Integrand size = 23, antiderivative size = 92 \[ \int \coth ^7(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {(a+b)^2 \coth ^2(c+d x)}{2 d}-\frac {a (a+2 b) \coth ^4(c+d x)}{4 d}-\frac {a^2 \coth ^6(c+d x)}{6 d}+\frac {(a+b)^2 \log (\cosh (c+d x))}{d}+\frac {(a+b)^2 \log (\tanh (c+d x))}{d} \]
-1/2*(a+b)^2*coth(d*x+c)^2/d-1/4*a*(a+2*b)*coth(d*x+c)^4/d-1/6*a^2*coth(d* x+c)^6/d+(a+b)^2*ln(cosh(d*x+c))/d+(a+b)^2*ln(tanh(d*x+c))/d
Time = 0.57 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.80 \[ \int \coth ^7(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {6 (a+b)^2 \coth ^2(c+d x)+3 a (a+2 b) \coth ^4(c+d x)+2 a^2 \coth ^6(c+d x)-12 (a+b)^2 (\log (\cosh (c+d x))+\log (\tanh (c+d x)))}{12 d} \]
-1/12*(6*(a + b)^2*Coth[c + d*x]^2 + 3*a*(a + 2*b)*Coth[c + d*x]^4 + 2*a^2 *Coth[c + d*x]^6 - 12*(a + b)^2*(Log[Cosh[c + d*x]] + Log[Tanh[c + d*x]])) /d
Time = 0.32 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 4153, 26, 354, 99, 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.
| \(\displaystyle \int \coth ^7(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a-b \tan (i c+i d x)^2\right )^2}{\tan (i c+i d x)^7}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (a-b \tan (i c+i d x)^2\right )^2}{\tan (i c+i d x)^7}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle -\frac {i \int \frac {i \coth ^7(c+d x) \left (b \tanh ^2(c+d x)+a\right )^2}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\int \frac {\coth ^7(c+d x) \left (b \tanh ^2(c+d x)+a\right )^2}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\coth ^4(c+d x) \left (b \tanh ^2(c+d x)+a\right )^2}{1-\tanh ^2(c+d x)}d\tanh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (a^2 \coth ^4(c+d x)+a (a+2 b) \coth ^3(c+d x)+(a+b)^2 \coth ^2(c+d x)+(a+b)^2 \coth (c+d x)-\frac {(a+b)^2}{\tanh ^2(c+d x)-1}\right )d\tanh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{3} a^2 \coth ^3(c+d x)-\frac {1}{2} a (a+2 b) \coth ^2(c+d x)-(a+b)^2 \coth (c+d x)+(a+b)^2 \log \left (\tanh ^2(c+d x)\right )-(a+b)^2 \log \left (1-\tanh ^2(c+d x)\right )}{2 d}\) |
(-((a + b)^2*Coth[c + d*x]) - (a*(a + 2*b)*Coth[c + d*x]^2)/2 - (a^2*Coth[ c + d*x]^3)/3 + (a + b)^2*Log[Tanh[c + d*x]^2] - (a + b)^2*Log[1 - Tanh[c + d*x]^2])/(2*d)
3.2.55.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00
| method | result | size |
| parallelrisch | \(\frac {-12 \left (a +b \right )^{2} \ln \left (1-\tanh \left (d x +c \right )\right )+12 \left (a +b \right )^{2} \ln \left (\tanh \left (d x +c \right )\right )-2 \coth \left (d x +c \right )^{6} a^{2}-3 a \coth \left (d x +c \right )^{4} \left (a +2 b \right )-6 \coth \left (d x +c \right )^{2} \left (a +b \right )^{2}-12 d x \left (a +b \right )^{2}}{12 d}\) | \(92\) |
| derivativedivides | \(-\frac {\left (\frac {1}{2} a^{2}+a b +\frac {1}{2} b^{2}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )+\left (\frac {1}{2} a^{2}+a b +\frac {1}{2} b^{2}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )-\frac {-a^{2}-2 a b -b^{2}}{2 \tanh \left (d x +c \right )^{2}}+\left (-a^{2}-2 a b -b^{2}\right ) \ln \left (\tanh \left (d x +c \right )\right )+\frac {a^{2}}{6 \tanh \left (d x +c \right )^{6}}+\frac {a \left (a +2 b \right )}{4 \tanh \left (d x +c \right )^{4}}}{d}\) | \(132\) |
| default | \(-\frac {\left (\frac {1}{2} a^{2}+a b +\frac {1}{2} b^{2}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )+\left (\frac {1}{2} a^{2}+a b +\frac {1}{2} b^{2}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )-\frac {-a^{2}-2 a b -b^{2}}{2 \tanh \left (d x +c \right )^{2}}+\left (-a^{2}-2 a b -b^{2}\right ) \ln \left (\tanh \left (d x +c \right )\right )+\frac {a^{2}}{6 \tanh \left (d x +c \right )^{6}}+\frac {a \left (a +2 b \right )}{4 \tanh \left (d x +c \right )^{4}}}{d}\) | \(132\) |
| risch | \(-a^{2} x -2 a b x -b^{2} x -\frac {2 a^{2} c}{d}-\frac {4 a b c}{d}-\frac {2 c \,b^{2}}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (9 a^{2} {\mathrm e}^{8 d x +8 c}+12 a b \,{\mathrm e}^{8 d x +8 c}+3 b^{2} {\mathrm e}^{8 d x +8 c}-18 a^{2} {\mathrm e}^{6 d x +6 c}-36 a b \,{\mathrm e}^{6 d x +6 c}-12 b^{2} {\mathrm e}^{6 d x +6 c}+34 a^{2} {\mathrm e}^{4 d x +4 c}+48 a b \,{\mathrm e}^{4 d x +4 c}+18 \,{\mathrm e}^{4 d x +4 c} b^{2}-18 a^{2} {\mathrm e}^{2 d x +2 c}-36 a b \,{\mathrm e}^{2 d x +2 c}-12 \,{\mathrm e}^{2 d x +2 c} b^{2}+9 a^{2}+12 a b +3 b^{2}\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}+\frac {a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}+\frac {2 a \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{2}}{d}\) | \(308\) |
1/12*(-12*(a+b)^2*ln(1-tanh(d*x+c))+12*(a+b)^2*ln(tanh(d*x+c))-2*coth(d*x+ c)^6*a^2-3*a*coth(d*x+c)^4*(a+2*b)-6*coth(d*x+c)^2*(a+b)^2-12*d*x*(a+b)^2) /d
Leaf count of result is larger than twice the leaf count of optimal. 3454 vs. \(2 (86) = 172\).
Time = 0.31 (sec) , antiderivative size = 3454, normalized size of antiderivative = 37.54 \[ \int \coth ^7(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\text {Too large to display} \]
-1/3*(3*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^12 + 36*(a^2 + 2*a*b + b^2)* d*x*cosh(d*x + c)*sinh(d*x + c)^11 + 3*(a^2 + 2*a*b + b^2)*d*x*sinh(d*x + c)^12 - 6*(3*(a^2 + 2*a*b + b^2)*d*x - 3*a^2 - 4*a*b - b^2)*cosh(d*x + c)^ 10 + 6*(33*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^2 - 3*(a^2 + 2*a*b + b^2) *d*x + 3*a^2 + 4*a*b + b^2)*sinh(d*x + c)^10 + 60*(11*(a^2 + 2*a*b + b^2)* d*x*cosh(d*x + c)^3 - (3*(a^2 + 2*a*b + b^2)*d*x - 3*a^2 - 4*a*b - b^2)*co sh(d*x + c))*sinh(d*x + c)^9 + 3*(15*(a^2 + 2*a*b + b^2)*d*x - 12*a^2 - 24 *a*b - 8*b^2)*cosh(d*x + c)^8 + 3*(495*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^4 + 15*(a^2 + 2*a*b + b^2)*d*x - 90*(3*(a^2 + 2*a*b + b^2)*d*x - 3*a^2 - 4*a*b - b^2)*cosh(d*x + c)^2 - 12*a^2 - 24*a*b - 8*b^2)*sinh(d*x + c)^8 + 24*(99*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^5 - 30*(3*(a^2 + 2*a*b + b^ 2)*d*x - 3*a^2 - 4*a*b - b^2)*cosh(d*x + c)^3 + (15*(a^2 + 2*a*b + b^2)*d* x - 12*a^2 - 24*a*b - 8*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(15*(a^2 + 2*a*b + b^2)*d*x - 17*a^2 - 24*a*b - 9*b^2)*cosh(d*x + c)^6 + 4*(693*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^6 - 315*(3*(a^2 + 2*a*b + b^2)*d*x - 3*a ^2 - 4*a*b - b^2)*cosh(d*x + c)^4 - 15*(a^2 + 2*a*b + b^2)*d*x + 21*(15*(a ^2 + 2*a*b + b^2)*d*x - 12*a^2 - 24*a*b - 8*b^2)*cosh(d*x + c)^2 + 17*a^2 + 24*a*b + 9*b^2)*sinh(d*x + c)^6 + 24*(99*(a^2 + 2*a*b + b^2)*d*x*cosh(d* x + c)^7 - 63*(3*(a^2 + 2*a*b + b^2)*d*x - 3*a^2 - 4*a*b - b^2)*cosh(d*x + c)^5 + 7*(15*(a^2 + 2*a*b + b^2)*d*x - 12*a^2 - 24*a*b - 8*b^2)*cosh(d...
Timed out. \[ \int \coth ^7(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (86) = 172\).
Time = 0.20 (sec) , antiderivative size = 390, normalized size of antiderivative = 4.24 \[ \int \coth ^7(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {1}{3} \, a^{2} {\left (3 \, x + \frac {3 \, c}{d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 18 \, e^{\left (-4 \, d x - 4 \, c\right )} + 34 \, e^{\left (-6 \, d x - 6 \, c\right )} - 18 \, e^{\left (-8 \, d x - 8 \, c\right )} + 9 \, e^{\left (-10 \, d x - 10 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + 2 \, a b {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + b^{2} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
1/3*a^2*(3*x + 3*c/d + 3*log(e^(-d*x - c) + 1)/d + 3*log(e^(-d*x - c) - 1) /d + 2*(9*e^(-2*d*x - 2*c) - 18*e^(-4*d*x - 4*c) + 34*e^(-6*d*x - 6*c) - 1 8*e^(-8*d*x - 8*c) + 9*e^(-10*d*x - 10*c))/(d*(6*e^(-2*d*x - 2*c) - 15*e^( -4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1))) + 2*a*b*(x + c/d + log(e^(-d*x - c) + 1 )/d + log(e^(-d*x - c) - 1)/d + 4*(e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) + e ^(-6*d*x - 6*c))/(d*(4*e^(-2*d*x - 2*c) - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1))) + b^2*(x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) - e ^(-4*d*x - 4*c) - 1)))
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (86) = 172\).
Time = 0.46 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.09 \[ \int \coth ^7(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (d x + c\right )} - 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {2 \, {\left (3 \, {\left (3 \, a^{2} + 4 \, a b + b^{2}\right )} e^{\left (10 \, d x + 10 \, c\right )} - 6 \, {\left (3 \, a^{2} + 6 \, a b + 2 \, b^{2}\right )} e^{\left (8 \, d x + 8 \, c\right )} + 2 \, {\left (17 \, a^{2} + 24 \, a b + 9 \, b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )} - 6 \, {\left (3 \, a^{2} + 6 \, a b + 2 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, {\left (3 \, a^{2} + 4 \, a b + b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{6}}}{3 \, d} \]
-1/3*(3*(a^2 + 2*a*b + b^2)*(d*x + c) - 3*(a^2 + 2*a*b + b^2)*log(abs(e^(2 *d*x + 2*c) - 1)) + 2*(3*(3*a^2 + 4*a*b + b^2)*e^(10*d*x + 10*c) - 6*(3*a^ 2 + 6*a*b + 2*b^2)*e^(8*d*x + 8*c) + 2*(17*a^2 + 24*a*b + 9*b^2)*e^(6*d*x + 6*c) - 6*(3*a^2 + 6*a*b + 2*b^2)*e^(4*d*x + 4*c) + 3*(3*a^2 + 4*a*b + b^ 2)*e^(2*d*x + 2*c))/(e^(2*d*x + 2*c) - 1)^6)/d
Time = 0.24 (sec) , antiderivative size = 362, normalized size of antiderivative = 3.93 \[ \int \coth ^7(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )\,\left (a^2+2\,a\,b+b^2\right )}{d}-\frac {2\,\left (3\,a^2+4\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {32\,a^2}{3\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-x\,{\left (a+b\right )}^2-\frac {2\,\left (9\,a^2+8\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,\left (13\,a^2+6\,b\,a\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4\,\left (11\,a^2+2\,b\,a\right )}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {32\,a^2}{d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )} \]
(log(exp(2*c)*exp(2*d*x) - 1)*(2*a*b + a^2 + b^2))/d - (2*(4*a*b + 3*a^2 + b^2))/(d*(exp(2*c + 2*d*x) - 1)) - (32*a^2)/(3*d*(15*exp(4*c + 4*d*x) - 6 *exp(2*c + 2*d*x) - 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) - 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1)) - x*(a + b)^2 - (2*(8*a*b + 9*a^2 + b^2))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (8*(6*a*b + 13*a^2 ))/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (4*(2*a*b + 11*a^2))/(d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp (6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (32*a^2)/(d*(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) - 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) - 1))