Integrand size = 18, antiderivative size = 83 \[ \int \csc (a+b x) \csc ^4(2 a+2 b x) \, dx=-\frac {35 \text {arctanh}(\cos (a+b x))}{128 b}-\frac {13 \cot (a+b x) \csc (a+b x)}{128 b}-\frac {\cot ^3(a+b x) \csc (a+b x)}{64 b}+\frac {3 \sec (a+b x)}{16 b}+\frac {\sec ^3(a+b x)}{48 b} \] Output:
-35/128*arctanh(cos(b*x+a))/b-13/128*cot(b*x+a)*csc(b*x+a)/b-1/64*cot(b*x+ a)^3*csc(b*x+a)/b+3/16*sec(b*x+a)/b+1/48*sec(b*x+a)^3/b
Leaf count is larger than twice the leaf count of optimal. \(268\) vs. \(2(83)=166\).
Time = 0.45 (sec) , antiderivative size = 268, normalized size of antiderivative = 3.23 \[ \int \csc (a+b x) \csc ^4(2 a+2 b x) \, dx=-\frac {\csc ^{10}(a+b x) \left (-204+658 \cos (2 (a+b x))-228 \cos (3 (a+b x))+140 \cos (4 (a+b x))-76 \cos (5 (a+b x))-210 \cos (6 (a+b x))+76 \cos (7 (a+b x))-315 \cos (3 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-105 \cos (5 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+105 \cos (7 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+3 \cos (a+b x) \left (76+105 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-105 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )+315 \cos (3 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )+105 \cos (5 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )-105 \cos (7 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )}{384 b \left (\csc ^2\left (\frac {1}{2} (a+b x)\right )-\sec ^2\left (\frac {1}{2} (a+b x)\right )\right )^3} \] Input:
Integrate[Csc[a + b*x]*Csc[2*a + 2*b*x]^4,x]
Output:
-1/384*(Csc[a + b*x]^10*(-204 + 658*Cos[2*(a + b*x)] - 228*Cos[3*(a + b*x) ] + 140*Cos[4*(a + b*x)] - 76*Cos[5*(a + b*x)] - 210*Cos[6*(a + b*x)] + 76 *Cos[7*(a + b*x)] - 315*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 105*Cos[5 *(a + b*x)]*Log[Cos[(a + b*x)/2]] + 105*Cos[7*(a + b*x)]*Log[Cos[(a + b*x) /2]] + 3*Cos[a + b*x]*(76 + 105*Log[Cos[(a + b*x)/2]] - 105*Log[Sin[(a + b *x)/2]]) + 315*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] + 105*Cos[5*(a + b*x )]*Log[Sin[(a + b*x)/2]] - 105*Cos[7*(a + b*x)]*Log[Sin[(a + b*x)/2]]))/(b *(Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2)^3)
Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4776, 3042, 3102, 25, 252, 252, 254, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \csc (a+b x) \csc ^4(2 a+2 b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (a+b x) \sin (2 a+2 b x)^4}dx\) |
\(\Big \downarrow \) 4776 |
\(\displaystyle \frac {1}{16} \int \csc ^5(a+b x) \sec ^4(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{16} \int \csc (a+b x)^5 \sec (a+b x)^4dx\) |
\(\Big \downarrow \) 3102 |
\(\displaystyle \frac {\int -\frac {\sec ^8(a+b x)}{\left (1-\sec ^2(a+b x)\right )^3}d\sec (a+b x)}{16 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\sec ^8(a+b x)}{\left (1-\sec ^2(a+b x)\right )^3}d\sec (a+b x)}{16 b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {\frac {7}{4} \int \frac {\sec ^6(a+b x)}{\left (1-\sec ^2(a+b x)\right )^2}d\sec (a+b x)-\frac {\sec ^7(a+b x)}{4 \left (1-\sec ^2(a+b x)\right )^2}}{16 b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {\frac {7}{4} \left (\frac {\sec ^5(a+b x)}{2 \left (1-\sec ^2(a+b x)\right )}-\frac {5}{2} \int \frac {\sec ^4(a+b x)}{1-\sec ^2(a+b x)}d\sec (a+b x)\right )-\frac {\sec ^7(a+b x)}{4 \left (1-\sec ^2(a+b x)\right )^2}}{16 b}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {\frac {7}{4} \left (\frac {\sec ^5(a+b x)}{2 \left (1-\sec ^2(a+b x)\right )}-\frac {5}{2} \int \left (-\sec ^2(a+b x)+\frac {1}{1-\sec ^2(a+b x)}-1\right )d\sec (a+b x)\right )-\frac {\sec ^7(a+b x)}{4 \left (1-\sec ^2(a+b x)\right )^2}}{16 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {7}{4} \left (\frac {\sec ^5(a+b x)}{2 \left (1-\sec ^2(a+b x)\right )}-\frac {5}{2} \left (\text {arctanh}(\sec (a+b x))-\frac {1}{3} \sec ^3(a+b x)-\sec (a+b x)\right )\right )-\frac {\sec ^7(a+b x)}{4 \left (1-\sec ^2(a+b x)\right )^2}}{16 b}\) |
Input:
Int[Csc[a + b*x]*Csc[2*a + 2*b*x]^4,x]
Output:
(-1/4*Sec[a + b*x]^7/(1 - Sec[a + b*x]^2)^2 + (7*(Sec[a + b*x]^5/(2*(1 - S ec[a + b*x]^2)) - (5*(ArcTanh[Sec[a + b*x]] - Sec[a + b*x] - Sec[a + b*x]^ 3/3))/2))/4)/(16*b)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_S ymbol] :> Simp[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])
Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_ Symbol] :> Simp[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] && I ntegerQ[p]
Time = 0.32 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {-\frac {1}{4 \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )^{3}}+\frac {7}{12 \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )^{3}}-\frac {35}{24 \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )}+\frac {35}{8 \cos \left (b x +a \right )}+\frac {35 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8}}{16 b}\) | \(89\) |
risch | \(\frac {105 \,{\mathrm e}^{13 i \left (b x +a \right )}-70 \,{\mathrm e}^{11 i \left (b x +a \right )}-329 \,{\mathrm e}^{9 i \left (b x +a \right )}+204 \,{\mathrm e}^{7 i \left (b x +a \right )}-329 \,{\mathrm e}^{5 i \left (b x +a \right )}-70 \,{\mathrm e}^{3 i \left (b x +a \right )}+105 \,{\mathrm e}^{i \left (b x +a \right )}}{192 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{3}}-\frac {35 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{128 b}+\frac {35 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{128 b}\) | \(145\) |
Input:
int(csc(b*x+a)*csc(2*b*x+2*a)^4,x,method=_RETURNVERBOSE)
Output:
1/16/b*(-1/4/sin(b*x+a)^4/cos(b*x+a)^3+7/12/sin(b*x+a)^2/cos(b*x+a)^3-35/2 4/sin(b*x+a)^2/cos(b*x+a)+35/8/cos(b*x+a)+35/8*ln(csc(b*x+a)-cot(b*x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (73) = 146\).
Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.78 \[ \int \csc (a+b x) \csc ^4(2 a+2 b x) \, dx=\frac {210 \, \cos \left (b x + a\right )^{6} - 350 \, \cos \left (b x + a\right )^{4} + 112 \, \cos \left (b x + a\right )^{2} - 105 \, {\left (\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 105 \, {\left (\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 16}{768 \, {\left (b \cos \left (b x + a\right )^{7} - 2 \, b \cos \left (b x + a\right )^{5} + b \cos \left (b x + a\right )^{3}\right )}} \] Input:
integrate(csc(b*x+a)*csc(2*b*x+2*a)^4,x, algorithm="fricas")
Output:
1/768*(210*cos(b*x + a)^6 - 350*cos(b*x + a)^4 + 112*cos(b*x + a)^2 - 105* (cos(b*x + a)^7 - 2*cos(b*x + a)^5 + cos(b*x + a)^3)*log(1/2*cos(b*x + a) + 1/2) + 105*(cos(b*x + a)^7 - 2*cos(b*x + a)^5 + cos(b*x + a)^3)*log(-1/2 *cos(b*x + a) + 1/2) + 16)/(b*cos(b*x + a)^7 - 2*b*cos(b*x + a)^5 + b*cos( b*x + a)^3)
\[ \int \csc (a+b x) \csc ^4(2 a+2 b x) \, dx=\int \csc {\left (a + b x \right )} \csc ^{4}{\left (2 a + 2 b x \right )}\, dx \] Input:
integrate(csc(b*x+a)*csc(2*b*x+2*a)**4,x)
Output:
Integral(csc(a + b*x)*csc(2*a + 2*b*x)**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 3846 vs. \(2 (73) = 146\).
Time = 0.21 (sec) , antiderivative size = 3846, normalized size of antiderivative = 46.34 \[ \int \csc (a+b x) \csc ^4(2 a+2 b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)*csc(2*b*x+2*a)^4,x, algorithm="maxima")
Output:
1/768*(4*(105*cos(13*b*x + 13*a) - 70*cos(11*b*x + 11*a) - 329*cos(9*b*x + 9*a) + 204*cos(7*b*x + 7*a) - 329*cos(5*b*x + 5*a) - 70*cos(3*b*x + 3*a) + 105*cos(b*x + a))*cos(14*b*x + 14*a) - 420*(cos(12*b*x + 12*a) + 3*cos(1 0*b*x + 10*a) - 3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4* a) + cos(2*b*x + 2*a) - 1)*cos(13*b*x + 13*a) + 4*(70*cos(11*b*x + 11*a) + 329*cos(9*b*x + 9*a) - 204*cos(7*b*x + 7*a) + 329*cos(5*b*x + 5*a) + 70*c os(3*b*x + 3*a) - 105*cos(b*x + a))*cos(12*b*x + 12*a) + 280*(3*cos(10*b*x + 10*a) - 3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(11*b*x + 11*a) + 12*(329*cos(9*b*x + 9*a) - 204* cos(7*b*x + 7*a) + 329*cos(5*b*x + 5*a) + 70*cos(3*b*x + 3*a) - 105*cos(b* x + a))*cos(10*b*x + 10*a) - 1316*(3*cos(8*b*x + 8*a) + 3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(9*b*x + 9*a) + 12*(204*c os(7*b*x + 7*a) - 329*cos(5*b*x + 5*a) - 70*cos(3*b*x + 3*a) + 105*cos(b*x + a))*cos(8*b*x + 8*a) + 816*(3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - c os(2*b*x + 2*a) + 1)*cos(7*b*x + 7*a) - 84*(47*cos(5*b*x + 5*a) + 10*cos(3 *b*x + 3*a) - 15*cos(b*x + a))*cos(6*b*x + 6*a) + 1316*(3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(5*b*x + 5*a) + 420*(2*cos(3*b*x + 3*a) - 3*co s(b*x + a))*cos(4*b*x + 4*a) + 280*(cos(2*b*x + 2*a) - 1)*cos(3*b*x + 3*a) - 420*cos(2*b*x + 2*a)*cos(b*x + a) + 105*(2*(cos(12*b*x + 12*a) + 3*cos( 10*b*x + 10*a) - 3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x ...
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (73) = 146\).
Time = 0.14 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.48 \[ \int \csc (a+b x) \csc ^4(2 a+2 b x) \, dx=\frac {\frac {3 \, {\left (\frac {24 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {210 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} - \frac {72 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {256 \, {\left (\frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {6 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 5\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{3}} + 420 \, \log \left (-\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1}\right )}{3072 \, b} \] Input:
integrate(csc(b*x+a)*csc(2*b*x+2*a)^4,x, algorithm="giac")
Output:
1/3072*(3*(24*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 210*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 - 1)*(cos(b*x + a) + 1)^2/(cos(b*x + a) - 1)^2 - 72*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 3*(cos(b*x + a) - 1)^2/(cos(b* x + a) + 1)^2 + 256*(9*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 6*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 5)/((cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)^3 + 420*log(-(cos(b*x + a) - 1)/(cos(b*x + a) + 1)))/b
Time = 18.96 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int \csc (a+b x) \csc ^4(2 a+2 b x) \, dx=\frac {\frac {35\,{\cos \left (a+b\,x\right )}^6}{128}-\frac {175\,{\cos \left (a+b\,x\right )}^4}{384}+\frac {7\,{\cos \left (a+b\,x\right )}^2}{48}+\frac {1}{48}}{b\,\left ({\cos \left (a+b\,x\right )}^7-2\,{\cos \left (a+b\,x\right )}^5+{\cos \left (a+b\,x\right )}^3\right )}-\frac {35\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{128\,b} \] Input:
int(1/(sin(a + b*x)*sin(2*a + 2*b*x)^4),x)
Output:
((7*cos(a + b*x)^2)/48 - (175*cos(a + b*x)^4)/384 + (35*cos(a + b*x)^6)/12 8 + 1/48)/(b*(cos(a + b*x)^3 - 2*cos(a + b*x)^5 + cos(a + b*x)^7)) - (35*a tanh(cos(a + b*x)))/(128*b)
\[ \int \csc (a+b x) \csc ^4(2 a+2 b x) \, dx=\int \csc \left (2 b x +2 a \right )^{4} \csc \left (b x +a \right )d x \] Input:
int(csc(b*x+a)*csc(2*b*x+2*a)^4,x)
Output:
int(csc(2*a + 2*b*x)**4*csc(a + b*x),x)