Integrand size = 21, antiderivative size = 51 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=-\frac {(2 a+b) \text {csch}^2(c+d x)}{2 d}-\frac {(a+b) \text {csch}^4(c+d x)}{4 d}+\frac {a \log (\sinh (c+d x))}{d} \] Output:
-1/2*(2*a+b)*csch(d*x+c)^2/d-1/4*(a+b)*csch(d*x+c)^4/d+a*ln(sinh(d*x+c))/d
Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.04 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=-\frac {b \coth ^4(c+d x)}{4 d}-\frac {a \left (4 \text {csch}^2(c+d x)+\text {csch}^4(c+d x)-4 \log (\sinh (c+d x))\right )}{4 d} \] Input:
Integrate[Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2),x]
Output:
-1/4*(b*Coth[c + d*x]^4)/d - (a*(4*Csch[c + d*x]^2 + Csch[c + d*x]^4 - 4*L og[Sinh[c + d*x]]))/(4*d)
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 26, 4626, 354, 86, 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.
\(\displaystyle \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \left (a+b \sec (i c+i d x)^2\right )}{\tan (i c+i d x)^5}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {b \sec (i c+i d x)^2+a}{\tan (i c+i d x)^5}dx\) |
\(\Big \downarrow \) 4626 |
\(\displaystyle -\frac {\int \frac {\cosh ^3(c+d x) \left (a \cosh ^2(c+d x)+b\right )}{\left (1-\cosh ^2(c+d x)\right )^3}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {\int \frac {\cosh ^2(c+d x) \left (a \cosh ^2(c+d x)+b\right )}{\left (1-\cosh ^2(c+d x)\right )^3}d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle -\frac {\int \left (-\frac {a}{\cosh ^2(c+d x)-1}+\frac {-2 a-b}{\left (\cosh ^2(c+d x)-1\right )^2}+\frac {-a-b}{\left (\cosh ^2(c+d x)-1\right )^3}\right )d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {a+b}{2 \left (1-\cosh ^2(c+d x)\right )^2}-\frac {2 a+b}{1-\cosh ^2(c+d x)}-a \log \left (1-\cosh ^2(c+d x)\right )}{2 d}\) |
Input:
Int[Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2),x]
Output:
-1/2*((a + b)/(2*(1 - Cosh[c + d*x]^2)^2) - (2*a + b)/(1 - Cosh[c + d*x]^2 ) - a*Log[1 - Cosh[c + d*x]^2])/d
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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> Module[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-(f *ff^(m + n*p - 1))^(-1) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff* x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n} , x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
Time = 4.65 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.31
method | result | size |
derivativedivides | \(\frac {a \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}-\frac {\coth \left (d x +c \right )^{4}}{4}\right )+b \left (-\frac {\cosh \left (d x +c \right )^{2}}{2 \sinh \left (d x +c \right )^{4}}+\frac {1}{4 \sinh \left (d x +c \right )^{4}}\right )}{d}\) | \(67\) |
default | \(\frac {a \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}-\frac {\coth \left (d x +c \right )^{4}}{4}\right )+b \left (-\frac {\cosh \left (d x +c \right )^{2}}{2 \sinh \left (d x +c \right )^{4}}+\frac {1}{4 \sinh \left (d x +c \right )^{4}}\right )}{d}\) | \(67\) |
risch | \(-x a -\frac {2 a c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (2 \,{\mathrm e}^{4 d x +4 c} a +{\mathrm e}^{4 d x +4 c} b -2 a \,{\mathrm e}^{2 d x +2 c}+2 a +b \right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a}{d}\) | \(97\) |
Input:
int(coth(d*x+c)^5*(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(ln(sinh(d*x+c))-1/2*coth(d*x+c)^2-1/4*coth(d*x+c)^4)+b*(-1/2/sinh( d*x+c)^4*cosh(d*x+c)^2+1/4/sinh(d*x+c)^4))
Leaf count of result is larger than twice the leaf count of optimal. 1099 vs. \(2 (47) = 94\).
Time = 0.23 (sec) , antiderivative size = 1099, normalized size of antiderivative = 21.55 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(coth(d*x+c)^5*(a+b*sech(d*x+c)^2),x, algorithm="fricas")
Output:
-(a*d*x*cosh(d*x + c)^8 + 8*a*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + a*d*x*si nh(d*x + c)^8 - 2*(2*a*d*x - 2*a - b)*cosh(d*x + c)^6 + 2*(14*a*d*x*cosh(d *x + c)^2 - 2*a*d*x + 2*a + b)*sinh(d*x + c)^6 + 4*(14*a*d*x*cosh(d*x + c) ^3 - 3*(2*a*d*x - 2*a - b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a*d*x - 2 *a)*cosh(d*x + c)^4 + 2*(35*a*d*x*cosh(d*x + c)^4 + 3*a*d*x - 15*(2*a*d*x - 2*a - b)*cosh(d*x + c)^2 - 2*a)*sinh(d*x + c)^4 + 8*(7*a*d*x*cosh(d*x + c)^5 - 5*(2*a*d*x - 2*a - b)*cosh(d*x + c)^3 + (3*a*d*x - 2*a)*cosh(d*x + c))*sinh(d*x + c)^3 + a*d*x - 2*(2*a*d*x - 2*a - b)*cosh(d*x + c)^2 + 2*(1 4*a*d*x*cosh(d*x + c)^6 - 15*(2*a*d*x - 2*a - b)*cosh(d*x + c)^4 - 2*a*d*x + 6*(3*a*d*x - 2*a)*cosh(d*x + c)^2 + 2*a + b)*sinh(d*x + c)^2 - (a*cosh( d*x + c)^8 + 8*a*cosh(d*x + c)*sinh(d*x + c)^7 + a*sinh(d*x + c)^8 - 4*a*c osh(d*x + c)^6 + 4*(7*a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^6 + 8*(7*a*cosh (d*x + c)^3 - 3*a*cosh(d*x + c))*sinh(d*x + c)^5 + 6*a*cosh(d*x + c)^4 + 2 *(35*a*cosh(d*x + c)^4 - 30*a*cosh(d*x + c)^2 + 3*a)*sinh(d*x + c)^4 + 8*( 7*a*cosh(d*x + c)^5 - 10*a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c))*sinh(d*x + c)^3 - 4*a*cosh(d*x + c)^2 + 4*(7*a*cosh(d*x + c)^6 - 15*a*cosh(d*x + c)^ 4 + 9*a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^2 + 8*(a*cosh(d*x + c)^7 - 3*a* cosh(d*x + c)^5 + 3*a*cosh(d*x + c)^3 - a*cosh(d*x + c))*sinh(d*x + c) + a )*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 4*(2*a*d*x*cosh(d *x + c)^7 - 3*(2*a*d*x - 2*a - b)*cosh(d*x + c)^5 + 2*(3*a*d*x - 2*a)*c...
Timed out. \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(coth(d*x+c)**5*(a+b*sech(d*x+c)**2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (47) = 94\).
Time = 0.05 (sec) , antiderivative size = 251, normalized size of antiderivative = 4.92 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=a {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 2 \, b {\left (\frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} + \frac {e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} \] Input:
integrate(coth(d*x+c)^5*(a+b*sech(d*x+c)^2),x, algorithm="maxima")
Output:
a*(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))) + 2*b*( e^(-2*d*x - 2*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)) + 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)))
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (47) = 94\).
Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.35 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=-\frac {12 \, {\left (d x + c\right )} a - 12 \, a \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {25 \, a e^{\left (8 \, d x + 8 \, c\right )} - 52 \, a e^{\left (6 \, d x + 6 \, c\right )} + 24 \, b e^{\left (6 \, d x + 6 \, c\right )} + 102 \, a e^{\left (4 \, d x + 4 \, c\right )} - 52 \, a e^{\left (2 \, d x + 2 \, c\right )} + 24 \, b e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}}}{12 \, d} \] Input:
integrate(coth(d*x+c)^5*(a+b*sech(d*x+c)^2),x, algorithm="giac")
Output:
-1/12*(12*(d*x + c)*a - 12*a*log(abs(e^(2*d*x + 2*c) - 1)) + (25*a*e^(8*d* x + 8*c) - 52*a*e^(6*d*x + 6*c) + 24*b*e^(6*d*x + 6*c) + 102*a*e^(4*d*x + 4*c) - 52*a*e^(2*d*x + 2*c) + 24*b*e^(2*d*x + 2*c) + 25*a)/(e^(2*d*x + 2*c ) - 1)^4)/d
Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 3.51 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\frac {a\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )}{d}-\frac {8\,\left (a+b\right )}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {2\,\left (2\,a+b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {4\,\left (a+b\right )}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-a\,x-\frac {2\,\left (4\,a+3\,b\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:
int(coth(c + d*x)^5*(a + b/cosh(c + d*x)^2),x)
Output:
(a*log(exp(2*c)*exp(2*d*x) - 1))/d - (8*(a + b))/(d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (2*(2*a + b))/(d*(exp(2*c + 2*d*x) - 1)) - (4*(a + b))/(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)) - a*x - (2*(4*a + 3*b))/(d*(exp (4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))
Time = 0.22 (sec) , antiderivative size = 389, normalized size of antiderivative = 7.63 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\frac {2 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}-1\right ) a +2 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}+1\right ) a -2 e^{8 d x +8 c} a d x -2 e^{8 d x +8 c} a -e^{8 d x +8 c} b -8 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) a -8 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) a +8 e^{6 d x +6 c} a d x +12 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a +12 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a -12 e^{4 d x +4 c} a d x -4 e^{4 d x +4 c} a -6 e^{4 d x +4 c} b -8 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a -8 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a +8 e^{2 d x +2 c} a d x +2 \,\mathrm {log}\left (e^{d x +c}-1\right ) a +2 \,\mathrm {log}\left (e^{d x +c}+1\right ) a -2 a d x -2 a -b}{2 d \left (e^{8 d x +8 c}-4 e^{6 d x +6 c}+6 e^{4 d x +4 c}-4 e^{2 d x +2 c}+1\right )} \] Input:
int(coth(d*x+c)^5*(a+b*sech(d*x+c)^2),x)
Output:
(2*e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*a + 2*e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*a - 2*e**(8*c + 8*d*x)*a*d*x - 2*e**(8*c + 8*d*x)*a - e**(8*c + 8*d*x)*b - 8*e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*a - 8*e**(6*c + 6*d *x)*log(e**(c + d*x) + 1)*a + 8*e**(6*c + 6*d*x)*a*d*x + 12*e**(4*c + 4*d* x)*log(e**(c + d*x) - 1)*a + 12*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a - 12*e**(4*c + 4*d*x)*a*d*x - 4*e**(4*c + 4*d*x)*a - 6*e**(4*c + 4*d*x)*b - 8*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a - 8*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a + 8*e**(2*c + 2*d*x)*a*d*x + 2*log(e**(c + d*x) - 1)*a + 2* log(e**(c + d*x) + 1)*a - 2*a*d*x - 2*a - b)/(2*d*(e**(8*c + 8*d*x) - 4*e* *(6*c + 6*d*x) + 6*e**(4*c + 4*d*x) - 4*e**(2*c + 2*d*x) + 1))