3.104 \(\int \frac{\csc ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=89 \[ \frac{4 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^7(c+d x)}{7 a^3 d}+\frac{7 \csc ^5(c+d x)}{5 a^3 d}-\frac{\csc ^3(c+d x)}{a^3 d} \]

[Out]

(3*Cot[c + d*x]^5)/(5*a^3*d) + (4*Cot[c + d*x]^7)/(7*a^3*d) - Csc[c + d*x]^3/(a^3*d) + (7*Csc[c + d*x]^5)/(5*a
^3*d) - (4*Csc[c + d*x]^7)/(7*a^3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.366936, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3872, 2875, 2873, 2607, 30, 2606, 270, 14} \[ \frac{4 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^7(c+d x)}{7 a^3 d}+\frac{7 \csc ^5(c+d x)}{5 a^3 d}-\frac{\csc ^3(c+d x)}{a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Cot[c + d*x]^5)/(5*a^3*d) + (4*Cot[c + d*x]^7)/(7*a^3*d) - Csc[c + d*x]^3/(a^3*d) + (7*Csc[c + d*x]^5)/(5*a
^3*d) - (4*Csc[c + d*x]^7)/(7*a^3*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2875

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

Rule 2873

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

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.563895, size = 137, normalized size = 1.54 \[ \frac{\csc (c) (602 \sin (c+d x)+602 \sin (2 (c+d x))+258 \sin (3 (c+d x))+43 \sin (4 (c+d x))-560 \sin (2 c+d x)+168 \sin (c+2 d x)-280 \sin (3 c+2 d x)-48 \sin (2 c+3 d x)-8 \sin (3 c+4 d x)-840 \sin (c)+448 \sin (d x)) \csc (c+d x) \sec ^3(c+d x)}{2240 a^3 d (\sec (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[c]*Csc[c + d*x]*Sec[c + d*x]^3*(-840*Sin[c] + 448*Sin[d*x] + 602*Sin[c + d*x] + 602*Sin[2*(c + d*x)] + 25
8*Sin[3*(c + d*x)] + 43*Sin[4*(c + d*x)] - 560*Sin[2*c + d*x] + 168*Sin[c + 2*d*x] - 280*Sin[3*c + 2*d*x] - 48
*Sin[2*c + 3*d*x] - 8*Sin[3*c + 4*d*x]))/(2240*a^3*d*(1 + Sec[c + d*x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.066, size = 60, normalized size = 0.7 \begin{align*}{\frac{1}{16\,d{a}^{3}} \left ( -{\frac{1}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{2}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-2\,\tan \left ( 1/2\,dx+c/2 \right ) - \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^2/(a+a*sec(d*x+c))^3,x)

[Out]

1/16/d/a^3*(-1/7*tan(1/2*d*x+1/2*c)^7+2/5*tan(1/2*d*x+1/2*c)^5-2*tan(1/2*d*x+1/2*c)-1/tan(1/2*d*x+1/2*c))

________________________________________________________________________________________

Maxima [A]  time = 1.03617, size = 122, normalized size = 1.37 \begin{align*} -\frac{\frac{\frac{70 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{14 \, \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 )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3}} + \frac{35 \,{\left (\cos \left (d x + c\right ) + 1\right )}}{a^{3} \sin \left (d x + c\right )}}{560 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/560*((70*sin(d*x + c)/(cos(d*x + c) + 1) - 14*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d*x + c)^7/(cos(d
*x + c) + 1)^7)/a^3 + 35*(cos(d*x + c) + 1)/(a^3*sin(d*x + c)))/d

________________________________________________________________________________________

Fricas [A]  time = 1.83632, size = 240, normalized size = 2.7 \begin{align*} \frac{\cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )^{2} - 18 \, \cos \left (d x + c\right ) - 6}{35 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/35*(cos(d*x + c)^4 + 3*cos(d*x + c)^3 - 15*cos(d*x + c)^2 - 18*cos(d*x + c) - 6)/((a^3*d*cos(d*x + c)^3 + 3*
a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**2/(a+a*sec(d*x+c))**3,x)

[Out]

Integral(csc(c + d*x)**2/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3

________________________________________________________________________________________

Giac [A]  time = 1.31231, size = 99, normalized size = 1.11 \begin{align*} -\frac{\frac{35}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} + \frac{5 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 14 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 70 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{21}}}{560 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/560*(35/(a^3*tan(1/2*d*x + 1/2*c)) + (5*a^18*tan(1/2*d*x + 1/2*c)^7 - 14*a^18*tan(1/2*d*x + 1/2*c)^5 + 70*a
^18*tan(1/2*d*x + 1/2*c))/a^21)/d