3.72 \(\int \csc ^3(a+b x) \csc ^4(2 a+2 b x) \, dx\)
Optimal. Leaf size=112 \[ \frac {35 \sec ^3(a+b x)}{256 b}+\frac {105 \sec (a+b x)}{256 b}-\frac {105 \tanh ^{-1}(\cos (a+b x))}{256 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b} \]
[Out]
-105/256*arctanh(cos(b*x+a))/b+105/256*sec(b*x+a)/b+35/256*sec(b*x+a)^3/b-21/256*csc(b*x+a)^2*sec(b*x+a)^3/b-3
/128*csc(b*x+a)^4*sec(b*x+a)^3/b-1/96*csc(b*x+a)^6*sec(b*x+a)^3/b
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Rubi [A] time = 0.09, antiderivative size = 112, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used =
{4288, 2622, 288, 302, 207} \[ \frac {35 \sec ^3(a+b x)}{256 b}+\frac {105 \sec (a+b x)}{256 b}-\frac {105 \tanh ^{-1}(\cos (a+b x))}{256 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*x]^3*Csc[2*a + 2*b*x]^4,x]
[Out]
(-105*ArcTanh[Cos[a + b*x]])/(256*b) + (105*Sec[a + b*x])/(256*b) + (35*Sec[a + b*x]^3)/(256*b) - (21*Csc[a +
b*x]^2*Sec[a + b*x]^3)/(256*b) - (3*Csc[a + b*x]^4*Sec[a + b*x]^3)/(128*b) - (Csc[a + b*x]^6*Sec[a + b*x]^3)/(
96*b)
Rule 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 288
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]
Rule 302
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]
Rule 2622
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Rule 4288
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 ^3(a+b x) \csc ^4(2 a+2 b x) \, dx &=\frac {1}{16} \int \csc ^7(a+b x) \sec ^4(a+b x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^{10}}{\left (-1+x^2\right )^4} \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}+\frac {3 \operatorname {Subst}\left (\int \frac {x^8}{\left (-1+x^2\right )^3} \, dx,x,\sec (a+b x)\right )}{32 b}\\ &=-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}+\frac {21 \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (a+b x)\right )}{128 b}\\ &=-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}+\frac {105 \operatorname {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{256 b}\\ &=-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}+\frac {105 \operatorname {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{256 b}\\ &=\frac {105 \sec (a+b x)}{256 b}+\frac {35 \sec ^3(a+b x)}{256 b}-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}+\frac {105 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{256 b}\\ &=-\frac {105 \tanh ^{-1}(\cos (a+b x))}{256 b}+\frac {105 \sec (a+b x)}{256 b}+\frac {35 \sec ^3(a+b x)}{256 b}-\frac {21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}\\ \end {align*}
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Mathematica [B] time = 0.85, size = 278, normalized size = 2.48 \[ \frac {\csc ^{12}(a+b x) \left (-4752 \cos (2 (a+b x))+1600 \cos (3 (a+b x))+504 \cos (4 (a+b x))+1680 \cos (6 (a+b x))-600 \cos (7 (a+b x))-630 \cos (8 (a+b x))+200 \cos (9 (a+b x))+2520 \cos (3 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-945 \cos (7 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+315 \cos (9 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-30 \cos (a+b x) \left (-63 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )+63 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+40\right )-2520 \cos (3 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )+945 \cos (7 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )-315 \cos (9 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )+1150\right )}{3072 b \left (\csc ^2\left (\frac {1}{2} (a+b x)\right )-\sec ^2\left (\frac {1}{2} (a+b x)\right )\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[a + b*x]^3*Csc[2*a + 2*b*x]^4,x]
[Out]
(Csc[a + b*x]^12*(1150 - 4752*Cos[2*(a + b*x)] + 1600*Cos[3*(a + b*x)] + 504*Cos[4*(a + b*x)] + 1680*Cos[6*(a
+ b*x)] - 600*Cos[7*(a + b*x)] - 630*Cos[8*(a + b*x)] + 200*Cos[9*(a + b*x)] + 2520*Cos[3*(a + b*x)]*Log[Cos[(
a + b*x)/2]] - 945*Cos[7*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 315*Cos[9*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 30*Co
s[a + b*x]*(40 + 63*Log[Cos[(a + b*x)/2]] - 63*Log[Sin[(a + b*x)/2]]) - 2520*Cos[3*(a + b*x)]*Log[Sin[(a + b*x
)/2]] + 945*Cos[7*(a + b*x)]*Log[Sin[(a + b*x)/2]] - 315*Cos[9*(a + b*x)]*Log[Sin[(a + b*x)/2]]))/(3072*b*(Csc
[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2)^3)
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fricas [A] time = 0.55, size = 194, normalized size = 1.73 \[ \frac {630 \, \cos \left (b x + a\right )^{8} - 1680 \, \cos \left (b x + a\right )^{6} + 1386 \, \cos \left (b x + a\right )^{4} - 288 \, \cos \left (b x + a\right )^{2} - 315 \, {\left (\cos \left (b x + a\right )^{9} - 3 \, \cos \left (b x + a\right )^{7} + 3 \, \cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 315 \, {\left (\cos \left (b x + a\right )^{9} - 3 \, \cos \left (b x + a\right )^{7} + 3 \, \cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 32}{1536 \, {\left (b \cos \left (b x + a\right )^{9} - 3 \, b \cos \left (b x + a\right )^{7} + 3 \, b \cos \left (b x + a\right )^{5} - b \cos \left (b x + a\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^3*csc(2*b*x+2*a)^4,x, algorithm="fricas")
[Out]
1/1536*(630*cos(b*x + a)^8 - 1680*cos(b*x + a)^6 + 1386*cos(b*x + a)^4 - 288*cos(b*x + a)^2 - 315*(cos(b*x + a
)^9 - 3*cos(b*x + a)^7 + 3*cos(b*x + a)^5 - cos(b*x + a)^3)*log(1/2*cos(b*x + a) + 1/2) + 315*(cos(b*x + a)^9
- 3*cos(b*x + a)^7 + 3*cos(b*x + a)^5 - cos(b*x + a)^3)*log(-1/2*cos(b*x + a) + 1/2) - 32)/(b*cos(b*x + a)^9 -
3*b*cos(b*x + a)^7 + 3*b*cos(b*x + a)^5 - b*cos(b*x + a)^3)
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^3*csc(2*b*x+2*a)^4,x, algorithm="giac")
[Out]
Timed out
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maple [A] time = 0.85, size = 120, normalized size = 1.07 \[ -\frac {1}{96 b \sin \left (b x +a \right )^{6} \cos \left (b x +a \right )^{3}}-\frac {3}{128 b \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )^{3}}+\frac {7}{128 b \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )^{3}}-\frac {35}{256 b \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )}+\frac {105}{256 b \cos \left (b x +a \right )}+\frac {105 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{256 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)^3*csc(2*b*x+2*a)^4,x)
[Out]
-1/96/b/sin(b*x+a)^6/cos(b*x+a)^3-3/128/b/sin(b*x+a)^4/cos(b*x+a)^3+7/128/b/sin(b*x+a)^2/cos(b*x+a)^3-35/256/b
/sin(b*x+a)^2/cos(b*x+a)+105/256/b/cos(b*x+a)+105/256/b*ln(csc(b*x+a)-cot(b*x+a))
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maxima [B] time = 1.57, size = 4268, normalized size = 38.11 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^3*csc(2*b*x+2*a)^4,x, algorithm="maxima")
[Out]
1/1536*(4*(315*cos(17*b*x + 17*a) - 840*cos(15*b*x + 15*a) - 252*cos(13*b*x + 13*a) + 2376*cos(11*b*x + 11*a)
- 1150*cos(9*b*x + 9*a) + 2376*cos(7*b*x + 7*a) - 252*cos(5*b*x + 5*a) - 840*cos(3*b*x + 3*a) + 315*cos(b*x +
a))*cos(18*b*x + 18*a) - 1260*(3*cos(16*b*x + 16*a) - 8*cos(12*b*x + 12*a) + 6*cos(10*b*x + 10*a) + 6*cos(8*b*
x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(2*b*x + 2*a) - 1)*cos(17*b*x + 17*a) + 12*(840*cos(15*b*x + 15*a) + 252*
cos(13*b*x + 13*a) - 2376*cos(11*b*x + 11*a) + 1150*cos(9*b*x + 9*a) - 2376*cos(7*b*x + 7*a) + 252*cos(5*b*x +
5*a) + 840*cos(3*b*x + 3*a) - 315*cos(b*x + a))*cos(16*b*x + 16*a) - 3360*(8*cos(12*b*x + 12*a) - 6*cos(10*b*
x + 10*a) - 6*cos(8*b*x + 8*a) + 8*cos(6*b*x + 6*a) - 3*cos(2*b*x + 2*a) + 1)*cos(15*b*x + 15*a) - 1008*(8*cos
(12*b*x + 12*a) - 6*cos(10*b*x + 10*a) - 6*cos(8*b*x + 8*a) + 8*cos(6*b*x + 6*a) - 3*cos(2*b*x + 2*a) + 1)*cos
(13*b*x + 13*a) + 32*(2376*cos(11*b*x + 11*a) - 1150*cos(9*b*x + 9*a) + 2376*cos(7*b*x + 7*a) - 252*cos(5*b*x
+ 5*a) - 840*cos(3*b*x + 3*a) + 315*cos(b*x + a))*cos(12*b*x + 12*a) - 9504*(6*cos(10*b*x + 10*a) + 6*cos(8*b*
x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(2*b*x + 2*a) - 1)*cos(11*b*x + 11*a) + 24*(1150*cos(9*b*x + 9*a) - 2376*
cos(7*b*x + 7*a) + 252*cos(5*b*x + 5*a) + 840*cos(3*b*x + 3*a) - 315*cos(b*x + a))*cos(10*b*x + 10*a) + 4600*(
6*cos(8*b*x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(2*b*x + 2*a) - 1)*cos(9*b*x + 9*a) - 72*(792*cos(7*b*x + 7*a)
- 84*cos(5*b*x + 5*a) - 280*cos(3*b*x + 3*a) + 105*cos(b*x + a))*cos(8*b*x + 8*a) + 9504*(8*cos(6*b*x + 6*a) -
3*cos(2*b*x + 2*a) + 1)*cos(7*b*x + 7*a) - 672*(12*cos(5*b*x + 5*a) + 40*cos(3*b*x + 3*a) - 15*cos(b*x + a))*
cos(6*b*x + 6*a) + 1008*(3*cos(2*b*x + 2*a) - 1)*cos(5*b*x + 5*a) + 3360*(3*cos(2*b*x + 2*a) - 1)*cos(3*b*x +
3*a) - 3780*cos(2*b*x + 2*a)*cos(b*x + a) + 315*(2*(3*cos(16*b*x + 16*a) - 8*cos(12*b*x + 12*a) + 6*cos(10*b*x
+ 10*a) + 6*cos(8*b*x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(2*b*x + 2*a) - 1)*cos(18*b*x + 18*a) - cos(18*b*x +
18*a)^2 + 6*(8*cos(12*b*x + 12*a) - 6*cos(10*b*x + 10*a) - 6*cos(8*b*x + 8*a) + 8*cos(6*b*x + 6*a) - 3*cos(2*
b*x + 2*a) + 1)*cos(16*b*x + 16*a) - 9*cos(16*b*x + 16*a)^2 + 16*(6*cos(10*b*x + 10*a) + 6*cos(8*b*x + 8*a) -
8*cos(6*b*x + 6*a) + 3*cos(2*b*x + 2*a) - 1)*cos(12*b*x + 12*a) - 64*cos(12*b*x + 12*a)^2 - 12*(6*cos(8*b*x +
8*a) - 8*cos(6*b*x + 6*a) + 3*cos(2*b*x + 2*a) - 1)*cos(10*b*x + 10*a) - 36*cos(10*b*x + 10*a)^2 + 12*(8*cos(6
*b*x + 6*a) - 3*cos(2*b*x + 2*a) + 1)*cos(8*b*x + 8*a) - 36*cos(8*b*x + 8*a)^2 + 16*(3*cos(2*b*x + 2*a) - 1)*c
os(6*b*x + 6*a) - 64*cos(6*b*x + 6*a)^2 - 9*cos(2*b*x + 2*a)^2 + 2*(3*sin(16*b*x + 16*a) - 8*sin(12*b*x + 12*a
) + 6*sin(10*b*x + 10*a) + 6*sin(8*b*x + 8*a) - 8*sin(6*b*x + 6*a) + 3*sin(2*b*x + 2*a))*sin(18*b*x + 18*a) -
sin(18*b*x + 18*a)^2 + 6*(8*sin(12*b*x + 12*a) - 6*sin(10*b*x + 10*a) - 6*sin(8*b*x + 8*a) + 8*sin(6*b*x + 6*a
) - 3*sin(2*b*x + 2*a))*sin(16*b*x + 16*a) - 9*sin(16*b*x + 16*a)^2 + 16*(6*sin(10*b*x + 10*a) + 6*sin(8*b*x +
8*a) - 8*sin(6*b*x + 6*a) + 3*sin(2*b*x + 2*a))*sin(12*b*x + 12*a) - 64*sin(12*b*x + 12*a)^2 - 12*(6*sin(8*b*
x + 8*a) - 8*sin(6*b*x + 6*a) + 3*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - 36*sin(10*b*x + 10*a)^2 + 12*(8*sin(6
*b*x + 6*a) - 3*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - 36*sin(8*b*x + 8*a)^2 - 64*sin(6*b*x + 6*a)^2 + 48*sin(6*
b*x + 6*a)*sin(2*b*x + 2*a) - 9*sin(2*b*x + 2*a)^2 + 6*cos(2*b*x + 2*a) - 1)*log(cos(b*x)^2 + 2*cos(b*x)*cos(a
) + cos(a)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(a) + sin(a)^2) - 315*(2*(3*cos(16*b*x + 16*a) - 8*cos(12*b*x + 12*a
) + 6*cos(10*b*x + 10*a) + 6*cos(8*b*x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(2*b*x + 2*a) - 1)*cos(18*b*x + 18*a
) - cos(18*b*x + 18*a)^2 + 6*(8*cos(12*b*x + 12*a) - 6*cos(10*b*x + 10*a) - 6*cos(8*b*x + 8*a) + 8*cos(6*b*x +
6*a) - 3*cos(2*b*x + 2*a) + 1)*cos(16*b*x + 16*a) - 9*cos(16*b*x + 16*a)^2 + 16*(6*cos(10*b*x + 10*a) + 6*cos
(8*b*x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(2*b*x + 2*a) - 1)*cos(12*b*x + 12*a) - 64*cos(12*b*x + 12*a)^2 - 12
*(6*cos(8*b*x + 8*a) - 8*cos(6*b*x + 6*a) + 3*cos(2*b*x + 2*a) - 1)*cos(10*b*x + 10*a) - 36*cos(10*b*x + 10*a)
^2 + 12*(8*cos(6*b*x + 6*a) - 3*cos(2*b*x + 2*a) + 1)*cos(8*b*x + 8*a) - 36*cos(8*b*x + 8*a)^2 + 16*(3*cos(2*b
*x + 2*a) - 1)*cos(6*b*x + 6*a) - 64*cos(6*b*x + 6*a)^2 - 9*cos(2*b*x + 2*a)^2 + 2*(3*sin(16*b*x + 16*a) - 8*s
in(12*b*x + 12*a) + 6*sin(10*b*x + 10*a) + 6*sin(8*b*x + 8*a) - 8*sin(6*b*x + 6*a) + 3*sin(2*b*x + 2*a))*sin(1
8*b*x + 18*a) - sin(18*b*x + 18*a)^2 + 6*(8*sin(12*b*x + 12*a) - 6*sin(10*b*x + 10*a) - 6*sin(8*b*x + 8*a) + 8
*sin(6*b*x + 6*a) - 3*sin(2*b*x + 2*a))*sin(16*b*x + 16*a) - 9*sin(16*b*x + 16*a)^2 + 16*(6*sin(10*b*x + 10*a)
+ 6*sin(8*b*x + 8*a) - 8*sin(6*b*x + 6*a) + 3*sin(2*b*x + 2*a))*sin(12*b*x + 12*a) - 64*sin(12*b*x + 12*a)^2
- 12*(6*sin(8*b*x + 8*a) - 8*sin(6*b*x + 6*a) + 3*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - 36*sin(10*b*x + 10*a)
^2 + 12*(8*sin(6*b*x + 6*a) - 3*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - 36*sin(8*b*x + 8*a)^2 - 64*sin(6*b*x + 6*
a)^2 + 48*sin(6*b*x + 6*a)*sin(2*b*x + 2*a) - 9*sin(2*b*x + 2*a)^2 + 6*cos(2*b*x + 2*a) - 1)*log(cos(b*x)^2 -
2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + sin(a)^2) + 4*(315*sin(17*b*x + 17*a) - 840*si
n(15*b*x + 15*a) - 252*sin(13*b*x + 13*a) + 2376*sin(11*b*x + 11*a) - 1150*sin(9*b*x + 9*a) + 2376*sin(7*b*x +
7*a) - 252*sin(5*b*x + 5*a) - 840*sin(3*b*x + 3*a) + 315*sin(b*x + a))*sin(18*b*x + 18*a) - 1260*(3*sin(16*b*
x + 16*a) - 8*sin(12*b*x + 12*a) + 6*sin(10*b*x + 10*a) + 6*sin(8*b*x + 8*a) - 8*sin(6*b*x + 6*a) + 3*sin(2*b*
x + 2*a))*sin(17*b*x + 17*a) + 12*(840*sin(15*b*x + 15*a) + 252*sin(13*b*x + 13*a) - 2376*sin(11*b*x + 11*a) +
1150*sin(9*b*x + 9*a) - 2376*sin(7*b*x + 7*a) + 252*sin(5*b*x + 5*a) + 840*sin(3*b*x + 3*a) - 315*sin(b*x + a
))*sin(16*b*x + 16*a) - 3360*(8*sin(12*b*x + 12*a) - 6*sin(10*b*x + 10*a) - 6*sin(8*b*x + 8*a) + 8*sin(6*b*x +
6*a) - 3*sin(2*b*x + 2*a))*sin(15*b*x + 15*a) - 1008*(8*sin(12*b*x + 12*a) - 6*sin(10*b*x + 10*a) - 6*sin(8*b
*x + 8*a) + 8*sin(6*b*x + 6*a) - 3*sin(2*b*x + 2*a))*sin(13*b*x + 13*a) + 32*(2376*sin(11*b*x + 11*a) - 1150*s
in(9*b*x + 9*a) + 2376*sin(7*b*x + 7*a) - 252*sin(5*b*x + 5*a) - 840*sin(3*b*x + 3*a) + 315*sin(b*x + a))*sin(
12*b*x + 12*a) - 9504*(6*sin(10*b*x + 10*a) + 6*sin(8*b*x + 8*a) - 8*sin(6*b*x + 6*a) + 3*sin(2*b*x + 2*a))*si
n(11*b*x + 11*a) + 24*(1150*sin(9*b*x + 9*a) - 2376*sin(7*b*x + 7*a) + 252*sin(5*b*x + 5*a) + 840*sin(3*b*x +
3*a) - 315*sin(b*x + a))*sin(10*b*x + 10*a) + 4600*(6*sin(8*b*x + 8*a) - 8*sin(6*b*x + 6*a) + 3*sin(2*b*x + 2*
a))*sin(9*b*x + 9*a) - 72*(792*sin(7*b*x + 7*a) - 84*sin(5*b*x + 5*a) - 280*sin(3*b*x + 3*a) + 105*sin(b*x + a
))*sin(8*b*x + 8*a) + 9504*(8*sin(6*b*x + 6*a) - 3*sin(2*b*x + 2*a))*sin(7*b*x + 7*a) - 672*(12*sin(5*b*x + 5*
a) + 40*sin(3*b*x + 3*a) - 15*sin(b*x + a))*sin(6*b*x + 6*a) + 3024*sin(5*b*x + 5*a)*sin(2*b*x + 2*a) + 10080*
sin(3*b*x + 3*a)*sin(2*b*x + 2*a) - 3780*sin(2*b*x + 2*a)*sin(b*x + a) + 1260*cos(b*x + a))/(b*cos(18*b*x + 18
*a)^2 + 9*b*cos(16*b*x + 16*a)^2 + 64*b*cos(12*b*x + 12*a)^2 + 36*b*cos(10*b*x + 10*a)^2 + 36*b*cos(8*b*x + 8*
a)^2 + 64*b*cos(6*b*x + 6*a)^2 + 9*b*cos(2*b*x + 2*a)^2 + b*sin(18*b*x + 18*a)^2 + 9*b*sin(16*b*x + 16*a)^2 +
64*b*sin(12*b*x + 12*a)^2 + 36*b*sin(10*b*x + 10*a)^2 + 36*b*sin(8*b*x + 8*a)^2 + 64*b*sin(6*b*x + 6*a)^2 - 48
*b*sin(6*b*x + 6*a)*sin(2*b*x + 2*a) + 9*b*sin(2*b*x + 2*a)^2 - 2*(3*b*cos(16*b*x + 16*a) - 8*b*cos(12*b*x + 1
2*a) + 6*b*cos(10*b*x + 10*a) + 6*b*cos(8*b*x + 8*a) - 8*b*cos(6*b*x + 6*a) + 3*b*cos(2*b*x + 2*a) - b)*cos(18
*b*x + 18*a) - 6*(8*b*cos(12*b*x + 12*a) - 6*b*cos(10*b*x + 10*a) - 6*b*cos(8*b*x + 8*a) + 8*b*cos(6*b*x + 6*a
) - 3*b*cos(2*b*x + 2*a) + b)*cos(16*b*x + 16*a) - 16*(6*b*cos(10*b*x + 10*a) + 6*b*cos(8*b*x + 8*a) - 8*b*cos
(6*b*x + 6*a) + 3*b*cos(2*b*x + 2*a) - b)*cos(12*b*x + 12*a) + 12*(6*b*cos(8*b*x + 8*a) - 8*b*cos(6*b*x + 6*a)
+ 3*b*cos(2*b*x + 2*a) - b)*cos(10*b*x + 10*a) - 12*(8*b*cos(6*b*x + 6*a) - 3*b*cos(2*b*x + 2*a) + b)*cos(8*b
*x + 8*a) - 16*(3*b*cos(2*b*x + 2*a) - b)*cos(6*b*x + 6*a) - 6*b*cos(2*b*x + 2*a) - 2*(3*b*sin(16*b*x + 16*a)
- 8*b*sin(12*b*x + 12*a) + 6*b*sin(10*b*x + 10*a) + 6*b*sin(8*b*x + 8*a) - 8*b*sin(6*b*x + 6*a) + 3*b*sin(2*b*
x + 2*a))*sin(18*b*x + 18*a) - 6*(8*b*sin(12*b*x + 12*a) - 6*b*sin(10*b*x + 10*a) - 6*b*sin(8*b*x + 8*a) + 8*b
*sin(6*b*x + 6*a) - 3*b*sin(2*b*x + 2*a))*sin(16*b*x + 16*a) - 16*(6*b*sin(10*b*x + 10*a) + 6*b*sin(8*b*x + 8*
a) - 8*b*sin(6*b*x + 6*a) + 3*b*sin(2*b*x + 2*a))*sin(12*b*x + 12*a) + 12*(6*b*sin(8*b*x + 8*a) - 8*b*sin(6*b*
x + 6*a) + 3*b*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - 12*(8*b*sin(6*b*x + 6*a) - 3*b*sin(2*b*x + 2*a))*sin(8*b
*x + 8*a) + b)
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mupad [B] time = 0.23, size = 100, normalized size = 0.89 \[ \frac {-\frac {105\,{\cos \left (a+b\,x\right )}^8}{256}+\frac {35\,{\cos \left (a+b\,x\right )}^6}{32}-\frac {231\,{\cos \left (a+b\,x\right )}^4}{256}+\frac {3\,{\cos \left (a+b\,x\right )}^2}{16}+\frac {1}{48}}{b\,\left (-{\cos \left (a+b\,x\right )}^9+3\,{\cos \left (a+b\,x\right )}^7-3\,{\cos \left (a+b\,x\right )}^5+{\cos \left (a+b\,x\right )}^3\right )}-\frac {105\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{256\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(sin(a + b*x)^3*sin(2*a + 2*b*x)^4),x)
[Out]
((3*cos(a + b*x)^2)/16 - (231*cos(a + b*x)^4)/256 + (35*cos(a + b*x)^6)/32 - (105*cos(a + b*x)^8)/256 + 1/48)/
(b*(cos(a + b*x)^3 - 3*cos(a + b*x)^5 + 3*cos(a + b*x)^7 - cos(a + b*x)^9)) - (105*atanh(cos(a + b*x)))/(256*b
)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^{3}{\left (a + b x \right )} \csc ^{4}{\left (2 a + 2 b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)**3*csc(2*b*x+2*a)**4,x)
[Out]
Integral(csc(a + b*x)**3*csc(2*a + 2*b*x)**4, x)
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