Integrand size = 16, antiderivative size = 37 \[ \int \text {sech}^3(c-b x) \sinh (a+b x) \, dx=-\frac {\cosh (a+c) \text {sech}^2(c-b x)}{2 b}-\frac {\sinh (a+c) \tanh (c-b x)}{b} \] Output:
-1/2*cosh(a+c)*sech(b*x-c)^2/b+sinh(a+c)*tanh(b*x-c)/b
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \text {sech}^3(c-b x) \sinh (a+b x) \, dx=-\frac {\text {sech}(c) \text {sech}^2(c-b x) (\cosh (a)+\sinh (a+c) \sinh (c-2 b x))}{2 b} \] Input:
Integrate[Sech[c - b*x]^3*Sinh[a + b*x],x]
Output:
-1/2*(Sech[c]*Sech[c - b*x]^2*(Cosh[a] + Sinh[a + c]*Sinh[c - 2*b*x]))/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.
\(\displaystyle \int \sinh (a+b x) \text {sech}^3(c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sinh (a+b x) \text {sech}^3(c-b x)dx\) |
Input:
Int[Sech[c - b*x]^3*Sinh[a + b*x],x]
Output:
$Aborted
Time = 0.50 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.19
method | result | size |
parallelrisch | \(\frac {-1-\cosh \left (2 b x -2 c \right )+2 \cosh \left (2 b x +a -c \right )}{2 b \left (1+\cosh \left (2 b x -2 c \right )\right )}\) | \(44\) |
risch | \(-\frac {\left ({\mathrm e}^{2 a +2 c}+2 \,{\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{3 a +3 c}}{\left ({\mathrm e}^{2 a +2 c}+{\mathrm e}^{2 b x +2 a}\right )^{2} b}\) | \(55\) |
Input:
int(sech(b*x-c)^3*sinh(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/2/b*(-1-cosh(2*b*x-2*c)+2*cosh(2*b*x+a-c))/(1+cosh(2*b*x-2*c))
Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (36) = 72\).
Time = 0.10 (sec) , antiderivative size = 447, normalized size of antiderivative = 12.08 \[ \int \text {sech}^3(c-b x) \sinh (a+b x) \, dx=-\frac {2 \, {\left (\cosh \left (b x + a\right ) \cosh \left (a + c\right )^{2} + \cosh \left (b x + a\right ) \cosh \left (a + c\right ) \sinh \left (a + c\right ) + {\left (\cosh \left (a + c\right )^{2} - \cosh \left (a + c\right ) \sinh \left (a + c\right ) - 2 \, \sinh \left (a + c\right )^{2}\right )} \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{3} \cosh \left (a + c\right )^{3} + 3 \, b \cosh \left (b x + a\right ) \cosh \left (a + c\right )^{3} + {\left (b \cosh \left (a + c\right )^{3} - 3 \, b \cosh \left (a + c\right )^{2} \sinh \left (a + c\right ) + 3 \, b \cosh \left (a + c\right ) \sinh \left (a + c\right )^{2} - b \sinh \left (a + c\right )^{3}\right )} \sinh \left (b x + a\right )^{3} - {\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (a + c\right )^{3} + 3 \, {\left (b \cosh \left (b x + a\right ) \cosh \left (a + c\right )^{3} - 3 \, b \cosh \left (b x + a\right ) \cosh \left (a + c\right )^{2} \sinh \left (a + c\right ) + 3 \, b \cosh \left (b x + a\right ) \cosh \left (a + c\right ) \sinh \left (a + c\right )^{2} - b \cosh \left (b x + a\right ) \sinh \left (a + c\right )^{3}\right )} \sinh \left (b x + a\right )^{2} + 3 \, {\left (b \cosh \left (b x + a\right )^{3} \cosh \left (a + c\right ) - b \cosh \left (b x + a\right ) \cosh \left (a + c\right )\right )} \sinh \left (a + c\right )^{2} + {\left (3 \, b \cosh \left (b x + a\right )^{2} \cosh \left (a + c\right )^{3} + b \cosh \left (a + c\right )^{3} - 3 \, {\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (a + c\right )^{3} + {\left (9 \, b \cosh \left (b x + a\right )^{2} \cosh \left (a + c\right ) - b \cosh \left (a + c\right )\right )} \sinh \left (a + c\right )^{2} - 3 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} \cosh \left (a + c\right )^{2} + b \cosh \left (a + c\right )^{2}\right )} \sinh \left (a + c\right )\right )} \sinh \left (b x + a\right ) - {\left (3 \, b \cosh \left (b x + a\right )^{3} \cosh \left (a + c\right )^{2} + b \cosh \left (b x + a\right ) \cosh \left (a + c\right )^{2}\right )} \sinh \left (a + c\right )} \] Input:
integrate(sech(b*x-c)^3*sinh(b*x+a),x, algorithm="fricas")
Output:
-2*(cosh(b*x + a)*cosh(a + c)^2 + cosh(b*x + a)*cosh(a + c)*sinh(a + c) + (cosh(a + c)^2 - cosh(a + c)*sinh(a + c) - 2*sinh(a + c)^2)*sinh(b*x + a)) /(b*cosh(b*x + a)^3*cosh(a + c)^3 + 3*b*cosh(b*x + a)*cosh(a + c)^3 + (b*c osh(a + c)^3 - 3*b*cosh(a + c)^2*sinh(a + c) + 3*b*cosh(a + c)*sinh(a + c) ^2 - b*sinh(a + c)^3)*sinh(b*x + a)^3 - (b*cosh(b*x + a)^3 - b*cosh(b*x + a))*sinh(a + c)^3 + 3*(b*cosh(b*x + a)*cosh(a + c)^3 - 3*b*cosh(b*x + a)*c osh(a + c)^2*sinh(a + c) + 3*b*cosh(b*x + a)*cosh(a + c)*sinh(a + c)^2 - b *cosh(b*x + a)*sinh(a + c)^3)*sinh(b*x + a)^2 + 3*(b*cosh(b*x + a)^3*cosh( a + c) - b*cosh(b*x + a)*cosh(a + c))*sinh(a + c)^2 + (3*b*cosh(b*x + a)^2 *cosh(a + c)^3 + b*cosh(a + c)^3 - 3*(b*cosh(b*x + a)^2 - b)*sinh(a + c)^3 + (9*b*cosh(b*x + a)^2*cosh(a + c) - b*cosh(a + c))*sinh(a + c)^2 - 3*(3* b*cosh(b*x + a)^2*cosh(a + c)^2 + b*cosh(a + c)^2)*sinh(a + c))*sinh(b*x + a) - (3*b*cosh(b*x + a)^3*cosh(a + c)^2 + b*cosh(b*x + a)*cosh(a + c)^2)* sinh(a + c))
\[ \int \text {sech}^3(c-b x) \sinh (a+b x) \, dx=\int \sinh {\left (a + b x \right )} \operatorname {sech}^{3}{\left (b x - c \right )}\, dx \] Input:
integrate(sech(b*x-c)**3*sinh(b*x+a),x)
Output:
Integral(sinh(a + b*x)*sech(b*x - c)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (36) = 72\).
Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.22 \[ \int \text {sech}^3(c-b x) \sinh (a+b x) \, dx=-\frac {2 \, e^{\left (-2 \, b x + 2 \, c\right )}}{b {\left (2 \, e^{\left (-2 \, b x + a + 3 \, c\right )} + e^{\left (-4 \, b x + a + 5 \, c\right )} + e^{\left (a + c\right )}\right )}} + \frac {e^{\left (2 \, a + 2 \, c\right )}}{b {\left (2 \, e^{\left (-2 \, b x + a + 3 \, c\right )} + e^{\left (-4 \, b x + a + 5 \, c\right )} + e^{\left (a + c\right )}\right )}} - \frac {1}{b {\left (2 \, e^{\left (-2 \, b x + a + 3 \, c\right )} + e^{\left (-4 \, b x + a + 5 \, c\right )} + e^{\left (a + c\right )}\right )}} \] Input:
integrate(sech(b*x-c)^3*sinh(b*x+a),x, algorithm="maxima")
Output:
-2*e^(-2*b*x + 2*c)/(b*(2*e^(-2*b*x + a + 3*c) + e^(-4*b*x + a + 5*c) + e^ (a + c))) + e^(2*a + 2*c)/(b*(2*e^(-2*b*x + a + 3*c) + e^(-4*b*x + a + 5*c ) + e^(a + c))) - 1/(b*(2*e^(-2*b*x + a + 3*c) + e^(-4*b*x + a + 5*c) + e^ (a + c)))
Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.35 \[ \int \text {sech}^3(c-b x) \sinh (a+b x) \, dx=-\frac {{\left (2 \, e^{\left (2 \, b x + 2 \, a + 3 \, c\right )} + e^{\left (2 \, a + 5 \, c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a\right )}}{b {\left (e^{\left (2 \, b x\right )} + e^{\left (2 \, c\right )}\right )}^{2}} \] Input:
integrate(sech(b*x-c)^3*sinh(b*x+a),x, algorithm="giac")
Output:
-(2*e^(2*b*x + 2*a + 3*c) + e^(2*a + 5*c) - e^(3*c))*e^(-a)/(b*(e^(2*b*x) + e^(2*c))^2)
Timed out. \[ \int \text {sech}^3(c-b x) \sinh (a+b x) \, dx=\int \frac {\mathrm {sinh}\left (a+b\,x\right )}{{\mathrm {cosh}\left (c-b\,x\right )}^3} \,d x \] Input:
int(sinh(a + b*x)/cosh(c - b*x)^3,x)
Output:
int(sinh(a + b*x)/cosh(c - b*x)^3, x)
Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.46 \[ \int \text {sech}^3(c-b x) \sinh (a+b x) \, dx=\frac {e^{c} \left (e^{4 b x +2 a}+e^{2 c}\right )}{e^{a} b \left (e^{4 b x}+2 e^{2 b x +2 c}+e^{4 c}\right )} \] Input:
int(sech(b*x-c)^3*sinh(b*x+a),x)
Output:
(e**c*(e**(2*a + 4*b*x) + e**(2*c)))/(e**a*b*(e**(4*b*x) + 2*e**(2*b*x + 2 *c) + e**(4*c)))