Integrand size = 18, antiderivative size = 85 \[ \int \text {csch}^3(c-b x) \sinh ^2(a+b x) \, dx=\frac {\text {arctanh}(\cosh (c-b x)) \cosh (2 (a+c))}{b}-\frac {\text {arctanh}(\cosh (c-b x)) \sinh ^2(a+c)}{2 b}+\frac {\coth (c-b x) \text {csch}(c-b x) \sinh ^2(a+c)}{2 b}-\frac {\text {csch}(c-b x) \sinh (2 (a+c))}{b} \] Output:
arctanh(cosh(b*x-c))*cosh(2*a+2*c)/b-1/2*arctanh(cosh(b*x-c))*sinh(a+c)^2/ b+1/2*coth(b*x-c)*csch(b*x-c)*sinh(a+c)^2/b+csch(b*x-c)*sinh(2*a+2*c)/b
Time = 4.46 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.91 \[ \int \text {csch}^3(c-b x) \sinh ^2(a+b x) \, dx=\frac {4 (1+3 \cosh (2 (a+c))) \log \left (\cosh \left (\frac {1}{2} (c-b x)\right )\right )-4 (1+3 \cosh (2 (a+c))) \log \left (\sinh \left (\frac {1}{2} (c-b x)\right )\right )+2 \text {csch}^2\left (\frac {1}{2} (c-b x)\right ) \sinh ^2(a+c)+2 \text {sech}^2\left (\frac {1}{2} (c-b x)\right ) \sinh ^2(a+c)-8 \text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {1}{2} (c-b x)\right ) \sinh (2 (a+c)) \sinh \left (\frac {b x}{2}\right )-8 \text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} (c-b x)\right ) \sinh (2 (a+c)) \sinh \left (\frac {b x}{2}\right )}{16 b} \] Input:
Integrate[Csch[c - b*x]^3*Sinh[a + b*x]^2,x]
Output:
(4*(1 + 3*Cosh[2*(a + c)])*Log[Cosh[(c - b*x)/2]] - 4*(1 + 3*Cosh[2*(a + c )])*Log[Sinh[(c - b*x)/2]] + 2*Csch[(c - b*x)/2]^2*Sinh[a + c]^2 + 2*Sech[ (c - b*x)/2]^2*Sinh[a + c]^2 - 8*Csch[c/2]*Csch[(c - b*x)/2]*Sinh[2*(a + c )]*Sinh[(b*x)/2] - 8*Sech[c/2]*Sech[(c - b*x)/2]*Sinh[2*(a + c)]*Sinh[(b*x )/2])/(16*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.
\(\displaystyle \int \sinh ^2(a+b x) \text {csch}^3(c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sinh ^2(a+b x) \text {csch}^3(c-b x)dx\) |
Input:
Int[Csch[c - b*x]^3*Sinh[a + b*x]^2,x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(294\) vs. \(2(89)=178\).
Time = 0.63 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.47
method | result | size |
risch | \(-\frac {\left (3 \,{\mathrm e}^{6 a +6 c}-5 \,{\mathrm e}^{2 b x +6 a +4 c}+2 \,{\mathrm e}^{4 a +4 c}+2 \,{\mathrm e}^{2 b x +4 a +2 c}-5 \,{\mathrm e}^{2 a +2 c}+3 \,{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x -c}}{4 \left ({\mathrm e}^{2 a +2 c}-{\mathrm e}^{2 b x +2 a}\right )^{2} b}-\frac {3 \ln \left (-{\mathrm e}^{a +c}+{\mathrm e}^{b x +a}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 a +4 c}}{8 b}-\frac {\ln \left (-{\mathrm e}^{a +c}+{\mathrm e}^{b x +a}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{2 a +2 c}}{4 b}-\frac {3 \ln \left (-{\mathrm e}^{a +c}+{\mathrm e}^{b x +a}\right ) {\mathrm e}^{-2 c -2 a}}{8 b}+\frac {3 \ln \left ({\mathrm e}^{a +c}+{\mathrm e}^{b x +a}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 a +4 c}}{8 b}+\frac {\ln \left ({\mathrm e}^{a +c}+{\mathrm e}^{b x +a}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{2 a +2 c}}{4 b}+\frac {3 \ln \left ({\mathrm e}^{a +c}+{\mathrm e}^{b x +a}\right ) {\mathrm e}^{-2 c -2 a}}{8 b}\) | \(295\) |
Input:
int(-csch(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*ln(-exp(a+c)+exp(b*x+a))/b*exp(-2*c-2*a)*exp(4*a+4*c)-1/4*ln(- exp(a+c)+exp(b*x+a))/b*exp(-2*c-2*a)*exp(2*a+2*c)-3/8*ln(-exp(a+c)+exp(b*x +a))/b*exp(-2*c-2*a)+3/8*ln(exp(a+c)+exp(b*x+a))/b*exp(-2*c-2*a)*exp(4*a+4 *c)+1/4*ln(exp(a+c)+exp(b*x+a))/b*exp(-2*c-2*a)*exp(2*a+2*c)+3/8*ln(exp(a+ c)+exp(b*x+a))/b*exp(-2*c-2*a)
Leaf count of result is larger than twice the leaf count of optimal. 5785 vs. \(2 (89) = 178\).
Time = 0.13 (sec) , antiderivative size = 5785, normalized size of antiderivative = 68.06 \[ \int \text {csch}^3(c-b x) \sinh ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(-csch(b*x-c)^3*sinh(b*x+a)^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \text {csch}^3(c-b x) \sinh ^2(a+b x) \, dx=- \int \sinh ^{2}{\left (a + b x \right )} \operatorname {csch}^{3}{\left (b x - c \right )}\, dx \] Input:
integrate(-csch(b*x-c)**3*sinh(b*x+a)**2,x)
Output:
-Integral(sinh(a + b*x)**2*csch(b*x - c)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (89) = 178\).
Time = 0.05 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.40 \[ \int \text {csch}^3(c-b x) \sinh ^2(a+b x) \, dx=\frac {{\left (3 \, e^{\left (4 \, a + 4 \, c\right )} + 2 \, e^{\left (2 \, a + 2 \, c\right )} + 3\right )} e^{\left (-2 \, a - 2 \, c\right )} \log \left (e^{\left (-b x + c\right )} + 1\right )}{8 \, b} - \frac {{\left (3 \, e^{\left (4 \, a + 4 \, c\right )} + 2 \, e^{\left (2 \, a + 2 \, c\right )} + 3\right )} e^{\left (-2 \, a - 2 \, c\right )} \log \left (e^{\left (-b x + c\right )} - 1\right )}{8 \, b} - \frac {{\left (5 \, e^{\left (4 \, a + 4 \, c\right )} - 2 \, e^{\left (2 \, a + 2 \, c\right )} - 3\right )} e^{\left (-b x - a\right )} - {\left (3 \, e^{\left (6 \, a + 6 \, c\right )} + 2 \, e^{\left (4 \, a + 4 \, c\right )} - 5 \, e^{\left (2 \, a + 2 \, c\right )}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{4 \, b {\left (2 \, e^{\left (-2 \, b x + a + 3 \, c\right )} - e^{\left (-4 \, b x + a + 5 \, c\right )} - e^{\left (a + c\right )}\right )}} \] Input:
integrate(-csch(b*x-c)^3*sinh(b*x+a)^2,x, algorithm="maxima")
Output:
1/8*(3*e^(4*a + 4*c) + 2*e^(2*a + 2*c) + 3)*e^(-2*a - 2*c)*log(e^(-b*x + c ) + 1)/b - 1/8*(3*e^(4*a + 4*c) + 2*e^(2*a + 2*c) + 3)*e^(-2*a - 2*c)*log( e^(-b*x + c) - 1)/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)))
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (89) = 178\).
Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.22 \[ \int \text {csch}^3(c-b x) \sinh ^2(a+b x) \, dx=\frac {{\left (3 \, e^{\left (4 \, a + 4 \, c\right )} + 2 \, e^{\left (2 \, a + 2 \, c\right )} + 3\right )} e^{\left (-2 \, a - 2 \, c\right )} \log \left (e^{\left (b x\right )} + e^{c}\right )}{8 \, b} - \frac {{\left (3 \, e^{\left (4 \, a + 4 \, c\right )} + 2 \, e^{\left (2 \, a + 2 \, c\right )} + 3\right )} e^{\left (-2 \, a - 2 \, c\right )} \log \left ({\left | e^{\left (b x\right )} - e^{c} \right |}\right )}{8 \, b} - \frac {{\left (3 \, e^{\left (3 \, b x\right )} - 5 \, e^{\left (3 \, b x + 4 \, a + 4 \, c\right )} + 2 \, e^{\left (3 \, b x + 2 \, a + 2 \, c\right )} + 3 \, e^{\left (b x + 4 \, a + 6 \, c\right )} + 2 \, e^{\left (b x + 2 \, a + 4 \, c\right )} - 5 \, e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-2 \, a - c\right )}}{4 \, b {\left (e^{\left (2 \, b x\right )} - e^{\left (2 \, c\right )}\right )}^{2}} \] Input:
integrate(-csch(b*x-c)^3*sinh(b*x+a)^2,x, algorithm="giac")
Output:
1/8*(3*e^(4*a + 4*c) + 2*e^(2*a + 2*c) + 3)*e^(-2*a - 2*c)*log(e^(b*x) + e ^c)/b - 1/8*(3*e^(4*a + 4*c) + 2*e^(2*a + 2*c) + 3)*e^(-2*a - 2*c)*log(abs (e^(b*x) - e^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. \[ \int \text {csch}^3(c-b x) \sinh ^2(a+b x) \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{{\mathrm {sinh}\left (c-b\,x\right )}^3} \,d x \] Input:
int(sinh(a + b*x)^2/sinh(c - b*x)^3,x)
Output:
int(sinh(a + b*x)^2/sinh(c - b*x)^3, x)
Time = 0.22 (sec) , antiderivative size = 538, normalized size of antiderivative = 6.33 \[ \int \text {csch}^3(c-b x) \sinh ^2(a+b x) \, dx=\frac {3 e^{4 b x +4 a +4 c} \mathrm {log}\left (e^{b x}+e^{c}\right )-3 e^{4 b x +4 a +4 c} \mathrm {log}\left (e^{b x}-e^{c}\right )+2 e^{4 b x +2 a +2 c} \mathrm {log}\left (e^{b x}+e^{c}\right )-2 e^{4 b x +2 a +2 c} \mathrm {log}\left (e^{b x}-e^{c}\right )+3 e^{4 b x} \mathrm {log}\left (e^{b x}+e^{c}\right )-3 e^{4 b x} \mathrm {log}\left (e^{b x}-e^{c}\right )+10 e^{3 b x +4 a +5 c}-4 e^{3 b x +2 a +3 c}-6 e^{3 b x +c}-6 e^{2 b x +4 a +6 c} \mathrm {log}\left (e^{b x}+e^{c}\right )+6 e^{2 b x +4 a +6 c} \mathrm {log}\left (e^{b x}-e^{c}\right )-4 e^{2 b x +2 a +4 c} \mathrm {log}\left (e^{b x}+e^{c}\right )+4 e^{2 b x +2 a +4 c} \mathrm {log}\left (e^{b x}-e^{c}\right )-6 e^{2 b x +2 c} \mathrm {log}\left (e^{b x}+e^{c}\right )+6 e^{2 b x +2 c} \mathrm {log}\left (e^{b x}-e^{c}\right )-6 e^{b x +4 a +7 c}-4 e^{b x +2 a +5 c}+10 e^{b x +3 c}+3 e^{4 a +8 c} \mathrm {log}\left (e^{b x}+e^{c}\right )-3 e^{4 a +8 c} \mathrm {log}\left (e^{b x}-e^{c}\right )+2 e^{2 a +6 c} \mathrm {log}\left (e^{b x}+e^{c}\right )-2 e^{2 a +6 c} \mathrm {log}\left (e^{b x}-e^{c}\right )+3 e^{4 c} \mathrm {log}\left (e^{b x}+e^{c}\right )-3 e^{4 c} \mathrm {log}\left (e^{b x}-e^{c}\right )}{8 e^{2 a +2 c} b \left (e^{4 b x}-2 e^{2 b x +2 c}+e^{4 c}\right )} \] Input:
int(-csch(b*x-c)^3*sinh(b*x+a)^2,x)
Output:
(3*e**(4*a + 4*b*x + 4*c)*log(e**(b*x) + e**c) - 3*e**(4*a + 4*b*x + 4*c)* log(e**(b*x) - e**c) + 2*e**(2*a + 4*b*x + 2*c)*log(e**(b*x) + e**c) - 2*e **(2*a + 4*b*x + 2*c)*log(e**(b*x) - e**c) + 3*e**(4*b*x)*log(e**(b*x) + e **c) - 3*e**(4*b*x)*log(e**(b*x) - e**c) + 10*e**(4*a + 3*b*x + 5*c) - 4*e **(2*a + 3*b*x + 3*c) - 6*e**(3*b*x + c) - 6*e**(4*a + 2*b*x + 6*c)*log(e* *(b*x) + e**c) + 6*e**(4*a + 2*b*x + 6*c)*log(e**(b*x) - e**c) - 4*e**(2*a + 2*b*x + 4*c)*log(e**(b*x) + e**c) + 4*e**(2*a + 2*b*x + 4*c)*log(e**(b* x) - e**c) - 6*e**(2*b*x + 2*c)*log(e**(b*x) + e**c) + 6*e**(2*b*x + 2*c)* log(e**(b*x) - e**c) - 6*e**(4*a + b*x + 7*c) - 4*e**(2*a + b*x + 5*c) + 1 0*e**(b*x + 3*c) + 3*e**(4*a + 8*c)*log(e**(b*x) + e**c) - 3*e**(4*a + 8*c )*log(e**(b*x) - e**c) + 2*e**(2*a + 6*c)*log(e**(b*x) + e**c) - 2*e**(2*a + 6*c)*log(e**(b*x) - e**c) + 3*e**(4*c)*log(e**(b*x) + e**c) - 3*e**(4*c )*log(e**(b*x) - e**c))/(8*e**(2*a + 2*c)*b*(e**(4*b*x) - 2*e**(2*b*x + 2* c) + e**(4*c)))