Integrand size = 21, antiderivative size = 52 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {(a+b)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {b (2 a+b) \text {sech}(c+d x)}{d}+\frac {b^2 \text {sech}^3(c+d x)}{3 d} \]
Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(52)=104\).
Time = 3.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.08 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {4 \left (b+a \cosh ^2(c+d x)\right )^2 \left (-b^2-3 b (2 a+b) \cosh ^2(c+d x)+3 (a+b)^2 \cosh ^3(c+d x) \left (\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )\right ) \text {sech}^3(c+d x)}{3 d (a+2 b+a \cosh (2 (c+d x)))^2} \]
(-4*(b + a*Cosh[c + d*x]^2)^2*(-b^2 - 3*b*(2*a + b)*Cosh[c + d*x]^2 + 3*(a + b)^2*Cosh[c + d*x]^3*(Log[Cosh[(c + d*x)/2]] - Log[Sinh[(c + d*x)/2]])) *Sech[c + d*x]^3)/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 26, 4621, 364, 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.
\(\displaystyle \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \left (a+b \sec (i c+i d x)^2\right )^2}{\sin (i c+i d x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\left (b \sec (i c+i d x)^2+a\right )^2}{\sin (i c+i d x)}dx\) |
\(\Big \downarrow \) 4621 |
\(\displaystyle -\frac {\int \frac {\left (a \cosh ^2(c+d x)+b\right )^2 \text {sech}^4(c+d x)}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 364 |
\(\displaystyle -\frac {\int \left (b^2 \text {sech}^4(c+d x)+b (2 a+b) \text {sech}^2(c+d x)-\frac {(a+b)^2}{\cosh ^2(c+d x)-1}\right )d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(a+b)^2 \text {arctanh}(\cosh (c+d x))-b (2 a+b) \text {sech}(c+d x)-\frac {1}{3} b^2 \text {sech}^3(c+d x)}{d}\) |
-(((a + b)^2*ArcTanh[Cosh[c + d*x]] - b*(2*a + b)*Sech[c + d*x] - (b^2*Sec h[c + d*x]^3)/3)/d)
3.1.13.3.1 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[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^2)^p/(c + d*x^2)), x], x ] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && (In tegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Time = 15.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.38
method | result | size |
derivativedivides | \(\frac {-2 a^{2} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+2 a b \left (\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {1}{3 \cosh \left (d x +c \right )^{3}}+\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(72\) |
default | \(\frac {-2 a^{2} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+2 a b \left (\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {1}{3 \cosh \left (d x +c \right )^{3}}+\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(72\) |
risch | \(\frac {2 b \,{\mathrm e}^{d x +c} \left (6 a \,{\mathrm e}^{4 d x +4 c}+3 b \,{\mathrm e}^{4 d x +4 c}+12 \,{\mathrm e}^{2 d x +2 c} a +10 b \,{\mathrm e}^{2 d x +2 c}+6 a +3 b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}-\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}-\frac {2 \ln \left ({\mathrm e}^{d x +c}+1\right ) a b}{d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{d}+\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}-1\right ) a b}{d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{d}\) | \(180\) |
1/d*(-2*a^2*arctanh(exp(d*x+c))+2*a*b*(1/cosh(d*x+c)-2*arctanh(exp(d*x+c)) )+b^2*(1/3/cosh(d*x+c)^3+1/cosh(d*x+c)-2*arctanh(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 1148 vs. \(2 (50) = 100\).
Time = 0.27 (sec) , antiderivative size = 1148, normalized size of antiderivative = 22.08 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\text {Too large to display} \]
1/3*(6*(2*a*b + b^2)*cosh(d*x + c)^5 + 30*(2*a*b + b^2)*cosh(d*x + c)*sinh (d*x + c)^4 + 6*(2*a*b + b^2)*sinh(d*x + c)^5 + 4*(6*a*b + 5*b^2)*cosh(d*x + c)^3 + 4*(15*(2*a*b + b^2)*cosh(d*x + c)^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^3 + 12*(5*(2*a*b + b^2)*cosh(d*x + c)^3 + (6*a*b + 5*b^2)*cosh(d*x + c) )*sinh(d*x + c)^2 + 6*(2*a*b + b^2)*cosh(d*x + c) - 3*((a^2 + 2*a*b + b^2) *cosh(d*x + c)^6 + 6*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + ( a^2 + 2*a*b + b^2)*sinh(d*x + c)^6 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 4*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)* cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 3 *(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 6*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2 + 6*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 2*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a ^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d *x + c) + 1) + 3*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 6*(a^2 + 2*a*b + b ^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^6 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 4*(5*(a^2 + 2*a*b + b^2)*cos h(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a ^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(d*x +...
\[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \operatorname {csch}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (50) = 100\).
Time = 0.20 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.79 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {1}{3} \, b^{2} {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} + 10 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )}\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} - 2 \, a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {2 \, e^{\left (-d x - c\right )}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + \frac {a^{2} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
-1/3*b^2*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)/d - 2*(3*e^( -d*x - c) + 10*e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c))/(d*(3*e^(-2*d*x - 2* c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1))) - 2*a*b*(log(e^(-d*x - c ) + 1)/d - log(e^(-d*x - c) - 1)/d - 2*e^(-d*x - c)/(d*(e^(-2*d*x - 2*c) + 1))) + a^2*log(tanh(1/2*d*x + 1/2*c))/d
Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (50) = 100\).
Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.67 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (6 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 3 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 4 \, b^{2}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}}}{6 \, d} \]
-1/6*(3*(a^2 + 2*a*b + b^2)*log(e^(d*x + c) + e^(-d*x - c) + 2) - 3*(a^2 + 2*a*b + b^2)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 4*(6*a*b*(e^(d*x + c) + e^(-d*x - c))^2 + 3*b^2*(e^(d*x + c) + e^(-d*x - c))^2 + 4*b^2)/(e^(d*x + c) + e^(-d*x - c))^3)/d
Time = 2.16 (sec) , antiderivative size = 232, normalized size of antiderivative = 4.46 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (b^2+2\,a\,b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^2\,\sqrt {-d^2}+b^2\,\sqrt {-d^2}+2\,a\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4}}\right )\,\sqrt {a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4}}{\sqrt {-d^2}}+\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
(2*exp(c + d*x)*(2*a*b + b^2))/(d*(exp(2*c + 2*d*x) + 1)) - (8*b^2*exp(c + d*x))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) - (2*atan((exp(d*x)*exp(c)*(a^2*(-d^2)^(1/2) + b^2*(-d^2)^(1/2) + 2*a* b*(-d^2)^(1/2)))/(d*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)^(1/2)))*(4 *a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)^(1/2))/(-d^2)^(1/2) + (8*b^2*exp (c + d*x))/(3*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))