Integrand size = 21, antiderivative size = 99 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {\sqrt {b} (3 a+b) \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{2 a^{3/2} (a+b)^2 d}-\frac {\text {arctanh}(\cosh (c+d x))}{(a+b)^2 d}-\frac {b \cosh (c+d x)}{2 a (a+b) d \left (b+a \cosh ^2(c+d x)\right )} \]
-arctanh(cosh(d*x+c))/(a+b)^2/d-1/2*b*cosh(d*x+c)/a/(a+b)/d/(b+a*cosh(d*x+ c)^2)+1/2*(3*a+b)*arctan(cosh(d*x+c)*a^(1/2)/b^(1/2))*b^(1/2)/a^(3/2)/(a+b )^2/d
Result contains complex when optimal does not.
Time = 2.15 (sec) , antiderivative size = 377, normalized size of antiderivative = 3.81 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^3(c+d x) \left (-\frac {2 b (a+b)}{a}+\frac {\sqrt {b} (3 a+b) \arctan \left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cosh (2 (c+d x))) \text {sech}(c+d x)}{a^{3/2}}+\frac {\sqrt {b} (3 a+b) \arctan \left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cosh (2 (c+d x))) \text {sech}(c+d x)}{a^{3/2}}-2 (a+2 b+a \cosh (2 (c+d x))) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {sech}(c+d x)+2 (a+2 b+a \cosh (2 (c+d x))) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right ) \text {sech}(c+d x)\right )}{8 (a+b)^2 d \left (a+b \text {sech}^2(c+d x)\right )^2} \]
((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*((-2*b*(a + b))/a + (Sqrt [b]*(3*a + b)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2] )*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech [c + d*x])/a^(3/2) + (Sqrt[b]*(3*a + b)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*S qrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*S qrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x])/a^(3/2) - 2*(a + 2*b + a*Cosh[2*(c + d*x)])*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] + 2*(a + 2*b + a*Cosh[2*(c + d *x)])*Log[Sinh[(c + d*x)/2]]*Sech[c + d*x]))/(8*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^2)
Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 26, 4621, 372, 397, 218, 219}
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.
\(\displaystyle \int \frac {\text {csch}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\sin (i c+i d x) \left (a+b \sec (i c+i d x)^2\right )^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\left (b \sec (i c+i d x)^2+a\right )^2 \sin (i c+i d x)}dx\) |
\(\Big \downarrow \) 4621 |
\(\displaystyle -\frac {\int \frac {\cosh ^4(c+d x)}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )^2}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 372 |
\(\displaystyle -\frac {\frac {b \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\int \frac {b-(2 a+b) \cosh ^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}d\cosh (c+d x)}{2 a (a+b)}}{d}\) |
\(\Big \downarrow \) 397 |
\(\displaystyle -\frac {\frac {b \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\frac {b (3 a+b) \int \frac {1}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{a+b}-\frac {2 a \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a+b}}{2 a (a+b)}}{d}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {\frac {b \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\frac {\sqrt {b} (3 a+b) \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}-\frac {2 a \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a+b}}{2 a (a+b)}}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {b \cosh (c+d x)}{2 a (a+b) \left (a \cosh ^2(c+d x)+b\right )}-\frac {\frac {\sqrt {b} (3 a+b) \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}-\frac {2 a \text {arctanh}(\cosh (c+d x))}{a+b}}{2 a (a+b)}}{d}\) |
-((-1/2*((Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(Sqrt [a]*(a + b)) - (2*a*ArcTanh[Cosh[c + d*x]])/(a + b))/(a*(a + b)) + (b*Cosh [c + d*x])/(2*a*(a + b)*(b + a*Cosh[c + d*x]^2)))/d)
3.1.37.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
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 = 26.04 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.69
method | result | size |
derivativedivides | \(\frac {\frac {4 b \left (\frac {-\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 a}-\frac {a +b}{4 a}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a +b \right ) \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a -2 b}{4 \sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{\left (a +b \right )^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a +b \right )^{2}}}{d}\) | \(167\) |
default | \(\frac {\frac {4 b \left (\frac {-\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 a}-\frac {a +b}{4 a}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a +b \right ) \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a -2 b}{4 \sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{\left (a +b \right )^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a +b \right )^{2}}}{d}\) | \(167\) |
risch | \(-\frac {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d a \left (a +b \right ) \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{d \left (a^{2}+2 a b +b^{2}\right )}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{d \left (a^{2}+2 a b +b^{2}\right )}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{4 a \left (a +b \right )^{2} d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{4 a^{2} \left (a +b \right )^{2} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{4 a \left (a +b \right )^{2} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{4 a^{2} \left (a +b \right )^{2} d}\) | \(319\) |
1/d*(4*b/(a+b)^2*((-1/4*(a-b)/a*tanh(1/2*d*x+1/2*c)^2-1/4*(a+b)/a)/(tanh(1 /2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh (1/2*d*x+1/2*c)^2*b+a+b)+1/8*(3*a+b)/a/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tan h(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2)))+1/(a+b)^2*ln(tanh(1/2*d*x+1/2*c) ))
Leaf count of result is larger than twice the leaf count of optimal. 1192 vs. \(2 (87) = 174\).
Time = 0.32 (sec) , antiderivative size = 2376, normalized size of antiderivative = 24.00 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[-1/4*(4*(a*b + b^2)*cosh(d*x + c)^3 + 12*(a*b + b^2)*cosh(d*x + c)*sinh(d *x + c)^2 + 4*(a*b + b^2)*sinh(d*x + c)^3 - ((3*a^2 + a*b)*cosh(d*x + c)^4 + 4*(3*a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a^2 + a*b)*sinh(d*x + c)^4 + 2*(3*a^2 + 7*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a^2 + a*b)*co sh(d*x + c)^2 + 3*a^2 + 7*a*b + 2*b^2)*sinh(d*x + c)^2 + 3*a^2 + a*b + 4*( (3*a^2 + a*b)*cosh(d*x + c)^3 + (3*a^2 + 7*a*b + 2*b^2)*cosh(d*x + c))*sin h(d*x + c))*sqrt(-b/a)*log((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a - 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d* x + c)^2 + a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a - 2*b)*cos h(d*x + c))*sinh(d*x + c) + 4*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh( d*x + c)^2 + a*sinh(d*x + c)^3 + a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + a)*sinh(d*x + c))*sqrt(-b/a) + a)/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*s inh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a* cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2 *b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 4*(a*b + b^2)*cosh(d*x + c) + 4*( a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cos h(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 4*(a^2 *cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x +...
\[ \int \frac {\text {csch}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\operatorname {csch}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
\[ \int \frac {\text {csch}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
-(b*e^(3*d*x + 3*c) + b*e^(d*x + c))/(a^3*d + a^2*b*d + (a^3*d*e^(4*c) + a ^2*b*d*e^(4*c))*e^(4*d*x) + 2*(a^3*d*e^(2*c) + 3*a^2*b*d*e^(2*c) + 2*a*b^2 *d*e^(2*c))*e^(2*d*x)) - log((e^(d*x + c) + 1)*e^(-c))/(a^2*d + 2*a*b*d + b^2*d) + log((e^(d*x + c) - 1)*e^(-c))/(a^2*d + 2*a*b*d + b^2*d) + 2*integ rate(1/2*((3*a*b*e^(3*c) + b^2*e^(3*c))*e^(3*d*x) - (3*a*b*e^c + b^2*e^c)* e^(d*x))/(a^4 + 2*a^3*b + a^2*b^2 + (a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b ^2*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) + 4*a^3*b*e^(2*c) + 5*a^2*b^2*e^(2* c) + 2*a*b^3*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\text {csch}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4}{\mathrm {sinh}\left (c+d\,x\right )\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \]