3.431 \(\int \frac{\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=138 \[ \frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{16 a^2 d} \]

[Out]

(-11*ArcTanh[Cos[c + d*x]])/(16*a^2*d) + (2*Cot[c + d*x])/(a^2*d) + (4*Cot[c + d*x]^3)/(3*a^2*d) + (2*Cot[c +
d*x]^5)/(5*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^2*d) -
(Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^2*d)

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Rubi [A]  time = 0.248182, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2869, 2757, 3768, 3770, 3767} \[ \frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{16 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^4*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]

[Out]

(-11*ArcTanh[Cos[c + d*x]])/(16*a^2*d) + (2*Cot[c + d*x])/(a^2*d) + (4*Cot[c + d*x]^3)/(3*a^2*d) + (2*Cot[c +
d*x]^5)/(5*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^2*d) -
(Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^2*d)

Rule 2869

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

Rule 2757

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int[Expan
dTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] &
& IGtQ[m, 0] && RationalQ[n]

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 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]

Rubi steps

\begin{align*} \int \frac{\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \csc ^7(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \csc ^5(c+d x)-2 a^2 \csc ^6(c+d x)+a^2 \csc ^7(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \csc ^5(c+d x) \, dx}{a^2}+\frac{\int \csc ^7(c+d x) \, dx}{a^2}-\frac{2 \int \csc ^6(c+d x) \, dx}{a^2}\\ &=-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac{3 \int \csc ^3(c+d x) \, dx}{4 a^2}+\frac{5 \int \csc ^5(c+d x) \, dx}{6 a^2}+\frac{2 \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}\\ &=\frac{2 \cot (c+d x)}{a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac{3 \int \csc (c+d x) \, dx}{8 a^2}+\frac{5 \int \csc ^3(c+d x) \, dx}{8 a^2}\\ &=-\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac{5 \int \csc (c+d x) \, dx}{16 a^2}\\ &=-\frac{11 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.806337, size = 229, normalized size = 1.66 \[ \frac{\csc ^6(c+d x) \left (3840 \sin (2 (c+d x))-1536 \sin (4 (c+d x))+256 \sin (6 (c+d x))-2820 \cos (c+d x)+1870 \cos (3 (c+d x))-330 \cos (5 (c+d x))+1650 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+2475 \cos (2 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-990 \cos (4 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+165 \cos (6 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-1650 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-2475 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+990 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-165 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{7680 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^4*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]

[Out]

(Csc[c + d*x]^6*(-2820*Cos[c + d*x] + 1870*Cos[3*(c + d*x)] - 330*Cos[5*(c + d*x)] - 1650*Log[Cos[(c + d*x)/2]
] + 2475*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 990*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 165*Cos[6*(c +
d*x)]*Log[Cos[(c + d*x)/2]] + 1650*Log[Sin[(c + d*x)/2]] - 2475*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 990*C
os[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 165*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 3840*Sin[2*(c + d*x)] - 1
536*Sin[4*(c + d*x)] + 256*Sin[6*(c + d*x)]))/(7680*a^2*d)

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Maple [A]  time = 0.196, size = 246, normalized size = 1.8 \begin{align*}{\frac{1}{384\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}-{\frac{1}{80\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{5}{128\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{5}{48\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{31}{128\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{5}{8\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{5}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{1}{80\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}-{\frac{5}{128\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{11}{16\,d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{384\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}+{\frac{5}{48\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{31}{128\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x)

[Out]

1/384/d/a^2*tan(1/2*d*x+1/2*c)^6-1/80/d/a^2*tan(1/2*d*x+1/2*c)^5+5/128/d/a^2*tan(1/2*d*x+1/2*c)^4-5/48/d/a^2*t
an(1/2*d*x+1/2*c)^3+31/128/d/a^2*tan(1/2*d*x+1/2*c)^2-5/8/d/a^2*tan(1/2*d*x+1/2*c)+5/8/d/a^2/tan(1/2*d*x+1/2*c
)+1/80/d/a^2/tan(1/2*d*x+1/2*c)^5-5/128/d/a^2/tan(1/2*d*x+1/2*c)^4+11/16/d/a^2*ln(tan(1/2*d*x+1/2*c))-1/384/d/
a^2/tan(1/2*d*x+1/2*c)^6+5/48/d/a^2/tan(1/2*d*x+1/2*c)^3-31/128/d/a^2/tan(1/2*d*x+1/2*c)^2

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Maxima [B]  time = 1.1117, size = 371, normalized size = 2.69 \begin{align*} -\frac{\frac{\frac{1200 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{465 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{75 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{24 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a^{2}} - \frac{1320 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac{{\left (\frac{24 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{75 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{465 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1200 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a^{2} \sin \left (d x + c\right )^{6}}}{1920 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/1920*((1200*sin(d*x + c)/(cos(d*x + c) + 1) - 465*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 200*sin(d*x + c)^3/
(cos(d*x + c) + 1)^3 - 75*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 24*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin
(d*x + c)^6/(cos(d*x + c) + 1)^6)/a^2 - 1320*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 - (24*sin(d*x + c)/(cos(
d*x + c) + 1) - 75*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 200*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 465*sin(d*x
+ c)^4/(cos(d*x + c) + 1)^4 + 1200*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5)*(cos(d*x + c) + 1)^6/(a^2*sin(d*x
+ c)^6))/d

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Fricas [A]  time = 1.19363, size = 555, normalized size = 4.02 \begin{align*} \frac{330 \, \cos \left (d x + c\right )^{5} - 880 \, \cos \left (d x + c\right )^{3} - 165 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 165 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 64 \,{\left (8 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 630 \, \cos \left (d x + c\right )}{480 \,{\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{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)^7/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/480*(330*cos(d*x + c)^5 - 880*cos(d*x + c)^3 - 165*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1
)*log(1/2*cos(d*x + c) + 1/2) + 165*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(-1/2*cos(d*
x + c) + 1/2) - 64*(8*cos(d*x + c)^5 - 20*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c) + 630*cos(d*x + c))/(
a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^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)**4*csc(d*x+c)**7/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.39306, size = 290, normalized size = 2.1 \begin{align*} \frac{\frac{1320 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac{3234 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1200 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 465 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 200 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 75 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, \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}} + \frac{5 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 24 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 75 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 200 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 465 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1200 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{12}}}{1920 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1920*(1320*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (3234*tan(1/2*d*x + 1/2*c)^6 - 1200*tan(1/2*d*x + 1/2*c)^5 +
465*tan(1/2*d*x + 1/2*c)^4 - 200*tan(1/2*d*x + 1/2*c)^3 + 75*tan(1/2*d*x + 1/2*c)^2 - 24*tan(1/2*d*x + 1/2*c)
+ 5)/(a^2*tan(1/2*d*x + 1/2*c)^6) + (5*a^10*tan(1/2*d*x + 1/2*c)^6 - 24*a^10*tan(1/2*d*x + 1/2*c)^5 + 75*a^10
*tan(1/2*d*x + 1/2*c)^4 - 200*a^10*tan(1/2*d*x + 1/2*c)^3 + 465*a^10*tan(1/2*d*x + 1/2*c)^2 - 1200*a^10*tan(1/
2*d*x + 1/2*c))/a^12)/d