\(\int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx\) [58]

Optimal result

Integrand size = 20, antiderivative size = 102 \[ \int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx=-\frac {5 \cot (a+b x)}{16 b}-\frac {5 \cot ^3(a+b x)}{64 b}-\frac {3 \cot ^5(a+b x)}{160 b}-\frac {\cot ^7(a+b x)}{448 b}+\frac {15 \tan (a+b x)}{64 b}+\frac {\tan ^3(a+b x)}{32 b}+\frac {\tan ^5(a+b x)}{320 b} \]

[Out]

-5/16*cot(b*x+a)/b-5/64*cot(b*x+a)^3/b-3/160*cot(b*x+a)^5/b-1/448*cot(b*x+a)^7/b+15/64*tan(b*x+a)/b+1/32*tan(b
*x+a)^3/b+1/320*tan(b*x+a)^5/b

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4373, 2700, 276} \[ \int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx=\frac {\tan ^5(a+b x)}{320 b}+\frac {\tan ^3(a+b x)}{32 b}+\frac {15 \tan (a+b x)}{64 b}-\frac {\cot ^7(a+b x)}{448 b}-\frac {3 \cot ^5(a+b x)}{160 b}-\frac {5 \cot ^3(a+b x)}{64 b}-\frac {5 \cot (a+b x)}{16 b} \]

[In]

Int[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^6,x]

[Out]

(-5*Cot[a + b*x])/(16*b) - (5*Cot[a + b*x]^3)/(64*b) - (3*Cot[a + b*x]^5)/(160*b) - Cot[a + b*x]^7/(448*b) + (
15*Tan[a + b*x])/(64*b) + Tan[a + b*x]^3/(32*b) + Tan[a + b*x]^5/(320*b)

Rule 276

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 2700

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

Rule 4373

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{64} \int \csc ^8(a+b x) \sec ^6(a+b x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^6}{x^8} \, dx,x,\tan (a+b x)\right )}{64 b} \\ & = \frac {\text {Subst}\left (\int \left (15+\frac {1}{x^8}+\frac {6}{x^6}+\frac {15}{x^4}+\frac {20}{x^2}+6 x^2+x^4\right ) \, dx,x,\tan (a+b x)\right )}{64 b} \\ & = -\frac {5 \cot (a+b x)}{16 b}-\frac {5 \cot ^3(a+b x)}{64 b}-\frac {3 \cot ^5(a+b x)}{160 b}-\frac {\cot ^7(a+b x)}{448 b}+\frac {15 \tan (a+b x)}{64 b}+\frac {\tan ^3(a+b x)}{32 b}+\frac {\tan ^5(a+b x)}{320 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.29 \[ \int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx=-\frac {281 \cot (a+b x)}{1120 b}-\frac {53 \cot (a+b x) \csc ^2(a+b x)}{1120 b}-\frac {27 \cot (a+b x) \csc ^4(a+b x)}{2240 b}-\frac {\cot (a+b x) \csc ^6(a+b x)}{448 b}+\frac {33 \tan (a+b x)}{160 b}+\frac {\sec ^2(a+b x) \tan (a+b x)}{40 b}+\frac {\sec ^4(a+b x) \tan (a+b x)}{320 b} \]

[In]

Integrate[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^6,x]

[Out]

(-281*Cot[a + b*x])/(1120*b) - (53*Cot[a + b*x]*Csc[a + b*x]^2)/(1120*b) - (27*Cot[a + b*x]*Csc[a + b*x]^4)/(2
240*b) - (Cot[a + b*x]*Csc[a + b*x]^6)/(448*b) + (33*Tan[a + b*x])/(160*b) + (Sec[a + b*x]^2*Tan[a + b*x])/(40
*b) + (Sec[a + b*x]^4*Tan[a + b*x])/(320*b)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.80 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {32 i \left (20 \,{\mathrm e}^{10 i \left (x b +a \right )}-5 \,{\mathrm e}^{8 i \left (x b +a \right )}-10 \,{\mathrm e}^{6 i \left (x b +a \right )}+4 \,{\mathrm e}^{4 i \left (x b +a \right )}+2 \,{\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{35 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{7} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{5}}\)	\(90\)
default	\(\frac {-\frac {1}{7 \sin \left (x b +a \right )^{7} \cos \left (x b +a \right )^{5}}+\frac {12}{35 \sin \left (x b +a \right )^{5} \cos \left (x b +a \right )^{5}}-\frac {24}{35 \sin \left (x b +a \right )^{5} \cos \left (x b +a \right )^{3}}+\frac {64}{35 \cos \left (x b +a \right )^{3} \sin \left (x b +a \right )^{3}}-\frac {128}{35 \cos \left (x b +a \right ) \sin \left (x b +a \right )^{3}}+\frac {512}{35 \sin \left (x b +a \right ) \cos \left (x b +a \right )}-\frac {1024 \cot \left (x b +a \right )}{35}}{64 b}\)	\(123\)

[In]

int(csc(b*x+a)^2*csc(2*b*x+2*a)^6,x,method=_RETURNVERBOSE)

[Out]

32/35*I*(20*exp(10*I*(b*x+a))-5*exp(8*I*(b*x+a))-10*exp(6*I*(b*x+a))+4*exp(4*I*(b*x+a))+2*exp(2*I*(b*x+a))-1)/
b/(exp(2*I*(b*x+a))-1)^7/(exp(2*I*(b*x+a))+1)^5

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.16 \[ \int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx=-\frac {1024 \, \cos \left (b x + a\right )^{12} - 3584 \, \cos \left (b x + a\right )^{10} + 4480 \, \cos \left (b x + a\right )^{8} - 2240 \, \cos \left (b x + a\right )^{6} + 280 \, \cos \left (b x + a\right )^{4} + 28 \, \cos \left (b x + a\right )^{2} + 7}{2240 \, {\left (b \cos \left (b x + a\right )^{11} - 3 \, b \cos \left (b x + a\right )^{9} + 3 \, b \cos \left (b x + a\right )^{7} - b \cos \left (b x + a\right )^{5}\right )} \sin \left (b x + a\right )} \]

[In]

integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^6,x, algorithm="fricas")

[Out]

-1/2240*(1024*cos(b*x + a)^12 - 3584*cos(b*x + a)^10 + 4480*cos(b*x + a)^8 - 2240*cos(b*x + a)^6 + 280*cos(b*x
+ a)^4 + 28*cos(b*x + a)^2 + 7)/((b*cos(b*x + a)^11 - 3*b*cos(b*x + a)^9 + 3*b*cos(b*x + a)^7 - b*cos(b*x + a
)^5)*sin(b*x + a))

Sympy [F]

\[ \int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx=\int \csc ^{2}{\left (a + b x \right )} \csc ^{6}{\left (2 a + 2 b x \right )}\, dx \]

[In]

integrate(csc(b*x+a)**2*csc(2*b*x+2*a)**6,x)

[Out]

Integral(csc(a + b*x)**2*csc(2*a + 2*b*x)**6, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2710 vs. \(2 (88) = 176\).

Time = 0.33 (sec) , antiderivative size = 2710, normalized size of antiderivative = 26.57 \[ \int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx=\text {Too large to display} \]

[In]

integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^6,x, algorithm="maxima")

[Out]

-32/35*((20*sin(10*b*x + 10*a) - 5*sin(8*b*x + 8*a) - 10*sin(6*b*x + 6*a) + 4*sin(4*b*x + 4*a) + 2*sin(2*b*x +
2*a))*cos(24*b*x + 24*a) - 2*(20*sin(10*b*x + 10*a) - 5*sin(8*b*x + 8*a) - 10*sin(6*b*x + 6*a) + 4*sin(4*b*x
+ 4*a) + 2*sin(2*b*x + 2*a))*cos(22*b*x + 22*a) - 4*(20*sin(10*b*x + 10*a) - 5*sin(8*b*x + 8*a) - 10*sin(6*b*x
+ 6*a) + 4*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*cos(20*b*x + 20*a) + 10*(20*sin(10*b*x + 10*a) - 5*sin(8*b*
x + 8*a) - 10*sin(6*b*x + 6*a) + 4*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*cos(18*b*x + 18*a) + 5*(20*sin(10*b*
x + 10*a) - 5*sin(8*b*x + 8*a) - 10*sin(6*b*x + 6*a) + 4*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*cos(16*b*x + 1
6*a) - 20*(20*sin(10*b*x + 10*a) - 5*sin(8*b*x + 8*a) - 10*sin(6*b*x + 6*a) + 4*sin(4*b*x + 4*a) + 2*sin(2*b*x
+ 2*a))*cos(14*b*x + 14*a) - (20*cos(10*b*x + 10*a) - 5*cos(8*b*x + 8*a) - 10*cos(6*b*x + 6*a) + 4*cos(4*b*x
+ 4*a) + 2*cos(2*b*x + 2*a) - 1)*sin(24*b*x + 24*a) + 2*(20*cos(10*b*x + 10*a) - 5*cos(8*b*x + 8*a) - 10*cos(6
*b*x + 6*a) + 4*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*sin(22*b*x + 22*a) + 4*(20*cos(10*b*x + 10*a) - 5*c
os(8*b*x + 8*a) - 10*cos(6*b*x + 6*a) + 4*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*sin(20*b*x + 20*a) - 10*(
20*cos(10*b*x + 10*a) - 5*cos(8*b*x + 8*a) - 10*cos(6*b*x + 6*a) + 4*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1
)*sin(18*b*x + 18*a) - 5*(20*cos(10*b*x + 10*a) - 5*cos(8*b*x + 8*a) - 10*cos(6*b*x + 6*a) + 4*cos(4*b*x + 4*a
) + 2*cos(2*b*x + 2*a) - 1)*sin(16*b*x + 16*a) + 20*(20*cos(10*b*x + 10*a) - 5*cos(8*b*x + 8*a) - 10*cos(6*b*x
+ 6*a) + 4*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*sin(14*b*x + 14*a))/(b*cos(24*b*x + 24*a)^2 + 4*b*cos(2
2*b*x + 22*a)^2 + 16*b*cos(20*b*x + 20*a)^2 + 100*b*cos(18*b*x + 18*a)^2 + 25*b*cos(16*b*x + 16*a)^2 + 400*b*c
os(14*b*x + 14*a)^2 + 400*b*cos(10*b*x + 10*a)^2 + 25*b*cos(8*b*x + 8*a)^2 + 100*b*cos(6*b*x + 6*a)^2 + 16*b*c
os(4*b*x + 4*a)^2 + 4*b*cos(2*b*x + 2*a)^2 + b*sin(24*b*x + 24*a)^2 + 4*b*sin(22*b*x + 22*a)^2 + 16*b*sin(20*b
*x + 20*a)^2 + 100*b*sin(18*b*x + 18*a)^2 + 25*b*sin(16*b*x + 16*a)^2 + 400*b*sin(14*b*x + 14*a)^2 + 400*b*sin
(10*b*x + 10*a)^2 + 25*b*sin(8*b*x + 8*a)^2 + 100*b*sin(6*b*x + 6*a)^2 + 16*b*sin(4*b*x + 4*a)^2 + 16*b*sin(4*
b*x + 4*a)*sin(2*b*x + 2*a) + 4*b*sin(2*b*x + 2*a)^2 - 2*(2*b*cos(22*b*x + 22*a) + 4*b*cos(20*b*x + 20*a) - 10
*b*cos(18*b*x + 18*a) - 5*b*cos(16*b*x + 16*a) + 20*b*cos(14*b*x + 14*a) - 20*b*cos(10*b*x + 10*a) + 5*b*cos(8
*b*x + 8*a) + 10*b*cos(6*b*x + 6*a) - 4*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(24*b*x + 24*a) + 4*
(4*b*cos(20*b*x + 20*a) - 10*b*cos(18*b*x + 18*a) - 5*b*cos(16*b*x + 16*a) + 20*b*cos(14*b*x + 14*a) - 20*b*co
s(10*b*x + 10*a) + 5*b*cos(8*b*x + 8*a) + 10*b*cos(6*b*x + 6*a) - 4*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a)
+ b)*cos(22*b*x + 22*a) - 8*(10*b*cos(18*b*x + 18*a) + 5*b*cos(16*b*x + 16*a) - 20*b*cos(14*b*x + 14*a) + 20*b
*cos(10*b*x + 10*a) - 5*b*cos(8*b*x + 8*a) - 10*b*cos(6*b*x + 6*a) + 4*b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*
a) - b)*cos(20*b*x + 20*a) + 20*(5*b*cos(16*b*x + 16*a) - 20*b*cos(14*b*x + 14*a) + 20*b*cos(10*b*x + 10*a) -
5*b*cos(8*b*x + 8*a) - 10*b*cos(6*b*x + 6*a) + 4*b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*a) - b)*cos(18*b*x + 1
8*a) - 10*(20*b*cos(14*b*x + 14*a) - 20*b*cos(10*b*x + 10*a) + 5*b*cos(8*b*x + 8*a) + 10*b*cos(6*b*x + 6*a) -
4*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(16*b*x + 16*a) - 40*(20*b*cos(10*b*x + 10*a) - 5*b*cos(8*
b*x + 8*a) - 10*b*cos(6*b*x + 6*a) + 4*b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*a) - b)*cos(14*b*x + 14*a) - 40*
(5*b*cos(8*b*x + 8*a) + 10*b*cos(6*b*x + 6*a) - 4*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(10*b*x +
10*a) + 10*(10*b*cos(6*b*x + 6*a) - 4*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(8*b*x + 8*a) - 20*(4*
b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*a) - b)*cos(6*b*x + 6*a) + 8*(2*b*cos(2*b*x + 2*a) - b)*cos(4*b*x + 4*a
) - 4*b*cos(2*b*x + 2*a) - 2*(2*b*sin(22*b*x + 22*a) + 4*b*sin(20*b*x + 20*a) - 10*b*sin(18*b*x + 18*a) - 5*b*
sin(16*b*x + 16*a) + 20*b*sin(14*b*x + 14*a) - 20*b*sin(10*b*x + 10*a) + 5*b*sin(8*b*x + 8*a) + 10*b*sin(6*b*x
+ 6*a) - 4*b*sin(4*b*x + 4*a) - 2*b*sin(2*b*x + 2*a))*sin(24*b*x + 24*a) + 4*(4*b*sin(20*b*x + 20*a) - 10*b*s
in(18*b*x + 18*a) - 5*b*sin(16*b*x + 16*a) + 20*b*sin(14*b*x + 14*a) - 20*b*sin(10*b*x + 10*a) + 5*b*sin(8*b*x
+ 8*a) + 10*b*sin(6*b*x + 6*a) - 4*b*sin(4*b*x + 4*a) - 2*b*sin(2*b*x + 2*a))*sin(22*b*x + 22*a) - 8*(10*b*si
n(18*b*x + 18*a) + 5*b*sin(16*b*x + 16*a) - 20*b*sin(14*b*x + 14*a) + 20*b*sin(10*b*x + 10*a) - 5*b*sin(8*b*x
+ 8*a) - 10*b*sin(6*b*x + 6*a) + 4*b*sin(4*b*x + 4*a) + 2*b*sin(2*b*x + 2*a))*sin(20*b*x + 20*a) + 20*(5*b*sin
(16*b*x + 16*a) - 20*b*sin(14*b*x + 14*a) + 20*b*sin(10*b*x + 10*a) - 5*b*sin(8*b*x + 8*a) - 10*b*sin(6*b*x +
6*a) + 4*b*sin(4*b*x + 4*a) + 2*b*sin(2*b*x + 2*a))*sin(18*b*x + 18*a) - 10*(20*b*sin(14*b*x + 14*a) - 20*b*si
n(10*b*x + 10*a) + 5*b*sin(8*b*x + 8*a) + 10*b*sin(6*b*x + 6*a) - 4*b*sin(4*b*x + 4*a) - 2*b*sin(2*b*x + 2*a))
*sin(16*b*x + 16*a) - 40*(20*b*sin(10*b*x + 10*a) - 5*b*sin(8*b*x + 8*a) - 10*b*sin(6*b*x + 6*a) + 4*b*sin(4*b
*x + 4*a) + 2*b*sin(2*b*x + 2*a))*sin(14*b*x + 14*a) - 40*(5*b*sin(8*b*x + 8*a) + 10*b*sin(6*b*x + 6*a) - 4*b*
sin(4*b*x + 4*a) - 2*b*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) + 20*(5*b*sin(6*b*x + 6*a) - 2*b*sin(4*b*x + 4*a)
- b*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - 40*(2*b*sin(4*b*x + 4*a) + b*sin(2*b*x + 2*a))*sin(6*b*x + 6*a) + b)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.75 \[ \int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx=\frac {7 \, \tan \left (b x + a\right )^{5} + 70 \, \tan \left (b x + a\right )^{3} - \frac {700 \, \tan \left (b x + a\right )^{6} + 175 \, \tan \left (b x + a\right )^{4} + 42 \, \tan \left (b x + a\right )^{2} + 5}{\tan \left (b x + a\right )^{7}} + 525 \, \tan \left (b x + a\right )}{2240 \, b} \]

[In]

integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^6,x, algorithm="giac")

[Out]

1/2240*(7*tan(b*x + a)^5 + 70*tan(b*x + a)^3 - (700*tan(b*x + a)^6 + 175*tan(b*x + a)^4 + 42*tan(b*x + a)^2 +
5)/tan(b*x + a)^7 + 525*tan(b*x + a))/b

Mupad [B] (verification not implemented)

Time = 19.76 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.81 \[ \int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx=\frac {15\,\mathrm {tan}\left (a+b\,x\right )}{64\,b}+\frac {{\mathrm {tan}\left (a+b\,x\right )}^3}{32\,b}+\frac {{\mathrm {tan}\left (a+b\,x\right )}^5}{320\,b}-\frac {{\mathrm {cot}\left (a+b\,x\right )}^7\,\left (\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^6}{16}+\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^4}{64}+\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^2}{160}+\frac {1}{448}\right )}{b} \]

[In]

int(1/(sin(a + b*x)^2*sin(2*a + 2*b*x)^6),x)

[Out]

(15*tan(a + b*x))/(64*b) + tan(a + b*x)^3/(32*b) + tan(a + b*x)^5/(320*b) - (cot(a + b*x)^7*((3*tan(a + b*x)^2
)/160 + (5*tan(a + b*x)^4)/64 + (5*tan(a + b*x)^6)/16 + 1/448))/b