3.1.57 \(\int \csc ^2(a+b x) \csc ^5(2 a+2 b x) \, dx\) [57]

Optimal. Leaf size=90 \[ -\frac {5 \cot ^2(a+b x)}{32 b}-\frac {5 \cot ^4(a+b x)}{128 b}-\frac {\cot ^6(a+b x)}{192 b}+\frac {5 \log (\tan (a+b x))}{16 b}+\frac {5 \tan ^2(a+b x)}{64 b}+\frac {\tan ^4(a+b x)}{128 b} \]

[Out]

-5/32*cot(b*x+a)^2/b-5/128*cot(b*x+a)^4/b-1/192*cot(b*x+a)^6/b+5/16*ln(tan(b*x+a))/b+5/64*tan(b*x+a)^2/b+1/128
*tan(b*x+a)^4/b

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Rubi [A]
time = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4373, 2700, 272, 45} \begin {gather*} \frac {\tan ^4(a+b x)}{128 b}+\frac {5 \tan ^2(a+b x)}{64 b}-\frac {\cot ^6(a+b x)}{192 b}-\frac {5 \cot ^4(a+b x)}{128 b}-\frac {5 \cot ^2(a+b x)}{32 b}+\frac {5 \log (\tan (a+b x))}{16 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^5,x]

[Out]

(-5*Cot[a + b*x]^2)/(32*b) - (5*Cot[a + b*x]^4)/(128*b) - Cot[a + b*x]^6/(192*b) + (5*Log[Tan[a + b*x]])/(16*b
) + (5*Tan[a + b*x]^2)/(64*b) + Tan[a + b*x]^4/(128*b)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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*} \int \csc ^2(a+b x) \csc ^5(2 a+2 b x) \, dx &=\frac {1}{32} \int \csc ^7(a+b x) \sec ^5(a+b x) \, dx\\ &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^5}{x^7} \, dx,x,\tan (a+b x)\right )}{32 b}\\ &=\frac {\text {Subst}\left (\int \frac {(1+x)^5}{x^4} \, dx,x,\tan ^2(a+b x)\right )}{64 b}\\ &=\frac {\text {Subst}\left (\int \left (5+\frac {1}{x^4}+\frac {5}{x^3}+\frac {10}{x^2}+\frac {10}{x}+x\right ) \, dx,x,\tan ^2(a+b x)\right )}{64 b}\\ &=-\frac {5 \cot ^2(a+b x)}{32 b}-\frac {5 \cot ^4(a+b x)}{128 b}-\frac {\cot ^6(a+b x)}{192 b}+\frac {5 \log (\tan (a+b x))}{16 b}+\frac {5 \tan ^2(a+b x)}{64 b}+\frac {\tan ^4(a+b x)}{128 b}\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 76, normalized size = 0.84 \begin {gather*} -\frac {36 \csc ^2(a+b x)+9 \csc ^4(a+b x)+2 \csc ^6(a+b x)+120 \log (\cos (a+b x))-120 \log (\sin (a+b x))-24 \sec ^2(a+b x)-3 \sec ^4(a+b x)}{384 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^5,x]

[Out]

-1/384*(36*Csc[a + b*x]^2 + 9*Csc[a + b*x]^4 + 2*Csc[a + b*x]^6 + 120*Log[Cos[a + b*x]] - 120*Log[Sin[a + b*x]
] - 24*Sec[a + b*x]^2 - 3*Sec[a + b*x]^4)/b

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Maple [A]
time = 0.12, size = 98, normalized size = 1.09

method	result	size
default	\(\frac {-\frac {1}{6 \sin \left (x b +a \right )^{6} \cos \left (x b +a \right )^{4}}+\frac {5}{12 \sin \left (x b +a \right )^{4} \cos \left (x b +a \right )^{4}}-\frac {5}{6 \sin \left (x b +a \right )^{4} \cos \left (x b +a \right )^{2}}+\frac {5}{2 \sin \left (x b +a \right )^{2} \cos \left (x b +a \right )^{2}}-\frac {5}{\sin \left (x b +a \right )^{2}}+10 \ln \left (\tan \left (x b +a \right )\right )}{32 b}\)	\(98\)
risch	\(\frac {15 \,{\mathrm e}^{18 i \left (x b +a \right )}-30 \,{\mathrm e}^{16 i \left (x b +a \right )}-40 \,{\mathrm e}^{14 i \left (x b +a \right )}+110 \,{\mathrm e}^{12 i \left (x b +a \right )}+18 \,{\mathrm e}^{10 i \left (x b +a \right )}+110 \,{\mathrm e}^{8 i \left (x b +a \right )}-40 \,{\mathrm e}^{6 i \left (x b +a \right )}-30 \,{\mathrm e}^{4 i \left (x b +a \right )}+15 \,{\mathrm e}^{2 i \left (x b +a \right )}}{24 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{6} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{4}}+\frac {5 \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{16 b}-\frac {5 \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{16 b}\)	\(167\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^2*csc(2*b*x+2*a)^5,x,method=_RETURNVERBOSE)

[Out]

1/32/b*(-1/6/sin(b*x+a)^6/cos(b*x+a)^4+5/12/sin(b*x+a)^4/cos(b*x+a)^4-5/6/sin(b*x+a)^4/cos(b*x+a)^2+5/2/sin(b*
x+a)^2/cos(b*x+a)^2-5/sin(b*x+a)^2+10*ln(tan(b*x+a)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 7650 vs. \(2 (78) = 156\).
time = 0.60, size = 7650, normalized size = 85.00 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^5,x, algorithm="maxima")

[Out]

1/96*(4*(15*cos(18*b*x + 18*a) - 30*cos(16*b*x + 16*a) - 40*cos(14*b*x + 14*a) + 110*cos(12*b*x + 12*a) + 18*c
os(10*b*x + 10*a) + 110*cos(8*b*x + 8*a) - 40*cos(6*b*x + 6*a) - 30*cos(4*b*x + 4*a) + 15*cos(2*b*x + 2*a))*co
s(20*b*x + 20*a) + 4*(15*cos(16*b*x + 16*a) + 200*cos(14*b*x + 14*a) - 190*cos(12*b*x + 12*a) - 216*cos(10*b*x
+ 10*a) - 190*cos(8*b*x + 8*a) + 200*cos(6*b*x + 6*a) + 15*cos(4*b*x + 4*a) - 60*cos(2*b*x + 2*a) + 15)*cos(1
8*b*x + 18*a) - 120*cos(18*b*x + 18*a)^2 - 12*(40*cos(14*b*x + 14*a) + 130*cos(12*b*x + 12*a) - 102*cos(10*b*x
+ 10*a) + 130*cos(8*b*x + 8*a) + 40*cos(6*b*x + 6*a) - 60*cos(4*b*x + 4*a) - 5*cos(2*b*x + 2*a) + 10)*cos(16*
b*x + 16*a) + 360*cos(16*b*x + 16*a)^2 + 32*(100*cos(12*b*x + 12*a) + 78*cos(10*b*x + 10*a) + 100*cos(8*b*x +
8*a) - 80*cos(6*b*x + 6*a) - 15*cos(4*b*x + 4*a) + 25*cos(2*b*x + 2*a) - 5)*cos(14*b*x + 14*a) - 1280*cos(14*b
*x + 14*a)^2 - 8*(642*cos(10*b*x + 10*a) - 220*cos(8*b*x + 8*a) - 400*cos(6*b*x + 6*a) + 195*cos(4*b*x + 4*a)
+ 95*cos(2*b*x + 2*a) - 55)*cos(12*b*x + 12*a) + 880*cos(12*b*x + 12*a)^2 - 24*(214*cos(8*b*x + 8*a) - 104*cos
(6*b*x + 6*a) - 51*cos(4*b*x + 4*a) + 36*cos(2*b*x + 2*a) - 3)*cos(10*b*x + 10*a) - 864*cos(10*b*x + 10*a)^2 +
40*(80*cos(6*b*x + 6*a) - 39*cos(4*b*x + 4*a) - 19*cos(2*b*x + 2*a) + 11)*cos(8*b*x + 8*a) + 880*cos(8*b*x +
8*a)^2 - 160*(3*cos(4*b*x + 4*a) - 5*cos(2*b*x + 2*a) + 1)*cos(6*b*x + 6*a) - 1280*cos(6*b*x + 6*a)^2 + 60*(co
s(2*b*x + 2*a) - 2)*cos(4*b*x + 4*a) + 360*cos(4*b*x + 4*a)^2 - 120*cos(2*b*x + 2*a)^2 + 15*(2*(2*cos(18*b*x +
18*a) + 3*cos(16*b*x + 16*a) - 8*cos(14*b*x + 14*a) - 2*cos(12*b*x + 12*a) + 12*cos(10*b*x + 10*a) - 2*cos(8*
b*x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*cos(20*b*x + 20*a) - cos(20*b*x
+ 20*a)^2 - 4*(3*cos(16*b*x + 16*a) - 8*cos(14*b*x + 14*a) - 2*cos(12*b*x + 12*a) + 12*cos(10*b*x + 10*a) - 2
*cos(8*b*x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*cos(18*b*x + 18*a) - 4*c
os(18*b*x + 18*a)^2 + 6*(8*cos(14*b*x + 14*a) + 2*cos(12*b*x + 12*a) - 12*cos(10*b*x + 10*a) + 2*cos(8*b*x + 8
*a) + 8*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - 2*cos(2*b*x + 2*a) + 1)*cos(16*b*x + 16*a) - 9*cos(16*b*x + 16
*a)^2 - 16*(2*cos(12*b*x + 12*a) - 12*cos(10*b*x + 10*a) + 2*cos(8*b*x + 8*a) + 8*cos(6*b*x + 6*a) - 3*cos(4*b
*x + 4*a) - 2*cos(2*b*x + 2*a) + 1)*cos(14*b*x + 14*a) - 64*cos(14*b*x + 14*a)^2 + 4*(12*cos(10*b*x + 10*a) -
2*cos(8*b*x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*cos(12*b*x + 12*a) - 4*
cos(12*b*x + 12*a)^2 + 24*(2*cos(8*b*x + 8*a) + 8*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - 2*cos(2*b*x + 2*a) +
1)*cos(10*b*x + 10*a) - 144*cos(10*b*x + 10*a)^2 - 4*(8*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - 2*cos(2*b*x +
2*a) + 1)*cos(8*b*x + 8*a) - 4*cos(8*b*x + 8*a)^2 + 16*(3*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*cos(6*b*
x + 6*a) - 64*cos(6*b*x + 6*a)^2 - 6*(2*cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - 9*cos(4*b*x + 4*a)^2 - 4*cos(
2*b*x + 2*a)^2 + 2*(2*sin(18*b*x + 18*a) + 3*sin(16*b*x + 16*a) - 8*sin(14*b*x + 14*a) - 2*sin(12*b*x + 12*a)
+ 12*sin(10*b*x + 10*a) - 2*sin(8*b*x + 8*a) - 8*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*s
in(20*b*x + 20*a) - sin(20*b*x + 20*a)^2 - 4*(3*sin(16*b*x + 16*a) - 8*sin(14*b*x + 14*a) - 2*sin(12*b*x + 12*
a) + 12*sin(10*b*x + 10*a) - 2*sin(8*b*x + 8*a) - 8*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a)
)*sin(18*b*x + 18*a) - 4*sin(18*b*x + 18*a)^2 + 6*(8*sin(14*b*x + 14*a) + 2*sin(12*b*x + 12*a) - 12*sin(10*b*x
+ 10*a) + 2*sin(8*b*x + 8*a) + 8*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - 2*sin(2*b*x + 2*a))*sin(16*b*x + 16*
a) - 9*sin(16*b*x + 16*a)^2 - 16*(2*sin(12*b*x + 12*a) - 12*sin(10*b*x + 10*a) + 2*sin(8*b*x + 8*a) + 8*sin(6*
b*x + 6*a) - 3*sin(4*b*x + 4*a) - 2*sin(2*b*x + 2*a))*sin(14*b*x + 14*a) - 64*sin(14*b*x + 14*a)^2 + 4*(12*sin
(10*b*x + 10*a) - 2*sin(8*b*x + 8*a) - 8*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*sin(12*b*
x + 12*a) - 4*sin(12*b*x + 12*a)^2 + 24*(2*sin(8*b*x + 8*a) + 8*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - 2*sin(
2*b*x + 2*a))*sin(10*b*x + 10*a) - 144*sin(10*b*x + 10*a)^2 - 4*(8*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - 2*s
in(2*b*x + 2*a))*sin(8*b*x + 8*a) - 4*sin(8*b*x + 8*a)^2 + 16*(3*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*sin(6*
b*x + 6*a) - 64*sin(6*b*x + 6*a)^2 - 9*sin(4*b*x + 4*a)^2 - 12*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - 4*sin(2*b*x
+ 2*a)^2 + 4*cos(2*b*x + 2*a) - 1)*log(cos(2*b*x)^2 + 2*cos(2*b*x)*cos(2*a) + cos(2*a)^2 + sin(2*b*x)^2 - 2*s
in(2*b*x)*sin(2*a) + sin(2*a)^2) - 15*(2*(2*cos(18*b*x + 18*a) + 3*cos(16*b*x + 16*a) - 8*cos(14*b*x + 14*a) -
2*cos(12*b*x + 12*a) + 12*cos(10*b*x + 10*a) - 2*cos(8*b*x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) +
2*cos(2*b*x + 2*a) - 1)*cos(20*b*x + 20*a) - cos(20*b*x + 20*a)^2 - 4*(3*cos(16*b*x + 16*a) - 8*cos(14*b*x +
14*a) - 2*cos(12*b*x + 12*a) + 12*cos(10*b*x + 10*a) - 2*cos(8*b*x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(4*b*x +
4*a) + 2*cos(2*b*x + 2*a) - 1)*cos(18*b*x + 18...

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (78) = 156\).
time = 3.05, size = 194, normalized size = 2.16 \begin {gather*} \frac {60 \, \cos \left (b x + a\right )^{8} - 150 \, \cos \left (b x + a\right )^{6} + 110 \, \cos \left (b x + a\right )^{4} - 15 \, \cos \left (b x + a\right )^{2} - 60 \, {\left (\cos \left (b x + a\right )^{10} - 3 \, \cos \left (b x + a\right )^{8} + 3 \, \cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) + 60 \, {\left (\cos \left (b x + a\right )^{10} - 3 \, \cos \left (b x + a\right )^{8} + 3 \, \cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - 3}{384 \, {\left (b \cos \left (b x + a\right )^{10} - 3 \, b \cos \left (b x + a\right )^{8} + 3 \, b \cos \left (b x + a\right )^{6} - b \cos \left (b x + a\right )^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^5,x, algorithm="fricas")

[Out]

1/384*(60*cos(b*x + a)^8 - 150*cos(b*x + a)^6 + 110*cos(b*x + a)^4 - 15*cos(b*x + a)^2 - 60*(cos(b*x + a)^10 -
3*cos(b*x + a)^8 + 3*cos(b*x + a)^6 - cos(b*x + a)^4)*log(cos(b*x + a)^2) + 60*(cos(b*x + a)^10 - 3*cos(b*x +
a)^8 + 3*cos(b*x + a)^6 - cos(b*x + a)^4)*log(-1/4*cos(b*x + a)^2 + 1/4) - 3)/(b*cos(b*x + a)^10 - 3*b*cos(b*
x + a)^8 + 3*b*cos(b*x + a)^6 - b*cos(b*x + a)^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \csc ^{2}{\left (a + b x \right )} \csc ^{5}{\left (2 a + 2 b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**2*csc(2*b*x+2*a)**5,x)

[Out]

Integral(csc(a + b*x)**2*csc(2*a + 2*b*x)**5, x)

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Giac [A]
time = 0.42, size = 94, normalized size = 1.04 \begin {gather*} -\frac {\frac {60 \, \sin \left (b x + a\right )^{8} - 90 \, \sin \left (b x + a\right )^{6} + 20 \, \sin \left (b x + a\right )^{4} + 5 \, \sin \left (b x + a\right )^{2} + 2}{{\left (\sin \left (b x + a\right )^{2} - 1\right )}^{2} \sin \left (b x + a\right )^{6}} + 60 \, \log \left (-\sin \left (b x + a\right )^{2} + 1\right ) - 120 \, \log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{384 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^5,x, algorithm="giac")

[Out]

-1/384*((60*sin(b*x + a)^8 - 90*sin(b*x + a)^6 + 20*sin(b*x + a)^4 + 5*sin(b*x + a)^2 + 2)/((sin(b*x + a)^2 -
1)^2*sin(b*x + a)^6) + 60*log(-sin(b*x + a)^2 + 1) - 120*log(abs(sin(b*x + a))))/b

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Mupad [B]
time = 0.29, size = 114, normalized size = 1.27 \begin {gather*} \frac {5\,\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{32\,b}-\frac {5\,\ln \left (\cos \left (a+b\,x\right )\right )}{16\,b}+\frac {-\frac {5\,{\cos \left (a+b\,x\right )}^8}{32}+\frac {25\,{\cos \left (a+b\,x\right )}^6}{64}-\frac {55\,{\cos \left (a+b\,x\right )}^4}{192}+\frac {5\,{\cos \left (a+b\,x\right )}^2}{128}+\frac {1}{128}}{b\,\left (-{\cos \left (a+b\,x\right )}^{10}+3\,{\cos \left (a+b\,x\right )}^8-3\,{\cos \left (a+b\,x\right )}^6+{\cos \left (a+b\,x\right )}^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(a + b*x)^2*sin(2*a + 2*b*x)^5),x)

[Out]

(5*log(sin(a + b*x)^2))/(32*b) - (5*log(cos(a + b*x)))/(16*b) + ((5*cos(a + b*x)^2)/128 - (55*cos(a + b*x)^4)/
192 + (25*cos(a + b*x)^6)/64 - (5*cos(a + b*x)^8)/32 + 1/128)/(b*(cos(a + b*x)^4 - 3*cos(a + b*x)^6 + 3*cos(a
+ b*x)^8 - cos(a + b*x)^10))

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