)^4 - 2*a^4*cos(d*x + c)^2 + 4*a^4)*sin(d*x + c))/(d*c
os(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(cot(d*x+c)**3*(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.39327, size = 130, normalized size = 1.27 \begin{align*} -\frac{3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + \frac{6 \,{\left (15 \, a^{4} \sin \left (d x + c\right )^{2} + 8 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )}}{\sin \left (d x + c\right )^{2}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(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 + 30*a^4*sin(d*x + c)^2 - 60*a^4*log(abs(sin(d*x + c))) +
6*(15*a^4*sin(d*x + c)^2 + 8*a^4*sin(d*x + c) + a^4)/sin(d*x + c)^2)/d