Integrand size = 21, antiderivative size = 100 \[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {b (4 a+3 b) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 a^{5/2} (a+b)^{3/2} d}+\frac {\sinh (c+d x)}{a^2 d}+\frac {b^2 \sinh (c+d x)}{2 a^2 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )} \]
-1/2*b*(4*a+3*b)*arctan(sinh(d*x+c)*a^(1/2)/(a+b)^(1/2))/a^(5/2)/(a+b)^(3/ 2)/d+sinh(d*x+c)/a^2/d+1/2*b^2*sinh(d*x+c)/a^2/(a+b)/d/(a+b+a*sinh(d*x+c)^ 2)
Time = 0.59 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {-\frac {b (4 a+3 b) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}+\sqrt {a} \sinh (c+d x) \left (2+\frac {b^2}{(a+b) \left (a+b+a \sinh ^2(c+d x)\right )}\right )}{2 a^{5/2} d} \]
(-((b*(4*a + 3*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(a + b)^(3/ 2)) + Sqrt[a]*Sinh[c + d*x]*(2 + b^2/((a + b)*(a + b + a*Sinh[c + d*x]^2)) ))/(2*a^(5/2)*d)
Time = 0.32 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 4635, 300, 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.
| \(\displaystyle \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (i c+i d x) \left (a+b \sec (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4635 |
| \(\displaystyle \frac {\int \frac {\left (\sinh ^2(c+d x)+1\right )^2}{\left (a \sinh ^2(c+d x)+a+b\right )^2}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (\frac {1}{a^2}-\frac {2 a b \sinh ^2(c+d x)+b (2 a+b)}{a^2 \left (a \sinh ^2(c+d x)+a+b\right )^2}\right )d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b (4 a+3 b) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 a^{5/2} (a+b)^{3/2}}+\frac {b^2 \sinh (c+d x)}{2 a^2 (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {\sinh (c+d x)}{a^2}}{d}\) |
(-1/2*(b*(4*a + 3*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(a^(5/2) *(a + b)^(3/2)) + Sinh[c + d*x]/a^2 + (b^2*Sinh[c + d*x])/(2*a^2*(a + b)*( a + b + a*Sinh[c + d*x]^2)))/d
3.1.85.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ ))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && In tegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(236\) vs. \(2(88)=176\).
Time = 0.93 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.37
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {1}{a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 b \left (\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{2 a +2 b}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (4 a +3 b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{2 a +2 b}\right )}{a^{2}}}{d}\) | \(237\) |
| default | \(\frac {-\frac {1}{a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {1}{a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 b \left (\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{2 a +2 b}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (4 a +3 b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{2 a +2 b}\right )}{a^{2}}}{d}\) | \(237\) |
| risch | \(\frac {{\mathrm e}^{d x +c}}{2 a^{2} d}-\frac {{\mathrm e}^{-d x -c}}{2 a^{2} d}+\frac {b^{2} {\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \,a^{2} \left (a +b \right ) \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{\sqrt {-a^{2}-a b}\, \left (a +b \right ) d a}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{2}}+\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{\sqrt {-a^{2}-a b}\, \left (a +b \right ) d a}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{2}}\) | \(351\) |
1/d*(-1/a^2/(1+tanh(1/2*d*x+1/2*c))-1/a^2/(tanh(1/2*d*x+1/2*c)-1)-2/a^2*b* ((1/2*b/(a+b)*tanh(1/2*d*x+1/2*c)^3-1/2*b/(a+b)*tanh(1/2*d*x+1/2*c))/(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*ta nh(1/2*d*x+1/2*c)^2*b+a+b)+1/2*(4*a+3*b)/(a+b)*(1/2/(a+b)^(1/2)/a^(1/2)*ar ctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)-2*b^(1/2))/a^(1/2))+1/2/(a+b)^ (1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)+2*b^(1/2))/a^( 1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 1636 vs. \(2 (88) = 176\).
Time = 0.30 (sec) , antiderivative size = 3154, normalized size of antiderivative = 31.54 \[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[1/4*(2*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^6 + 12*(a^4 + 2*a^3*b + a^ 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(a^4 + 2*a^3*b + a^2*b^2)*sinh(d* x + c)^6 + 2*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^4 + 2*(a ^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3 + 15*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d* x + c)^2)*sinh(d*x + c)^4 - 2*a^4 - 4*a^3*b - 2*a^2*b^2 + 8*(5*(a^4 + 2*a^ 3*b + a^2*b^2)*cosh(d*x + c)^3 + (a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*co sh(d*x + c))*sinh(d*x + c)^3 - 2*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*co sh(d*x + c)^2 + 2*(15*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^4 - a^4 - 6* a^3*b - 11*a^2*b^2 - 6*a*b^3 + 6*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*co sh(d*x + c)^2)*sinh(d*x + c)^2 - ((4*a^2*b + 3*a*b^2)*cosh(d*x + c)^5 + 5* (4*a^2*b + 3*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^4 + (4*a^2*b + 3*a*b^2)*si nh(d*x + c)^5 + 2*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c)^3 + 2*(4*a^2* b + 11*a*b^2 + 6*b^3 + 5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c )^3 + 2*(5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^3 + 3*(4*a^2*b + 11*a*b^2 + 6 *b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (4*a^2*b + 3*a*b^2)*cosh(d*x + c) + (5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^4 + 4*a^2*b + 3*a*b^2 + 6*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(-a^2 - a*b)*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*(3*a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - 3*a - 2*b)*sinh( d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (3*a + 2*b)*cosh(d*x + c))*sinh(d*x...
\[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
\[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
-1/2*(a^2 + a*b - (a^2*e^(6*c) + a*b*e^(6*c))*e^(6*d*x) - (a^2*e^(4*c) + 5 *a*b*e^(4*c) + 6*b^2*e^(4*c))*e^(4*d*x) + (a^2*e^(2*c) + 5*a*b*e^(2*c) + 6 *b^2*e^(2*c))*e^(2*d*x))/((a^4*d*e^(5*c) + a^3*b*d*e^(5*c))*e^(5*d*x) + 2* (a^4*d*e^(3*c) + 3*a^3*b*d*e^(3*c) + 2*a^2*b^2*d*e^(3*c))*e^(3*d*x) + (a^4 *d*e^c + a^3*b*d*e^c)*e^(d*x)) - 1/2*integrate(2*((4*a*b*e^(3*c) + 3*b^2*e ^(3*c))*e^(3*d*x) + (4*a*b*e^c + 3*b^2*e^c)*e^(d*x))/(a^4 + a^3*b + (a^4*e ^(4*c) + a^3*b*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) + 3*a^3*b*e^(2*c) + 2*a ^2*b^2*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^2} \,d x \]