Integrand size = 21, antiderivative size = 31 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=-\frac {(a+b) \text {csch}^2(c+d x)}{2 d}+\frac {a \log (\sinh (c+d x))}{d} \] Output:
-1/2*(a+b)*csch(d*x+c)^2/d+a*ln(sinh(d*x+c))/d
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=-\frac {b \text {csch}^2(c+d x)}{2 d}-\frac {a \left (\text {csch}^2(c+d x)-2 \log (\sinh (c+d x))\right )}{2 d} \] Input:
Integrate[Coth[c + d*x]^3*(a + b*Sech[c + d*x]^2),x]
Output:
-1/2*(b*Csch[c + d*x]^2)/d - (a*(Csch[c + d*x]^2 - 2*Log[Sinh[c + d*x]]))/ (2*d)
Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32, 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, 353, 49, 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.
\(\displaystyle \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \left (a+b \sec (i c+i d x)^2\right )}{\tan (i c+i d x)^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {b \sec (i c+i d x)^2+a}{\tan (i c+i d x)^3}dx\) |
\(\Big \downarrow \) 4626 |
\(\displaystyle \frac {\int \frac {\cosh (c+d x) \left (a \cosh ^2(c+d x)+b\right )}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {\int \frac {a \cosh ^2(c+d x)+b}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\frac {a}{\cosh ^2(c+d x)-1}+\frac {a+b}{\left (\cosh ^2(c+d x)-1\right )^2}\right )d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {a+b}{1-\cosh ^2(c+d x)}+a \log \left (1-\cosh ^2(c+d x)\right )}{2 d}\) |
Input:
Int[Coth[c + d*x]^3*(a + b*Sech[c + d*x]^2),x]
Output:
((a + b)/(1 - Cosh[c + d*x]^2) + a*Log[1 - Cosh[c + d*x]^2])/(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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(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]
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 = 1.72 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {a \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}\right )-\frac {b}{2 \sinh \left (d x +c \right )^{2}}}{d}\) | \(37\) |
default | \(\frac {a \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}\right )-\frac {b}{2 \sinh \left (d x +c \right )^{2}}}{d}\) | \(37\) |
risch | \(-x a -\frac {2 a c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a +b \right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a}{d}\) | \(60\) |
Input:
int(coth(d*x+c)^3*(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)-1/2*b/sinh(d*x+c)^2)
Leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (29) = 58\).
Time = 0.23 (sec) , antiderivative size = 378, normalized size of antiderivative = 12.19 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=-\frac {a d x \cosh \left (d x + c\right )^{4} + 4 \, a d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a d x \sinh \left (d x + c\right )^{4} + a d x - 2 \, {\left (a d x - a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a d x \cosh \left (d x + c\right )^{2} - a d x + a + b\right )} \sinh \left (d x + c\right )^{2} - {\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, a \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (a d x \cosh \left (d x + c\right )^{3} - {\left (a d x - a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \] Input:
integrate(coth(d*x+c)^3*(a+b*sech(d*x+c)^2),x, algorithm="fricas")
Output:
-(a*d*x*cosh(d*x + c)^4 + 4*a*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*x*si nh(d*x + c)^4 + a*d*x - 2*(a*d*x - a - b)*cosh(d*x + c)^2 + 2*(3*a*d*x*cos h(d*x + c)^2 - a*d*x + a + b)*sinh(d*x + c)^2 - (a*cosh(d*x + c)^4 + 4*a*c osh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*a*cosh(d*x + c)^2 + 2 *(3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^2 + 4*(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*(a*d*x*cosh(d*x + c)^3 - (a*d*x - a - b)*cosh(d*x + c))*sinh( d*x + c))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*sinh(d*x + c)^3 + d*sinh( d*x + c)^4 - 2*d*cosh(d*x + c)^2 + 2*(3*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 4*(d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c) + d)
\[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \coth ^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(coth(d*x+c)**3*(a+b*sech(d*x+c)**2),x)
Output:
Integral((a + b*sech(c + d*x)**2)*coth(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (29) = 58\).
Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.48 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=a {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac {2 \, b}{d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}} \] Input:
integrate(coth(d*x+c)^3*(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 + 2*e^(-2*d *x - 2*c)/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1))) - 2*b/(d*(e^(d* x + c) - e^(-d*x - c))^2)
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (29) = 58\).
Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.71 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (d x + c\right )} a - 2 \, a \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \] Input:
integrate(coth(d*x+c)^3*(a+b*sech(d*x+c)^2),x, algorithm="giac")
Output:
-1/2*(2*(d*x + c)*a - 2*a*log(abs(e^(2*d*x + 2*c) - 1)) + (3*a*e^(4*d*x + 4*c) - 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + 3*a)/(e^(2*d*x + 2*c) - 1)^2)/d
Time = 2.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.45 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\frac {a\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )}{d}-a\,x-\frac {2\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:
int(coth(c + d*x)^3*(a + b/cosh(c + d*x)^2),x)
Output:
(a*log(exp(2*c)*exp(2*d*x) - 1))/d - a*x - (2*(a + b))/(d*(exp(2*c + 2*d*x ) - 1)) - (2*(a + b))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))
Time = 0.22 (sec) , antiderivative size = 212, normalized size of antiderivative = 6.84 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\frac {e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a +e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a -e^{4 d x +4 c} a d x -e^{4 d x +4 c} a -e^{4 d x +4 c} b -2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a -2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a +2 e^{2 d x +2 c} a d x +\mathrm {log}\left (e^{d x +c}-1\right ) a +\mathrm {log}\left (e^{d x +c}+1\right ) a -a d x -a -b}{d \left (e^{4 d x +4 c}-2 e^{2 d x +2 c}+1\right )} \] Input:
int(coth(d*x+c)^3*(a+b*sech(d*x+c)^2),x)
Output:
(e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a + e**(4*c + 4*d*x)*log(e**(c + d *x) + 1)*a - e**(4*c + 4*d*x)*a*d*x - e**(4*c + 4*d*x)*a - e**(4*c + 4*d*x )*b - 2*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a - 2*e**(2*c + 2*d*x)*log( e**(c + d*x) + 1)*a + 2*e**(2*c + 2*d*x)*a*d*x + log(e**(c + d*x) - 1)*a + log(e**(c + d*x) + 1)*a - a*d*x - a - b)/(d*(e**(4*c + 4*d*x) - 2*e**(2*c + 2*d*x) + 1))