Integrand size = 17, antiderivative size = 88 \[ \int \cosh ^2(a+b x) \text {sech}^3(c+b x) \, dx=\frac {\arctan (\sinh (c+b x)) \cosh (2 (a-c))}{b}-\frac {\arctan (\sinh (c+b x)) \sinh ^2(a-c)}{2 b}-\frac {\text {sech}(c+b x) \sinh (2 (a-c))}{b}-\frac {\text {sech}(c+b x) \sinh ^2(a-c) \tanh (c+b x)}{2 b} \] Output:
arctan(sinh(b*x+c))*cosh(2*a-2*c)/b-1/2*arctan(sinh(b*x+c))*sinh(a-c)^2/b- sech(b*x+c)*sinh(2*a-2*c)/b-1/2*sech(b*x+c)*sinh(a-c)^2*tanh(b*x+c)/b
Time = 0.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.23 \[ \int \cosh ^2(a+b x) \text {sech}^3(c+b x) \, dx=\frac {12 \arctan \left (\tanh \left (\frac {1}{2} (c+b x)\right )\right ) \cosh (2 (a-c))+\text {sech}^2(c+b x) \left (2 \arctan \left (\tanh \left (\frac {1}{2} (c+b x)\right )\right )+2 \arctan \left (\tanh \left (\frac {1}{2} (c+b x)\right )\right ) \cosh (2 (c+b x))-3 \sinh (2 a-3 c-b x)-5 \sinh (2 a-c+b x)+2 \sinh (c+b x)\right )}{8 b} \] Input:
Integrate[Cosh[a + b*x]^2*Sech[c + b*x]^3,x]
Output:
(12*ArcTan[Tanh[(c + b*x)/2]]*Cosh[2*(a - c)] + Sech[c + b*x]^2*(2*ArcTan[ Tanh[(c + b*x)/2]] + 2*ArcTan[Tanh[(c + b*x)/2]]*Cosh[2*(c + b*x)] - 3*Sin h[2*a - 3*c - b*x] - 5*Sinh[2*a - c + b*x] + 2*Sinh[c + b*x]))/(8*b)
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.
| \(\displaystyle \int \cosh ^2(a+b x) \text {sech}^3(b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \cosh ^2(a+b x) \text {sech}^3(b x+c)dx\) |
Input:
Int[Cosh[a + b*x]^2*Sech[c + b*x]^3,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 4.22 (sec) , antiderivative size = 321, normalized size of antiderivative = 3.65
| method | result | size |
| risch | \(-\frac {\left (5 \,{\mathrm e}^{2 b x +6 a +2 c}-2 \,{\mathrm e}^{2 b x +4 a +4 c}-3 \,{\mathrm e}^{2 b x +2 a +6 c}+3 \,{\mathrm e}^{6 a}+2 \,{\mathrm e}^{4 a +2 c}-5 \,{\mathrm e}^{2 a +4 c}\right ) {\mathrm e}^{b x -c}}{4 \left ({\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{2} b}+\frac {3 i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 a}}{8 b}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{2 a +2 c}}{4 b}+\frac {3 i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 c}}{8 b}-\frac {3 i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 a}}{8 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{2 a +2 c}}{4 b}-\frac {3 i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 c}}{8 b}\) | \(321\) |
Input:
int(cosh(b*x+a)^2*sech(b*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-1/4/(exp(2*b*x+2*a+2*c)+exp(2*a))^2/b*(5*exp(2*b*x+6*a+2*c)-2*exp(2*b*x+4 *a+4*c)-3*exp(2*b*x+2*a+6*c)+3*exp(6*a)+2*exp(4*a+2*c)-5*exp(2*a+4*c))*exp (b*x-c)+3/8*I*ln(exp(b*x+a)+I*exp(a-c))/b*exp(-2*c-2*a)*exp(4*a)+1/4*I*ln( exp(b*x+a)+I*exp(a-c))/b*exp(-2*c-2*a)*exp(2*a+2*c)+3/8*I*ln(exp(b*x+a)+I* exp(a-c))/b*exp(-2*c-2*a)*exp(4*c)-3/8*I*ln(exp(b*x+a)-I*exp(a-c))/b*exp(- 2*c-2*a)*exp(4*a)-1/4*I*ln(exp(b*x+a)-I*exp(a-c))/b*exp(-2*c-2*a)*exp(2*a+ 2*c)-3/8*I*ln(exp(b*x+a)-I*exp(a-c))/b*exp(-2*c-2*a)*exp(4*c)
Leaf count of result is larger than twice the leaf count of optimal. 2173 vs. \(2 (84) = 168\).
Time = 0.12 (sec) , antiderivative size = 2173, normalized size of antiderivative = 24.69 \[ \int \cosh ^2(a+b x) \text {sech}^3(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cosh(b*x+a)^2*sech(b*x+c)^3,x, algorithm="fricas")
Output:
-1/4*((5*cosh(b*x + c)^3 + 3*cosh(b*x + c))*sinh(-a + c)^4 + (5*cosh(-a + c)^4 - 2*cosh(-a + c)^2 - 3)*cosh(b*x + c)^3 + (5*cosh(-a + c)^4 - 20*cosh (-a + c)*sinh(-a + c)^3 + 5*sinh(-a + c)^4 + 2*(15*cosh(-a + c)^2 - 1)*sin h(-a + c)^2 - 2*cosh(-a + c)^2 - 4*(5*cosh(-a + c)^3 - cosh(-a + c))*sinh( -a + c) - 3)*sinh(b*x + c)^3 - 4*(5*cosh(b*x + c)^3*cosh(-a + c) + 3*cosh( b*x + c)*cosh(-a + c))*sinh(-a + c)^3 - 3*(20*cosh(b*x + c)*cosh(-a + c)*s inh(-a + c)^3 - 5*cosh(b*x + c)*sinh(-a + c)^4 - 2*(15*cosh(-a + c)^2 - 1) *cosh(b*x + c)*sinh(-a + c)^2 + 4*(5*cosh(-a + c)^3 - cosh(-a + c))*cosh(b *x + c)*sinh(-a + c) - (5*cosh(-a + c)^4 - 2*cosh(-a + c)^2 - 3)*cosh(b*x + c))*sinh(b*x + c)^2 + 2*((15*cosh(-a + c)^2 - 1)*cosh(b*x + c)^3 + (9*co sh(-a + c)^2 + 1)*cosh(b*x + c))*sinh(-a + c)^2 - ((3*cosh(-a + c)^4 + 2*c osh(-a + c)^2 + 3)*cosh(b*x + c)^4 + (3*cosh(-a + c)^4 - 12*cosh(-a + c)*s inh(-a + c)^3 + 3*sinh(-a + c)^4 + 2*(9*cosh(-a + c)^2 + 1)*sinh(-a + c)^2 + 2*cosh(-a + c)^2 - 4*(3*cosh(-a + c)^3 + cosh(-a + c))*sinh(-a + c) + 3 )*sinh(b*x + c)^4 + 3*(cosh(b*x + c)^4 + 2*cosh(b*x + c)^2 + 1)*sinh(-a + c)^4 + 3*cosh(-a + c)^4 - 4*(12*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c)^3 - 3*cosh(b*x + c)*sinh(-a + c)^4 - 2*(9*cosh(-a + c)^2 + 1)*cosh(b*x + c)* sinh(-a + c)^2 + 4*(3*cosh(-a + c)^3 + cosh(-a + c))*cosh(b*x + c)*sinh(-a + c) - (3*cosh(-a + c)^4 + 2*cosh(-a + c)^2 + 3)*cosh(b*x + c))*sinh(b*x + c)^3 - 12*(cosh(b*x + c)^4*cosh(-a + c) + 2*cosh(b*x + c)^2*cosh(-a +...
\[ \int \cosh ^2(a+b x) \text {sech}^3(c+b x) \, dx=\int \cosh ^{2}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (b x + c \right )}\, dx \] Input:
integrate(cosh(b*x+a)**2*sech(b*x+c)**3,x)
Output:
Integral(cosh(a + b*x)**2*sech(b*x + c)**3, x)
Time = 0.13 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.77 \[ \int \cosh ^2(a+b x) \text {sech}^3(c+b x) \, dx=-\frac {{\left (3 \, e^{\left (4 \, a\right )} + 2 \, e^{\left (2 \, a + 2 \, c\right )} + 3 \, e^{\left (4 \, c\right )}\right )} \arctan \left (e^{\left (-b x - c\right )}\right ) e^{\left (-2 \, a - 2 \, c\right )}}{4 \, b} - \frac {{\left (5 \, e^{\left (4 \, a + 2 \, c\right )} - 2 \, e^{\left (2 \, a + 4 \, c\right )} - 3 \, e^{\left (6 \, c\right )}\right )} e^{\left (-b x - a\right )} + {\left (3 \, e^{\left (6 \, a\right )} + 2 \, e^{\left (4 \, a + 2 \, c\right )} - 5 \, e^{\left (2 \, a + 4 \, c\right )}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{4 \, b {\left (2 \, e^{\left (-2 \, b x + a + 3 \, c\right )} + e^{\left (-4 \, b x + a + c\right )} + e^{\left (a + 5 \, c\right )}\right )}} \] Input:
integrate(cosh(b*x+a)^2*sech(b*x+c)^3,x, algorithm="maxima")
Output:
-1/4*(3*e^(4*a) + 2*e^(2*a + 2*c) + 3*e^(4*c))*arctan(e^(-b*x - c))*e^(-2* a - 2*c)/b - 1/4*((5*e^(4*a + 2*c) - 2*e^(2*a + 4*c) - 3*e^(6*c))*e^(-b*x - a) + (3*e^(6*a) + 2*e^(4*a + 2*c) - 5*e^(2*a + 4*c))*e^(-3*b*x - 3*a))/( b*(2*e^(-2*b*x + a + 3*c) + e^(-4*b*x + a + c) + e^(a + 5*c)))
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.62 \[ \int \cosh ^2(a+b x) \text {sech}^3(c+b x) \, dx=\frac {{\left (3 \, e^{\left (4 \, a\right )} + 2 \, e^{\left (2 \, a + 2 \, c\right )} + 3 \, e^{\left (4 \, c\right )}\right )} \arctan \left (e^{\left (b x + c\right )}\right ) e^{\left (-2 \, a - 2 \, c\right )}}{4 \, b} - \frac {{\left (5 \, e^{\left (3 \, b x + 4 \, a + 2 \, c\right )} - 2 \, e^{\left (3 \, b x + 2 \, a + 4 \, c\right )} - 3 \, e^{\left (3 \, b x + 6 \, c\right )} + 3 \, e^{\left (b x + 4 \, a\right )} + 2 \, e^{\left (b x + 2 \, a + 2 \, c\right )} - 5 \, e^{\left (b x + 4 \, c\right )}\right )} e^{\left (-2 \, a - c\right )}}{4 \, b {\left (e^{\left (2 \, b x + 2 \, c\right )} + 1\right )}^{2}} \] Input:
integrate(cosh(b*x+a)^2*sech(b*x+c)^3,x, algorithm="giac")
Output:
1/4*(3*e^(4*a) + 2*e^(2*a + 2*c) + 3*e^(4*c))*arctan(e^(b*x + c))*e^(-2*a - 2*c)/b - 1/4*(5*e^(3*b*x + 4*a + 2*c) - 2*e^(3*b*x + 2*a + 4*c) - 3*e^(3 *b*x + 6*c) + 3*e^(b*x + 4*a) + 2*e^(b*x + 2*a + 2*c) - 5*e^(b*x + 4*c))*e ^(-2*a - c)/(b*(e^(2*b*x + 2*c) + 1)^2)
Timed out. \[ \int \cosh ^2(a+b x) \text {sech}^3(c+b x) \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{{\mathrm {cosh}\left (c+b\,x\right )}^3} \,d x \] Input:
int(cosh(a + b*x)^2/cosh(c + b*x)^3,x)
Output:
int(cosh(a + b*x)^2/cosh(c + b*x)^3, x)
Time = 0.29 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.44 \[ \int \cosh ^2(a+b x) \text {sech}^3(c+b x) \, dx=\frac {3 e^{4 b x +4 a +4 c} \mathit {atan} \left (e^{b x +c}\right )+2 e^{4 b x +2 a +6 c} \mathit {atan} \left (e^{b x +c}\right )+3 e^{4 b x +8 c} \mathit {atan} \left (e^{b x +c}\right )+6 e^{2 b x +4 a +2 c} \mathit {atan} \left (e^{b x +c}\right )+4 e^{2 b x +2 a +4 c} \mathit {atan} \left (e^{b x +c}\right )+6 e^{2 b x +6 c} \mathit {atan} \left (e^{b x +c}\right )+3 e^{4 a} \mathit {atan} \left (e^{b x +c}\right )+2 e^{2 a +2 c} \mathit {atan} \left (e^{b x +c}\right )+3 e^{4 c} \mathit {atan} \left (e^{b x +c}\right )-5 e^{3 b x +4 a +3 c}+2 e^{3 b x +2 a +5 c}+3 e^{3 b x +7 c}-3 e^{b x +4 a +c}-2 e^{b x +2 a +3 c}+5 e^{b x +5 c}}{4 e^{2 a +2 c} b \left (e^{4 b x +4 c}+2 e^{2 b x +2 c}+1\right )} \] Input:
int(cosh(b*x+a)^2*sech(b*x+c)^3,x)
Output:
(3*e**(4*a + 4*b*x + 4*c)*atan(e**(b*x + c)) + 2*e**(2*a + 4*b*x + 6*c)*at an(e**(b*x + c)) + 3*e**(4*b*x + 8*c)*atan(e**(b*x + c)) + 6*e**(4*a + 2*b *x + 2*c)*atan(e**(b*x + c)) + 4*e**(2*a + 2*b*x + 4*c)*atan(e**(b*x + c)) + 6*e**(2*b*x + 6*c)*atan(e**(b*x + c)) + 3*e**(4*a)*atan(e**(b*x + c)) + 2*e**(2*a + 2*c)*atan(e**(b*x + c)) + 3*e**(4*c)*atan(e**(b*x + c)) - 5*e **(4*a + 3*b*x + 3*c) + 2*e**(2*a + 3*b*x + 5*c) + 3*e**(3*b*x + 7*c) - 3* e**(4*a + b*x + c) - 2*e**(2*a + b*x + 3*c) + 5*e**(b*x + 5*c))/(4*e**(2*a + 2*c)*b*(e**(4*b*x + 4*c) + 2*e**(2*b*x + 2*c) + 1))