Integrand size = 21, antiderivative size = 84 \[ \int \frac {\sinh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{2 a^{5/2} d}+\frac {3 \cosh (c+d x)}{2 a^2 d}-\frac {\cosh ^3(c+d x)}{2 a d \left (b+a \cosh ^2(c+d x)\right )} \]
3/2*cosh(d*x+c)/a^2/d-1/2*cosh(d*x+c)^3/a/d/(b+a*cosh(d*x+c)^2)-3/2*arctan (cosh(d*x+c)*a^(1/2)/b^(1/2))*b^(1/2)/a^(5/2)/d
Result contains complex when optimal does not.
Time = 3.03 (sec) , antiderivative size = 479, normalized size of antiderivative = 5.70 \[ \int \frac {\sinh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}^4(c+d x) \left (\frac {32 \cosh (c) \cosh (d x)}{a^2}+\frac {32 b \cosh (c+d x)}{a^2 (a+2 b+a \cosh (2 (c+d x)))}+\frac {2 \left (-\left (\left (a^2+24 b^2\right ) \arctan \left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )\right )-a^2 \arctan \left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )-24 b^2 \arctan \left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )+a^2 \arctan \left (\frac {\sqrt {a}-i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+a^2 \arctan \left (\frac {\sqrt {a}+i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+16 \sqrt {a} b^{3/2} \sinh (c) \sinh (d x)\right )}{a^{5/2} b^{3/2}}\right )}{128 d \left (a+b \text {sech}^2(c+d x)\right )^2} \]
((a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]^4*((32*Cosh[c]*Cosh[d*x]) /a^2 + (32*b*Cosh[c + d*x])/(a^2*(a + 2*b + a*Cosh[2*(c + d*x)])) + (2*(-( (a^2 + 24*b^2)*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*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]] - 24 *b^2*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*ArcTan[(Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + a^2*ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)/ 2])/Sqrt[b]] + 16*Sqrt[a]*b^(3/2)*Sinh[c]*Sinh[d*x]))/(a^(5/2)*b^(3/2))))/ (128*d*(a + b*Sech[c + d*x]^2)^2)
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 26, 4621, 252, 262, 218}
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.
\(\displaystyle \int \frac {\sinh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
\(\Big \downarrow \) 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 |
\(\displaystyle -i \int \frac {\sin (i c+i d x)}{\left (b \sec (i c+i d x)^2+a\right )^2}dx\) |
\(\Big \downarrow \) 4621 |
\(\displaystyle \frac {\int \frac {\cosh ^4(c+d x)}{\left (a \cosh ^2(c+d x)+b\right )^2}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {\frac {3 \int \frac {\cosh ^2(c+d x)}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{2 a}-\frac {\cosh ^3(c+d x)}{2 a \left (a \cosh ^2(c+d x)+b\right )}}{d}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {\frac {3 \left (\frac {\cosh (c+d x)}{a}-\frac {b \int \frac {1}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{a}\right )}{2 a}-\frac {\cosh ^3(c+d x)}{2 a \left (a \cosh ^2(c+d x)+b\right )}}{d}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {3 \left (\frac {\cosh (c+d x)}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{a^{3/2}}\right )}{2 a}-\frac {\cosh ^3(c+d x)}{2 a \left (a \cosh ^2(c+d x)+b\right )}}{d}\) |
((3*(-((Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/a^(3/2)) + Cosh[c + d*x]/a))/(2*a) - Cosh[c + d*x]^3/(2*a*(b + a*Cosh[c + d*x]^2)))/d
3.1.36.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, 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 = 17.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(-\frac {-\frac {1}{a^{2} \operatorname {sech}\left (d x +c \right )}-\frac {b \left (\frac {\operatorname {sech}\left (d x +c \right )}{2 a +2 b \operatorname {sech}\left (d x +c \right )^{2}}+\frac {3 \arctan \left (\frac {b \,\operatorname {sech}\left (d x +c \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}}{d}\) | \(70\) |
default | \(-\frac {-\frac {1}{a^{2} \operatorname {sech}\left (d x +c \right )}-\frac {b \left (\frac {\operatorname {sech}\left (d x +c \right )}{2 a +2 b \operatorname {sech}\left (d x +c \right )^{2}}+\frac {3 \arctan \left (\frac {b \,\operatorname {sech}\left (d x +c \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}}{d}\) | \(70\) |
risch | \(\frac {{\mathrm e}^{d x +c}}{2 a^{2} d}+\frac {{\mathrm e}^{-d x -c}}{2 a^{2} d}+\frac {{\mathrm e}^{d x +c} b \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d \,a^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{4 a^{3} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{4 a^{3} d}\) | \(183\) |
-1/d*(-1/a^2/sech(d*x+c)-1/a^2*b*(1/2*sech(d*x+c)/(a+b*sech(d*x+c)^2)+3/2/ (a*b)^(1/2)*arctan(b*sech(d*x+c)/(a*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (70) = 140\).
Time = 0.29 (sec) , antiderivative size = 1780, normalized size of antiderivative = 21.19 \[ \int \frac {\sinh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[1/4*(2*a*cosh(d*x + c)^6 + 12*a*cosh(d*x + c)*sinh(d*x + c)^5 + 2*a*sinh( d*x + c)^6 + 6*(a + 2*b)*cosh(d*x + c)^4 + 6*(5*a*cosh(d*x + c)^2 + a + 2* b)*sinh(d*x + c)^4 + 8*(5*a*cosh(d*x + c)^3 + 3*(a + 2*b)*cosh(d*x + c))*s inh(d*x + c)^3 + 6*(a + 2*b)*cosh(d*x + c)^2 + 6*(5*a*cosh(d*x + c)^4 + 6* (a + 2*b)*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 3*(a*cosh(d*x + c)^ 5 + 5*a*cosh(d*x + c)*sinh(d*x + c)^4 + a*sinh(d*x + c)^5 + 2*(a + 2*b)*co sh(d*x + c)^3 + 2*(5*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^3 + 2*(5*a *cosh(d*x + c)^3 + 3*(a + 2*b)*cosh(d*x + c))*sinh(d*x + c)^2 + a*cosh(d*x + c) + (5*a*cosh(d*x + c)^4 + 6*(a + 2*b)*cosh(d*x + c)^2 + a)*sinh(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)*cosh(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)*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) + a)) + 12*(a*cosh(d*x + c)^5 + 2*(a + 2*b)*cosh (d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + 2*a)/(a^3*d*cosh(d* x + c)^5 + 5*a^3*d*cosh(d*x + c)*sinh(d*x + c)^4 + a^3*d*sinh(d*x + c)^...
\[ \int \frac {\sinh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\sinh {\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
\[ \int \frac {\sinh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
1/2*(3*(a*e^(4*c) + 2*b*e^(4*c))*e^(4*d*x) + 3*(a*e^(2*c) + 2*b*e^(2*c))*e ^(2*d*x) + a*e^(6*d*x + 6*c) + a)/(a^3*d*e^(5*d*x + 5*c) + a^3*d*e^(d*x + c) + 2*(a^3*d*e^(3*c) + 2*a^2*b*d*e^(3*c))*e^(3*d*x)) - 1/2*integrate(6*(b *e^(3*d*x + 3*c) - b*e^(d*x + c))/(a^3*e^(4*d*x + 4*c) + a^3 + 2*(a^3*e^(2 *c) + 2*a^2*b*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\sinh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Time = 2.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.85 \[ \int \frac {\sinh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {b\,\mathrm {cosh}\left (c+d\,x\right )}{2\,\left (d\,a^3\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\,d\,a^2\right )}+\frac {\mathrm {cosh}\left (c+d\,x\right )}{a^2\,d}-\frac {3\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\mathrm {cosh}\left (c+d\,x\right )}{\sqrt {b}}\right )}{2\,a^{5/2}\,d} \]