Integrand size = 21, antiderivative size = 154 \[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {3 b \left (4 (a+b)^2+(2 a+b)^2\right ) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 a^{7/2} (a+b)^{5/2} d}+\frac {\sinh (c+d x)}{a^3 d}-\frac {b^3 \sinh (c+d x)}{4 a^3 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac {3 b^2 (4 a+3 b) \sinh (c+d x)}{8 a^3 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )} \] Output:
-3/8*b*(4*(a+b)^2+(2*a+b)^2)*arctan(a^(1/2)*sinh(d*x+c)/(a+b)^(1/2))/a^(7/ 2)/(a+b)^(5/2)/d+sinh(d*x+c)/a^3/d-1/4*b^3*sinh(d*x+c)/a^3/(a+b)/d/(a+b+a* sinh(d*x+c)^2)^2+3/8*b^2*(4*a+3*b)*sinh(d*x+c)/a^3/(a+b)^2/d/(a+b+a*sinh(d *x+c)^2)
Time = 1.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90 \[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {-\frac {3 b \left (8 a^2+12 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}+\sqrt {a} \sinh (c+d x) \left (8+\frac {12 b^2}{(a+b) \left (a+b+a \sinh ^2(c+d x)\right )}-\frac {b^3 \left (5 (a+b)+3 a \sinh ^2(c+d x)\right )}{(a+b)^2 \left (a+b+a \sinh ^2(c+d x)\right )^2}\right )}{8 a^{7/2} d} \] Input:
Integrate[Cosh[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]
Output:
((-3*b*(8*a^2 + 12*a*b + 5*b^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b] ])/(a + b)^(5/2) + Sqrt[a]*Sinh[c + d*x]*(8 + (12*b^2)/((a + b)*(a + b + a *Sinh[c + d*x]^2)) - (b^3*(5*(a + b) + 3*a*Sinh[c + d*x]^2))/((a + b)^2*(a + b + a*Sinh[c + d*x]^2)^2)))/(8*a^(7/2)*d)
Time = 0.38 (sec) , antiderivative size = 146, 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.
| \(\displaystyle \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (i c+i d x) \left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4635 |
| \(\displaystyle \frac {\int \frac {\left (\sinh ^2(c+d x)+1\right )^3}{\left (a \sinh ^2(c+d x)+a+b\right )^3}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (\frac {1}{a^3}-\frac {3 a^2 b \sinh ^4(c+d x)+3 a b (2 a+b) \sinh ^2(c+d x)+b \left (3 a^2+3 b a+b^2\right )}{a^3 \left (a \sinh ^2(c+d x)+a+b\right )^3}\right )d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {3 b \left (4 (a+b)^2+(2 a+b)^2\right ) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 a^{7/2} (a+b)^{5/2}}-\frac {b^3 \sinh (c+d x)}{4 a^3 (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2}+\frac {3 b^2 (4 a+3 b) \sinh (c+d x)}{8 a^3 (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {\sinh (c+d x)}{a^3}}{d}\) |
Input:
Int[Cosh[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]
Output:
((-3*b*(4*(a + b)^2 + (2*a + b)^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(8*a^(7/2)*(a + b)^(5/2)) + Sinh[c + d*x]/a^3 - (b^3*Sinh[c + d*x])/ (4*a^3*(a + b)*(a + b + a*Sinh[c + d*x]^2)^2) + (3*b^2*(4*a + 3*b)*Sinh[c + d*x])/(8*a^3*(a + b)^2*(a + b + a*Sinh[c + d*x]^2)))/d
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. \(334\) vs. \(2(140)=280\).
Time = 2.32 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.18
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 b \left (\frac {\frac {b \left (12 a +7 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a +8 b}+\frac {3 b \left (4 a^{2}-7 a b -7 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a +b \right )^{2}}-\frac {3 b \left (4 a^{2}-7 a b -7 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a +b \right )^{2}}-\frac {b \left (12 a +7 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {3 \left (8 a^{2}+12 a b +5 b^{2}\right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right )}\right )}{a^{3}}-\frac {1}{a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(335\) |
| default | \(\frac {-\frac {1}{a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 b \left (\frac {\frac {b \left (12 a +7 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a +8 b}+\frac {3 b \left (4 a^{2}-7 a b -7 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a +b \right )^{2}}-\frac {3 b \left (4 a^{2}-7 a b -7 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a +b \right )^{2}}-\frac {b \left (12 a +7 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {3 \left (8 a^{2}+12 a b +5 b^{2}\right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right )}\right )}{a^{3}}-\frac {1}{a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(335\) |
| risch | \(\frac {{\mathrm e}^{d x +c}}{2 a^{3} d}-\frac {{\mathrm e}^{-d x -c}}{2 a^{3} d}+\frac {b^{2} {\mathrm e}^{d x +c} \left (12 a^{2} {\mathrm e}^{6 d x +6 c}+9 a b \,{\mathrm e}^{6 d x +6 c}+12 a^{2} {\mathrm e}^{4 d x +4 c}+49 a b \,{\mathrm e}^{4 d x +4 c}+28 b^{2} {\mathrm e}^{4 d x +4 c}-12 a^{2} {\mathrm e}^{2 d x +2 c}-49 a b \,{\mathrm e}^{2 d x +2 c}-28 b^{2} {\mathrm e}^{2 d x +2 c}-12 a^{2}-9 a b \right )}{4 a^{3} \left (a +b \right )^{2} d \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}-\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}-\frac {9 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d \,a^{2}}-\frac {15 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d \,a^{3}}+\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}+\frac {9 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d \,a^{2}}+\frac {15 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d \,a^{3}}\) | \(587\) |
Input:
int(cosh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/a^3/(tanh(1/2*d*x+1/2*c)+1)-2/a^3*b*((1/8*b*(12*a+7*b)/(a+b)*tanh( 1/2*d*x+1/2*c)^7+3/8*b*(4*a^2-7*a*b-7*b^2)/(a+b)^2*tanh(1/2*d*x+1/2*c)^5-3 /8*b*(4*a^2-7*a*b-7*b^2)/(a+b)^2*tanh(1/2*d*x+1/2*c)^3-1/8*b*(12*a+7*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*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2+3/8*(8*a^2+12*a *b+5*b^2)/(a^2+2*a*b+b^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))+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))))-1/a^3/(tanh(1 /2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 5415 vs. \(2 (140) = 280\).
Time = 0.48 (sec) , antiderivative size = 9857, normalized size of antiderivative = 64.01 \[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(cosh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(cosh(d*x+c)/(a+b*sech(d*x+c)**2)**3,x)
Output:
Timed out
\[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \] Input:
integrate(cosh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
Output:
-1/4*(2*a^4 + 4*a^3*b + 2*a^2*b^2 - 2*(a^4*e^(10*c) + 2*a^3*b*e^(10*c) + a ^2*b^2*e^(10*c))*e^(10*d*x) - (6*a^4*e^(8*c) + 28*a^3*b*e^(8*c) + 50*a^2*b ^2*e^(8*c) + 25*a*b^3*e^(8*c))*e^(8*d*x) - (4*a^4*e^(6*c) + 24*a^3*b*e^(6* c) + 80*a^2*b^2*e^(6*c) + 129*a*b^3*e^(6*c) + 60*b^4*e^(6*c))*e^(6*d*x) + (4*a^4*e^(4*c) + 24*a^3*b*e^(4*c) + 80*a^2*b^2*e^(4*c) + 129*a*b^3*e^(4*c) + 60*b^4*e^(4*c))*e^(4*d*x) + (6*a^4*e^(2*c) + 28*a^3*b*e^(2*c) + 50*a^2* b^2*e^(2*c) + 25*a*b^3*e^(2*c))*e^(2*d*x))/((a^7*d*e^(9*c) + 2*a^6*b*d*e^( 9*c) + a^5*b^2*d*e^(9*c))*e^(9*d*x) + 4*(a^7*d*e^(7*c) + 4*a^6*b*d*e^(7*c) + 5*a^5*b^2*d*e^(7*c) + 2*a^4*b^3*d*e^(7*c))*e^(7*d*x) + 2*(3*a^7*d*e^(5* c) + 14*a^6*b*d*e^(5*c) + 27*a^5*b^2*d*e^(5*c) + 24*a^4*b^3*d*e^(5*c) + 8* a^3*b^4*d*e^(5*c))*e^(5*d*x) + 4*(a^7*d*e^(3*c) + 4*a^6*b*d*e^(3*c) + 5*a^ 5*b^2*d*e^(3*c) + 2*a^4*b^3*d*e^(3*c))*e^(3*d*x) + (a^7*d*e^c + 2*a^6*b*d* e^c + a^5*b^2*d*e^c)*e^(d*x)) - 1/2*integrate(3/2*((8*a^2*b*e^(3*c) + 12*a *b^2*e^(3*c) + 5*b^3*e^(3*c))*e^(3*d*x) + (8*a^2*b*e^c + 12*a*b^2*e^c + 5* b^3*e^c)*e^(d*x))/(a^6 + 2*a^5*b + a^4*b^2 + (a^6*e^(4*c) + 2*a^5*b*e^(4*c ) + a^4*b^2*e^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) + 4*a^5*b*e^(2*c) + 5*a^4* b^2*e^(2*c) + 2*a^3*b^3*e^(2*c))*e^(2*d*x)), x)
Exception generated. \[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cosh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^3} \,d x \] Input:
int(cosh(c + d*x)/(a + b/cosh(c + d*x)^2)^3,x)
Output:
int(cosh(c + d*x)/(a + b/cosh(c + d*x)^2)^3, x)
Time = 0.41 (sec) , antiderivative size = 9351, normalized size of antiderivative = 60.72 \[ \int \frac {\cosh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(cosh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x)
Output:
(48*e**(9*c + 9*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b ) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**4*b + 72*e**(9*c + 9*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2* sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt( b)*sqrt(a + b) + a + 2*b)))*a**3*b**2 + 30*e**(9*c + 9*d*x)*sqrt(b)*sqrt(a )*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/ (sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**2*b**3 + 192*e**(7*c + 7*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)* atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a** 4*b + 672*e**(7*c + 7*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt (a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**3*b**2 + 696*e**(7*c + 7*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sq rt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**2*b**3 + 240*e**(7*c + 7*d*x)*sqr t(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a*b**4 + 288*e **(5*c + 5*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2* b)))*a**4*b + 1200*e**(5*c + 5*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqr t(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(...