3.1100 \(\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx\)

Optimal. Leaf size=88 \[ -\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot ^3(c+d x)}{3 d}+\frac{b \cot (c+d x)}{d}+b x \]

[Out]

b*x - (3*a*ArcTanh[Cos[c + d*x]])/(8*d) + (b*Cot[c + d*x])/d - (b*Cot[c + d*x]^3)/(3*d) + (3*a*Cot[c + d*x]*Cs
c[c + d*x])/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x])/(4*d)

________________________________________________________________________________________

Rubi [A]  time = 0.111163, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2838, 2611, 3770, 3473, 8} \[ -\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot ^3(c+d x)}{3 d}+\frac{b \cot (c+d x)}{d}+b x \]

Antiderivative was successfully verified.

[In]

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

[Out]

b*x - (3*a*ArcTanh[Cos[c + d*x]])/(8*d) + (b*Cot[c + d*x])/d - (b*Cot[c + d*x]^3)/(3*d) + (3*a*Cot[c + d*x]*Cs
c[c + d*x])/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x])/(4*d)

Rule 2838

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cot ^4(c+d x) \csc (c+d x) \, dx+b \int \cot ^4(c+d x) \, dx\\ &=-\frac{b \cot ^3(c+d x)}{3 d}-\frac{a \cot ^3(c+d x) \csc (c+d x)}{4 d}-\frac{1}{4} (3 a) \int \cot ^2(c+d x) \csc (c+d x) \, dx-b \int \cot ^2(c+d x) \, dx\\ &=\frac{b \cot (c+d x)}{d}-\frac{b \cot ^3(c+d x)}{3 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{1}{8} (3 a) \int \csc (c+d x) \, dx+b \int 1 \, dx\\ &=b x-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{b \cot (c+d x)}{d}-\frac{b \cot ^3(c+d x)}{3 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a \cot ^3(c+d x) \csc (c+d x)}{4 d}\\ \end{align*}

Mathematica [C]  time = 0.0507189, size = 153, normalized size = 1.74 \[ -\frac{a \csc ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{5 a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{a \sec ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{3 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}-\frac{3 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}-\frac{b \cot ^3(c+d x) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (b*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1,
-1/2, -Tan[c + d*x]^2])/(3*d) - (3*a*Log[Cos[(c + d*x)/2]])/(8*d) + (3*a*Log[Sin[(c + d*x)/2]])/(8*d) - (5*a*
Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)

________________________________________________________________________________________

Maple [A]  time = 0.063, size = 128, normalized size = 1.5 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d}}+{\frac{3\,\cos \left ( dx+c \right ) a}{8\,d}}+{\frac{3\,a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{b \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{b\cot \left ( dx+c \right ) }{d}}+bx+{\frac{cb}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.63938, size = 144, normalized size = 1.64 \begin{align*} \frac{16 \,{\left (3 \, d x + 3 \, c + \frac{3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} b - 3 \, a{\left (\frac{2 \,{\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/48*(16*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*b - 3*a*(2*(5*cos(d*x + c)^3 - 3*cos(d*x + c))/
(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)))/d

________________________________________________________________________________________

Fricas [B]  time = 1.81543, size = 494, normalized size = 5.61 \begin{align*} \frac{48 \, b d x \cos \left (d x + c\right )^{4} - 96 \, b d x \cos \left (d x + c\right )^{2} - 30 \, a \cos \left (d x + c\right )^{3} + 48 \, b d x + 18 \, a \cos \left (d x + c\right ) - 9 \,{\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 9 \,{\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 16 \,{\left (4 \, b \cos \left (d x + c\right )^{3} - 3 \, b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \,{\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)^4*csc(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/48*(48*b*d*x*cos(d*x + c)^4 - 96*b*d*x*cos(d*x + c)^2 - 30*a*cos(d*x + c)^3 + 48*b*d*x + 18*a*cos(d*x + c) -
9*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a)*log(1/2*cos(d*x + c) + 1/2) + 9*(a*cos(d*x + c)^4 - 2*a*cos(d*x
+ c)^2 + a)*log(-1/2*cos(d*x + c) + 1/2) - 16*(4*b*cos(d*x + c)^3 - 3*b*cos(d*x + c))*sin(d*x + c))/(d*cos(d*
x + c)^4 - 2*d*cos(d*x + c)^2 + d)

________________________________________________________________________________________

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)**4*csc(d*x+c)**5*(a+b*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.43739, size = 207, normalized size = 2.35 \begin{align*} \frac{3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 8 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 24 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 192 \,{\left (d x + c\right )} b + 72 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 120 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{150 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 120 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 24 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 8 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}}{192 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/192*(3*a*tan(1/2*d*x + 1/2*c)^4 + 8*b*tan(1/2*d*x + 1/2*c)^3 - 24*a*tan(1/2*d*x + 1/2*c)^2 + 192*(d*x + c)*b
+ 72*a*log(abs(tan(1/2*d*x + 1/2*c))) - 120*b*tan(1/2*d*x + 1/2*c) - (150*a*tan(1/2*d*x + 1/2*c)^4 - 120*b*ta
n(1/2*d*x + 1/2*c)^3 - 24*a*tan(1/2*d*x + 1/2*c)^2 + 8*b*tan(1/2*d*x + 1/2*c) + 3*a)/tan(1/2*d*x + 1/2*c)^4)/d