Integrand size = 18, antiderivative size = 86 \[ \int \text {sech}^3(c-b x) \sinh ^2(a+b x) \, dx=\frac {\arctan (\sinh (c-b x)) \cosh ^2(a+c)}{2 b}-\frac {\arctan (\sinh (c-b x)) \cosh (2 (a+c))}{b}-\frac {\text {sech}(c-b x) \sinh (2 (a+c))}{b}+\frac {\cosh ^2(a+c) \text {sech}(c-b x) \tanh (c-b x)}{2 b} \] Output:
-1/2*arctan(sinh(b*x-c))*cosh(a+c)^2/b+arctan(sinh(b*x-c))*cosh(2*a+2*c)/b -sech(b*x-c)*sinh(2*a+2*c)/b-1/2*cosh(a+c)^2*sech(b*x-c)*tanh(b*x-c)/b
Time = 0.33 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.28 \[ \int \text {sech}^3(c-b x) \sinh ^2(a+b x) \, dx=\frac {-12 \arctan \left (\tanh \left (\frac {1}{2} (c-b x)\right )\right ) \cosh (2 (a+c))+\text {sech}^2(c-b x) \left (2 \arctan \left (\tanh \left (\frac {1}{2} (c-b x)\right )\right )+2 \arctan \left (\tanh \left (\frac {1}{2} (c-b x)\right )\right ) \cosh (2 (c-b x))+2 \sinh (c-b x)-3 \sinh (2 a+3 c-b x)-5 \sinh (2 a+c+b x)\right )}{8 b} \] Input:
Integrate[Sech[c - b*x]^3*Sinh[a + b*x]^2,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)] + 2*Si nh[c - b*x] - 3*Sinh[2*a + 3*c - b*x] - 5*Sinh[2*a + 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.
\(\displaystyle \int \sinh ^2(a+b x) \text {sech}^3(c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sinh ^2(a+b x) \text {sech}^3(c-b x)dx\) |
Input:
Int[Sech[c - b*x]^3*Sinh[a + b*x]^2,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 4.70 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.62
method | result | size |
risch | \(-\frac {\left (3 \,{\mathrm e}^{6 a +6 c}+5 \,{\mathrm e}^{2 b x +6 a +4 c}-2 \,{\mathrm e}^{4 a +4 c}+2 \,{\mathrm e}^{2 b x +4 a +2 c}-5 \,{\mathrm e}^{2 a +2 c}-3 \,{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x -c}}{4 \left ({\mathrm e}^{2 a +2 c}+{\mathrm e}^{2 b x +2 a}\right )^{2} b}+\frac {3 i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a +c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 a +4 c}}{8 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a +c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{2 a +2 c}}{4 b}+\frac {3 i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a +c}\right ) {\mathrm e}^{-2 c -2 a}}{8 b}-\frac {3 i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a +c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 a +4 c}}{8 b}+\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a +c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{2 a +2 c}}{4 b}-\frac {3 i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a +c}\right ) {\mathrm e}^{-2 c -2 a}}{8 b}\) | \(311\) |
Input:
int(sech(b*x-c)^3*sinh(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/4/(exp(2*a+2*c)+exp(2*b*x+2*a))^2/b*(3*exp(6*a+6*c)+5*exp(2*b*x+6*a+4*c )-2*exp(4*a+4*c)+2*exp(2*b*x+4*a+2*c)-5*exp(2*a+2*c)-3*exp(2*b*x+2*a))*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+4*c)-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)-3/8*I*ln(exp(b*x+a)-I*exp(a+c))/b*exp(-2*c-2 *a)*exp(4*a+4*c)+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)
Leaf count of result is larger than twice the leaf count of optimal. 3915 vs. \(2 (90) = 180\).
Time = 0.12 (sec) , antiderivative size = 3915, normalized size of antiderivative = 45.52 \[ \int \text {sech}^3(c-b x) \sinh ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(sech(b*x-c)^3*sinh(b*x+a)^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \text {sech}^3(c-b x) \sinh ^2(a+b x) \, dx=\int \sinh ^{2}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (b x - c \right )}\, dx \] Input:
integrate(sech(b*x-c)**3*sinh(b*x+a)**2,x)
Output:
Integral(sinh(a + b*x)**2*sech(b*x - c)**3, x)
Time = 0.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.77 \[ \int \text {sech}^3(c-b x) \sinh ^2(a+b x) \, dx=-\frac {{\left (3 \, e^{\left (4 \, a + 4 \, c\right )} - 2 \, e^{\left (2 \, a + 2 \, c\right )} + 3\right )} \arctan \left (e^{\left (-b x + c\right )}\right ) e^{\left (-2 \, a - 2 \, c\right )}}{4 \, b} - \frac {{\left (5 \, e^{\left (4 \, a + 4 \, c\right )} + 2 \, e^{\left (2 \, a + 2 \, c\right )} - 3\right )} e^{\left (-b x - a\right )} + {\left (3 \, e^{\left (6 \, a + 6 \, c\right )} - 2 \, e^{\left (4 \, a + 4 \, c\right )} - 5 \, e^{\left (2 \, a + 2 \, c\right )}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{4 \, b {\left (2 \, e^{\left (-2 \, b x + a + 3 \, c\right )} + e^{\left (-4 \, b x + a + 5 \, c\right )} + e^{\left (a + c\right )}\right )}} \] Input:
integrate(sech(b*x-c)^3*sinh(b*x+a)^2,x, algorithm="maxima")
Output:
-1/4*(3*e^(4*a + 4*c) - 2*e^(2*a + 2*c) + 3)*arctan(e^(-b*x + c))*e^(-2*a - 2*c)/b - 1/4*((5*e^(4*a + 4*c) + 2*e^(2*a + 2*c) - 3)*e^(-b*x - a) + (3* e^(6*a + 6*c) - 2*e^(4*a + 4*c) - 5*e^(2*a + 2*c))*e^(-3*b*x - 3*a))/(b*(2 *e^(-2*b*x + a + 3*c) + e^(-4*b*x + a + 5*c) + e^(a + c)))
Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.65 \[ \int \text {sech}^3(c-b x) \sinh ^2(a+b x) \, dx=\frac {{\left (3 \, e^{\left (4 \, a + 4 \, c\right )} - 2 \, e^{\left (2 \, a + 2 \, c\right )} + 3\right )} \arctan \left (e^{\left (b x - c\right )}\right ) e^{\left (-2 \, a - 2 \, c\right )}}{4 \, b} + \frac {{\left (3 \, e^{\left (3 \, b x\right )} - 5 \, e^{\left (3 \, b x + 4 \, a + 4 \, c\right )} - 2 \, e^{\left (3 \, b x + 2 \, a + 2 \, c\right )} - 3 \, e^{\left (b x + 4 \, a + 6 \, c\right )} + 2 \, e^{\left (b x + 2 \, a + 4 \, c\right )} + 5 \, e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-2 \, a - c\right )}}{4 \, b {\left (e^{\left (2 \, b x\right )} + e^{\left (2 \, c\right )}\right )}^{2}} \] Input:
integrate(sech(b*x-c)^3*sinh(b*x+a)^2,x, algorithm="giac")
Output:
1/4*(3*e^(4*a + 4*c) - 2*e^(2*a + 2*c) + 3)*arctan(e^(b*x - c))*e^(-2*a - 2*c)/b + 1/4*(3*e^(3*b*x) - 5*e^(3*b*x + 4*a + 4*c) - 2*e^(3*b*x + 2*a + 2 *c) - 3*e^(b*x + 4*a + 6*c) + 2*e^(b*x + 2*a + 4*c) + 5*e^(b*x + 2*c))*e^( -2*a - c)/(b*(e^(2*b*x) + e^(2*c))^2)
Timed out. \[ \int \text {sech}^3(c-b x) \sinh ^2(a+b x) \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{{\mathrm {cosh}\left (c-b\,x\right )}^3} \,d x \] Input:
int(sinh(a + b*x)^2/cosh(c - b*x)^3,x)
Output:
int(sinh(a + b*x)^2/cosh(c - b*x)^3, x)
Time = 0.27 (sec) , antiderivative size = 339, normalized size of antiderivative = 3.94 \[ \int \text {sech}^3(c-b x) \sinh ^2(a+b x) \, dx=\frac {3 e^{4 b x +4 a +4 c} \mathit {atan} \left (\frac {e^{b x}}{e^{c}}\right )-2 e^{4 b x +2 a +2 c} \mathit {atan} \left (\frac {e^{b x}}{e^{c}}\right )+3 e^{4 b x} \mathit {atan} \left (\frac {e^{b x}}{e^{c}}\right )+6 e^{2 b x +4 a +6 c} \mathit {atan} \left (\frac {e^{b x}}{e^{c}}\right )-4 e^{2 b x +2 a +4 c} \mathit {atan} \left (\frac {e^{b x}}{e^{c}}\right )+6 e^{2 b x +2 c} \mathit {atan} \left (\frac {e^{b x}}{e^{c}}\right )+3 e^{4 a +8 c} \mathit {atan} \left (\frac {e^{b x}}{e^{c}}\right )-2 e^{2 a +6 c} \mathit {atan} \left (\frac {e^{b x}}{e^{c}}\right )+3 e^{4 c} \mathit {atan} \left (\frac {e^{b x}}{e^{c}}\right )-5 e^{3 b x +4 a +5 c}-2 e^{3 b x +2 a +3 c}+3 e^{3 b x +c}-3 e^{b x +4 a +7 c}+2 e^{b x +2 a +5 c}+5 e^{b x +3 c}}{4 e^{2 a +2 c} b \left (e^{4 b x}+2 e^{2 b x +2 c}+e^{4 c}\right )} \] Input:
int(sech(b*x-c)^3*sinh(b*x+a)^2,x)
Output:
(3*e**(4*a + 4*b*x + 4*c)*atan(e**(b*x)/e**c) - 2*e**(2*a + 4*b*x + 2*c)*a tan(e**(b*x)/e**c) + 3*e**(4*b*x)*atan(e**(b*x)/e**c) + 6*e**(4*a + 2*b*x + 6*c)*atan(e**(b*x)/e**c) - 4*e**(2*a + 2*b*x + 4*c)*atan(e**(b*x)/e**c) + 6*e**(2*b*x + 2*c)*atan(e**(b*x)/e**c) + 3*e**(4*a + 8*c)*atan(e**(b*x)/ e**c) - 2*e**(2*a + 6*c)*atan(e**(b*x)/e**c) + 3*e**(4*c)*atan(e**(b*x)/e* *c) - 5*e**(4*a + 3*b*x + 5*c) - 2*e**(2*a + 3*b*x + 3*c) + 3*e**(3*b*x + c) - 3*e**(4*a + b*x + 7*c) + 2*e**(2*a + b*x + 5*c) + 5*e**(b*x + 3*c))/( 4*e**(2*a + 2*c)*b*(e**(4*b*x) + 2*e**(2*b*x + 2*c) + e**(4*c)))