Integrand size = 18, antiderivative size = 80 \[ \int \text {sech}^4(c-b x) \sinh ^2(a+b x) \, dx=-\frac {\text {sech}^2(c-b x) \sinh (2 (a+c))}{2 b}+\frac {\cosh ^2(a+c) \tanh (c-b x)}{b}-\frac {\cosh (2 (a+c)) \tanh (c-b x)}{b}-\frac {\cosh ^2(a+c) \tanh ^3(c-b x)}{3 b} \] Output:
-1/2*sech(b*x-c)^2*sinh(2*a+2*c)/b-cosh(a+c)^2*tanh(b*x-c)/b+cosh(2*a+2*c) *tanh(b*x-c)/b+1/3*cosh(a+c)^2*tanh(b*x-c)^3/b
Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04 \[ \int \text {sech}^4(c-b x) \sinh ^2(a+b x) \, dx=-\frac {\text {sech}(c) \text {sech}^3(c-b x) (3 \sinh (b x)-\sinh (2 c-3 b x)+\sinh (2 a+4 c-3 b x)+3 \sinh (2 a+2 c-b x)+3 \sinh (2 a+b x)-\sinh (2 a+3 b x))}{12 b} \] Input:
Integrate[Sech[c - b*x]^4*Sinh[a + b*x]^2,x]
Output:
-1/12*(Sech[c]*Sech[c - b*x]^3*(3*Sinh[b*x] - Sinh[2*c - 3*b*x] + Sinh[2*a + 4*c - 3*b*x] + 3*Sinh[2*a + 2*c - b*x] + 3*Sinh[2*a + b*x] - Sinh[2*a + 3*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.
\(\displaystyle \int \sinh ^2(a+b x) \text {sech}^4(c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sinh ^2(a+b x) \text {sech}^4(c-b x)dx\) |
Input:
Int[Sech[c - b*x]^4*Sinh[a + b*x]^2,x]
Output:
$Aborted
Time = 2.94 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80
method | result | size |
parallelrisch | \(\frac {2 \sinh \left (3 b x +2 a -c \right )-3 \sinh \left (b x -c \right )-\sinh \left (3 b x -3 c \right )}{3 b \left (\cosh \left (3 b x -3 c \right )+3 \cosh \left (b x -c \right )\right )}\) | \(64\) |
risch | \(-\frac {2 \left ({\mathrm e}^{4 a +4 c}+3 \,{\mathrm e}^{2 b x +4 a +2 c}-{\mathrm e}^{2 a +2 c}+3 \,{\mathrm e}^{4 b x +4 a}-3 \,{\mathrm e}^{2 b x +2 a}+1\right ) {\mathrm e}^{4 a +4 c}}{3 \left ({\mathrm e}^{2 a +2 c}+{\mathrm e}^{2 b x +2 a}\right )^{3} b}\) | \(90\) |
Input:
int(sech(b*x-c)^4*sinh(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/3/b*(2*sinh(3*b*x+2*a-c)-3*sinh(b*x-c)-sinh(3*b*x-3*c))/(cosh(3*b*x-3*c) +3*cosh(b*x-c))
Leaf count of result is larger than twice the leaf count of optimal. 1015 vs. \(2 (84) = 168\).
Time = 0.10 (sec) , antiderivative size = 1015, normalized size of antiderivative = 12.69 \[ \int \text {sech}^4(c-b x) \sinh ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(sech(b*x-c)^4*sinh(b*x+a)^2,x, algorithm="fricas")
Output:
-2/3*(4*cosh(b*x + a)^2*cosh(a + c)*sinh(a + c)^3 + (5*cosh(b*x + a)^2 - 3 )*sinh(a + c)^4 - 3*cosh(a + c)^4 + (5*cosh(a + c)^4 - cosh(a + c)^2)*cosh (b*x + a)^2 + (5*cosh(a + c)^4 + 4*cosh(a + c)*sinh(a + c)^3 + 5*sinh(a + c)^4 - (2*cosh(a + c)^2 + 1)*sinh(a + c)^2 - cosh(a + c)^2 + 2*(2*cosh(a + c)^3 - cosh(a + c))*sinh(a + c))*sinh(b*x + a)^2 - ((2*cosh(a + c)^2 + 1) *cosh(b*x + a)^2 - 6*cosh(a + c)^2 - 3)*sinh(a + c)^2 + 3*cosh(a + c)^2 - 2*(4*cosh(b*x + a)*cosh(a + c)*sinh(a + c)^3 - cosh(b*x + a)*sinh(a + c)^4 + (10*cosh(a + c)^2 - 1)*cosh(b*x + a)*sinh(a + c)^2 + 2*(2*cosh(a + c)^3 - cosh(a + c))*cosh(b*x + a)*sinh(a + c) - (cosh(a + c)^4 + cosh(a + c)^2 )*cosh(b*x + a))*sinh(b*x + a) + 2*((2*cosh(a + c)^3 - cosh(a + c))*cosh(b *x + a)^2 + 3*cosh(a + c))*sinh(a + c))/(b*cosh(b*x + a)^4*cosh(a + c)^4 + 4*b*cosh(b*x + a)^2*cosh(a + c)^4 + 3*b*cosh(a + c)^4 + (b*cosh(a + c)^4 - 4*b*cosh(a + c)^3*sinh(a + c) + 6*b*cosh(a + c)^2*sinh(a + c)^2 - 4*b*co sh(a + c)*sinh(a + c)^3 + b*sinh(a + c)^4)*sinh(b*x + a)^4 + (b*cosh(b*x + a)^4 - 4*b*cosh(b*x + a)^2 + 3*b)*sinh(a + c)^4 + 4*(b*cosh(b*x + a)*cosh (a + c)^4 - 4*b*cosh(b*x + a)*cosh(a + c)^3*sinh(a + c) + 6*b*cosh(b*x + a )*cosh(a + c)^2*sinh(a + c)^2 - 4*b*cosh(b*x + a)*cosh(a + c)*sinh(a + c)^ 3 + b*cosh(b*x + a)*sinh(a + c)^4)*sinh(b*x + a)^3 - 4*(b*cosh(b*x + a)^4* cosh(a + c) - b*cosh(b*x + a)^2*cosh(a + c))*sinh(a + c)^3 + 2*(3*b*cosh(b *x + a)^2*cosh(a + c)^4 + 18*b*cosh(b*x + a)^2*cosh(a + c)^2*sinh(a + c...
\[ \int \text {sech}^4(c-b x) \sinh ^2(a+b x) \, dx=\int \sinh ^{2}{\left (a + b x \right )} \operatorname {sech}^{4}{\left (b x - c \right )}\, dx \] Input:
integrate(sech(b*x-c)**4*sinh(b*x+a)**2,x)
Output:
Integral(sinh(a + b*x)**2*sech(b*x - c)**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (84) = 168\).
Time = 0.04 (sec) , antiderivative size = 334, normalized size of antiderivative = 4.18 \[ \int \text {sech}^4(c-b x) \sinh ^2(a+b x) \, dx=-\frac {2 \, {\left (e^{\left (4 \, a + 4 \, c\right )} - e^{\left (2 \, a + 2 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} + 3 \, e^{\left (-4 \, b x + 2 \, a + 6 \, c\right )} + e^{\left (-6 \, b x + 2 \, a + 8 \, c\right )} + e^{\left (2 \, a + 2 \, c\right )}\right )}} + \frac {2 \, e^{\left (-4 \, b x + 4 \, c\right )}}{b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} + 3 \, e^{\left (-4 \, b x + 2 \, a + 6 \, c\right )} + e^{\left (-6 \, b x + 2 \, a + 8 \, c\right )} + e^{\left (2 \, a + 2 \, c\right )}\right )}} + \frac {2 \, e^{\left (4 \, a + 4 \, c\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} + 3 \, e^{\left (-4 \, b x + 2 \, a + 6 \, c\right )} + e^{\left (-6 \, b x + 2 \, a + 8 \, c\right )} + e^{\left (2 \, a + 2 \, c\right )}\right )}} - \frac {2 \, e^{\left (2 \, a + 2 \, c\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} + 3 \, e^{\left (-4 \, b x + 2 \, a + 6 \, c\right )} + e^{\left (-6 \, b x + 2 \, a + 8 \, c\right )} + e^{\left (2 \, a + 2 \, c\right )}\right )}} + \frac {2}{3 \, b {\left (3 \, e^{\left (-2 \, b x + 2 \, a + 4 \, c\right )} + 3 \, e^{\left (-4 \, b x + 2 \, a + 6 \, c\right )} + e^{\left (-6 \, b x + 2 \, a + 8 \, c\right )} + e^{\left (2 \, a + 2 \, c\right )}\right )}} \] Input:
integrate(sech(b*x-c)^4*sinh(b*x+a)^2,x, algorithm="maxima")
Output:
-2*(e^(4*a + 4*c) - e^(2*a + 2*c))*e^(-2*b*x - 2*a)/(b*(3*e^(-2*b*x + 2*a + 4*c) + 3*e^(-4*b*x + 2*a + 6*c) + e^(-6*b*x + 2*a + 8*c) + e^(2*a + 2*c) )) + 2*e^(-4*b*x + 4*c)/(b*(3*e^(-2*b*x + 2*a + 4*c) + 3*e^(-4*b*x + 2*a + 6*c) + e^(-6*b*x + 2*a + 8*c) + e^(2*a + 2*c))) + 2/3*e^(4*a + 4*c)/(b*(3 *e^(-2*b*x + 2*a + 4*c) + 3*e^(-4*b*x + 2*a + 6*c) + e^(-6*b*x + 2*a + 8*c ) + e^(2*a + 2*c))) - 2/3*e^(2*a + 2*c)/(b*(3*e^(-2*b*x + 2*a + 4*c) + 3*e ^(-4*b*x + 2*a + 6*c) + e^(-6*b*x + 2*a + 8*c) + e^(2*a + 2*c))) + 2/3/(b* (3*e^(-2*b*x + 2*a + 4*c) + 3*e^(-4*b*x + 2*a + 6*c) + e^(-6*b*x + 2*a + 8 *c) + e^(2*a + 2*c)))
Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.08 \[ \int \text {sech}^4(c-b x) \sinh ^2(a+b x) \, dx=-\frac {2 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a + 4 \, c\right )} + 3 \, e^{\left (2 \, b x + 4 \, a + 6 \, c\right )} - 3 \, e^{\left (2 \, b x + 2 \, a + 4 \, c\right )} + e^{\left (4 \, a + 8 \, c\right )} - e^{\left (2 \, a + 6 \, c\right )} + e^{\left (4 \, c\right )}\right )} e^{\left (-2 \, a\right )}}{3 \, b {\left (e^{\left (2 \, b x\right )} + e^{\left (2 \, c\right )}\right )}^{3}} \] Input:
integrate(sech(b*x-c)^4*sinh(b*x+a)^2,x, algorithm="giac")
Output:
-2/3*(3*e^(4*b*x + 4*a + 4*c) + 3*e^(2*b*x + 4*a + 6*c) - 3*e^(2*b*x + 2*a + 4*c) + e^(4*a + 8*c) - e^(2*a + 6*c) + e^(4*c))*e^(-2*a)/(b*(e^(2*b*x) + e^(2*c))^3)
Timed out. \[ \int \text {sech}^4(c-b x) \sinh ^2(a+b x) \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{{\mathrm {cosh}\left (c-b\,x\right )}^4} \,d x \] Input:
int(sinh(a + b*x)^2/cosh(c - b*x)^4,x)
Output:
int(sinh(a + b*x)^2/cosh(c - b*x)^4, x)
Time = 0.23 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.21 \[ \int \text {sech}^4(c-b x) \sinh ^2(a+b x) \, dx=\frac {2 e^{2 c} \left (e^{6 b x +4 a}+3 e^{2 b x +2 a +2 c}+e^{2 a +4 c}-e^{2 c}\right )}{3 e^{2 a} b \left (e^{6 b x}+3 e^{4 b x +2 c}+3 e^{2 b x +4 c}+e^{6 c}\right )} \] Input:
int(sech(b*x-c)^4*sinh(b*x+a)^2,x)
Output:
(2*e**(2*c)*(e**(4*a + 6*b*x) + 3*e**(2*a + 2*b*x + 2*c) + e**(2*a + 4*c) - e**(2*c)))/(3*e**(2*a)*b*(e**(6*b*x) + 3*e**(4*b*x + 2*c) + 3*e**(2*b*x + 4*c) + e**(6*c)))