3.598 \(\int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=157 \[ \frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{25 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-2 a^2 x \]

[Out]

-2*a^2*x - (25*a^2*ArcTanh[Cos[c + d*x]])/(16*d) + (a^2*Cos[c + d*x])/d - (2*a^2*Cot[c + d*x])/d + (2*a^2*Cot[
c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (7*a^2*Cot[c + d
*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)

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Rubi [A]  time = 0.234526, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2872, 3770, 3767, 8, 3768, 2638} \[ \frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{25 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-2 a^2 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]

[Out]

-2*a^2*x - (25*a^2*ArcTanh[Cos[c + d*x]])/(16*d) + (a^2*Cos[c + d*x])/d - (2*a^2*Cot[c + d*x])/d + (2*a^2*Cot[
c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (7*a^2*Cot[c + d
*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)

Rule 2872

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))

Rule 3770

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 2638

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

Rubi steps

\begin{align*} \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{\int \left (-2 a^8+2 a^8 \csc (c+d x)+6 a^8 \csc ^2(c+d x)-6 a^8 \csc ^4(c+d x)-2 a^8 \csc ^5(c+d x)+2 a^8 \csc ^6(c+d x)+a^8 \csc ^7(c+d x)-a^8 \sin (c+d x)\right ) \, dx}{a^6}\\ &=-2 a^2 x+a^2 \int \csc ^7(c+d x) \, dx-a^2 \int \sin (c+d x) \, dx+\left (2 a^2\right ) \int \csc (c+d x) \, dx-\left (2 a^2\right ) \int \csc ^5(c+d x) \, dx+\left (2 a^2\right ) \int \csc ^6(c+d x) \, dx+\left (6 a^2\right ) \int \csc ^2(c+d x) \, dx-\left (6 a^2\right ) \int \csc ^4(c+d x) \, dx\\ &=-2 a^2 x-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{1}{6} \left (5 a^2\right ) \int \csc ^5(c+d x) \, dx-\frac{1}{2} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac{\left (6 a^2\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-2 a^2 x-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{1}{8} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{4} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=-2 a^2 x-\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}+\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{1}{16} \left (5 a^2\right ) \int \csc (c+d x) \, dx\\ &=-2 a^2 x-\frac{25 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}\\ \end{align*}

Mathematica [A]  time = 1.49425, size = 270, normalized size = 1.72 \[ -\frac{a^2 \sin (c+d x) (\sin (c+d x)+1)^2 \left (-1920 \cot (c+d x)+\csc ^2\left (\frac{1}{2} (c+d x)\right ) (1472-210 \csc (c+d x))+\csc ^6\left (\frac{1}{2} (c+d x)\right ) (5 \csc (c+d x)+12)-2 \csc ^4\left (\frac{1}{2} (c+d x)\right ) (15 \csc (c+d x)+82)-2 (327 \cos (c+d x)+92 \cos (2 (c+d x))+241) \sec ^6\left (\frac{1}{2} (c+d x)\right )-320 \sin ^6\left (\frac{1}{2} (c+d x)\right ) \csc ^7(c+d x)+480 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^5(c+d x)+840 \sin ^2\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)+120 \csc (c+d x) \left (32 (c+d x)-25 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+25 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{1920 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]

[Out]

-(a^2*(-1920*Cot[c + d*x] + Csc[(c + d*x)/2]^2*(1472 - 210*Csc[c + d*x]) + Csc[(c + d*x)/2]^6*(12 + 5*Csc[c +
d*x]) - 2*Csc[(c + d*x)/2]^4*(82 + 15*Csc[c + d*x]) + 120*Csc[c + d*x]*(32*(c + d*x) + 25*Log[Cos[(c + d*x)/2]
] - 25*Log[Sin[(c + d*x)/2]]) - 2*(241 + 327*Cos[c + d*x] + 92*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^6 + 840*Csc[
c + d*x]^3*Sin[(c + d*x)/2]^2 + 480*Csc[c + d*x]^5*Sin[(c + d*x)/2]^4 - 320*Csc[c + d*x]^7*Sin[(c + d*x)/2]^6)
*Sin[c + d*x]*(1 + Sin[c + d*x])^2)/(1920*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)

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Maple [A]  time = 0.083, size = 205, normalized size = 1.3 \begin{align*} -{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d}}+{\frac{25\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{48\,d}}+{\frac{25\,{a}^{2}\cos \left ( dx+c \right ) }{16\,d}}+{\frac{25\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{2\,{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{2\,{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-2\,{\frac{{a}^{2}\cot \left ( dx+c \right ) }{d}}-2\,{a}^{2}x-2\,{\frac{c{a}^{2}}{d}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x)

[Out]

-5/24/d*a^2/sin(d*x+c)^4*cos(d*x+c)^7+5/16/d*a^2/sin(d*x+c)^2*cos(d*x+c)^7+5/16*a^2*cos(d*x+c)^5/d+25/48*a^2*c
os(d*x+c)^3/d+25/16*a^2*cos(d*x+c)/d+25/16/d*a^2*ln(csc(d*x+c)-cot(d*x+c))-2/5*a^2*cot(d*x+c)^5/d+2/3*a^2*cot(
d*x+c)^3/d-2*a^2*cot(d*x+c)/d-2*a^2*x-2/d*c*a^2-1/6/d*a^2/sin(d*x+c)^6*cos(d*x+c)^7

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Maxima [A]  time = 1.64454, size = 297, normalized size = 1.89 \begin{align*} -\frac{64 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} - 5 \, a^{2}{\left (\frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 30 \, a^{2}{\left (\frac{2 \,{\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/480*(64*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a^2 - 5*a^2*(2*(33*cos(
d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) +
15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 30*a^2*(2*(9*cos(d*x + c)^3 - 7*cos(d*x + c))/(cos(d*x
+ c)^4 - 2*cos(d*x + c)^2 + 1) - 16*cos(d*x + c) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d

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Fricas [B]  time = 1.25213, size = 782, normalized size = 4.98 \begin{align*} -\frac{960 \, a^{2} d x \cos \left (d x + c\right )^{6} - 480 \, a^{2} \cos \left (d x + c\right )^{7} - 2880 \, a^{2} d x \cos \left (d x + c\right )^{4} + 1650 \, a^{2} \cos \left (d x + c\right )^{5} + 2880 \, a^{2} d x \cos \left (d x + c\right )^{2} - 2000 \, a^{2} \cos \left (d x + c\right )^{3} - 960 \, a^{2} d x + 750 \, a^{2} \cos \left (d x + c\right ) + 375 \,{\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 375 \,{\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 64 \,{\left (23 \, a^{2} \cos \left (d x + c\right )^{5} - 35 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, 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)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/480*(960*a^2*d*x*cos(d*x + c)^6 - 480*a^2*cos(d*x + c)^7 - 2880*a^2*d*x*cos(d*x + c)^4 + 1650*a^2*cos(d*x +
c)^5 + 2880*a^2*d*x*cos(d*x + c)^2 - 2000*a^2*cos(d*x + c)^3 - 960*a^2*d*x + 750*a^2*cos(d*x + c) + 375*(a^2*
cos(d*x + c)^6 - 3*a^2*cos(d*x + c)^4 + 3*a^2*cos(d*x + c)^2 - a^2)*log(1/2*cos(d*x + c) + 1/2) - 375*(a^2*cos
(d*x + c)^6 - 3*a^2*cos(d*x + c)^4 + 3*a^2*cos(d*x + c)^2 - a^2)*log(-1/2*cos(d*x + c) + 1/2) - 64*(23*a^2*cos
(d*x + c)^5 - 35*a^2*cos(d*x + c)^3 + 15*a^2*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^
4 + 3*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)**6*csc(d*x+c)**7*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.35652, size = 350, normalized size = 2.23 \begin{align*} \frac{5 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 24 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 280 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 255 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3840 \,{\left (d x + c\right )} a^{2} + 3000 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 2640 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{3840 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} - \frac{7350 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 2640 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 255 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 280 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6}}}{1920 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1920*(5*a^2*tan(1/2*d*x + 1/2*c)^6 + 24*a^2*tan(1/2*d*x + 1/2*c)^5 - 15*a^2*tan(1/2*d*x + 1/2*c)^4 - 280*a^2
*tan(1/2*d*x + 1/2*c)^3 - 255*a^2*tan(1/2*d*x + 1/2*c)^2 - 3840*(d*x + c)*a^2 + 3000*a^2*log(abs(tan(1/2*d*x +
1/2*c))) + 2640*a^2*tan(1/2*d*x + 1/2*c) + 3840*a^2/(tan(1/2*d*x + 1/2*c)^2 + 1) - (7350*a^2*tan(1/2*d*x + 1/
2*c)^6 + 2640*a^2*tan(1/2*d*x + 1/2*c)^5 - 255*a^2*tan(1/2*d*x + 1/2*c)^4 - 280*a^2*tan(1/2*d*x + 1/2*c)^3 - 1
5*a^2*tan(1/2*d*x + 1/2*c)^2 + 24*a^2*tan(1/2*d*x + 1/2*c) + 5*a^2)/tan(1/2*d*x + 1/2*c)^6)/d