Integrand size = 17, antiderivative size = 89 \[ \int \text {csch}^3(c+b x) \sinh ^2(a+b x) \, dx=-\frac {\text {arctanh}(\cosh (c+b x)) \cosh (2 (a-c))}{b}+\frac {\text {arctanh}(\cosh (c+b x)) \sinh ^2(a-c)}{2 b}-\frac {\coth (c+b x) \text {csch}(c+b x) \sinh ^2(a-c)}{2 b}-\frac {\text {csch}(c+b x) \sinh (2 (a-c))}{b} \] 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
Leaf count is larger than twice the leaf count of optimal. \(246\) vs. \(2(89)=178\).
Time = 1.56 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.76 \[ \int \text {csch}^3(c+b x) \sinh ^2(a+b x) \, dx=\frac {\text {csch}^2\left (\frac {1}{2} (c+b x)\right )-\cosh (2 (a-c)) \text {csch}^2\left (\frac {1}{2} (c+b x)\right )-4 \log \left (\cosh \left (\frac {1}{2} (c+b x)\right )\right )-12 \cosh (2 (a-c)) \log \left (\cosh \left (\frac {1}{2} (c+b x)\right )\right )+4 \log \left (\sinh \left (\frac {1}{2} (c+b x)\right )\right )+12 \cosh (2 (a-c)) \log \left (\sinh \left (\frac {1}{2} (c+b x)\right )\right )+\text {sech}^2\left (\frac {1}{2} (c+b x)\right )-\cosh (2 (a-c)) \text {sech}^2\left (\frac {1}{2} (c+b x)\right )-4 \cosh \left (2 a-2 c-\frac {b x}{2}\right ) \left (\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {1}{2} (c+b x)\right )+\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} (c+b x)\right )\right )+4 \cosh \left (2 a-2 c+\frac {b x}{2}\right ) \left (\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {1}{2} (c+b x)\right )+\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} (c+b x)\right )\right )}{16 b} \] Input:
Integrate[Csch[c + b*x]^3*Sinh[a + b*x]^2,x]
Output:
(Csch[(c + b*x)/2]^2 - Cosh[2*(a - c)]*Csch[(c + b*x)/2]^2 - 4*Log[Cosh[(c + b*x)/2]] - 12*Cosh[2*(a - c)]*Log[Cosh[(c + b*x)/2]] + 4*Log[Sinh[(c + b*x)/2]] + 12*Cosh[2*(a - c)]*Log[Sinh[(c + b*x)/2]] + Sech[(c + b*x)/2]^2 - Cosh[2*(a - c)]*Sech[(c + b*x)/2]^2 - 4*Cosh[2*a - 2*c - (b*x)/2]*(Csch [c/2]*Csch[(c + b*x)/2] + Sech[c/2]*Sech[(c + b*x)/2]) + 4*Cosh[2*a - 2*c + (b*x)/2]*(Csch[c/2]*Csch[(c + b*x)/2] + Sech[c/2]*Sech[(c + b*x)/2]))/(1 6*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.
\(\displaystyle \int \sinh ^2(a+b x) \text {csch}^3(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sinh ^2(a+b x) \text {csch}^3(b x+c)dx\) |
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. \(304\) vs. \(2(85)=170\).
Time = 0.97 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.43
method | result | size |
risch | \(\frac {\left (-5 \,{\mathrm e}^{2 b x +6 a +2 c}+2 \,{\mathrm e}^{2 b x +4 a +4 c}+3 \,{\mathrm e}^{2 b x +2 a +6 c}+3 \,{\mathrm e}^{6 a}+2 \,{\mathrm e}^{4 a +2 c}-5 \,{\mathrm e}^{2 a +4 c}\right ) {\mathrm e}^{b x -c}}{4 \left (-{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{2} b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 a}}{8 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{2 a +2 c}}{4 b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 c}}{8 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 a}}{8 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{2 a +2 c}}{4 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 c}}{8 b}\) | \(305\) |
Input:
int(csch(b*x+c)^3*sinh(b*x+a)^2,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))*ex p(b*x-c)+3/8*ln(exp(b*x+a)-exp(a-c))/b*exp(-2*c-2*a)*exp(4*a)+1/4*ln(exp(b *x+a)-exp(a-c))/b*exp(-2*c-2*a)*exp(2*a+2*c)+3/8*ln(exp(b*x+a)-exp(a-c))/b *exp(-2*c-2*a)*exp(4*c)-3/8*ln(exp(b*x+a)+exp(a-c))/b*exp(-2*c-2*a)*exp(4* a)-1/4*ln(exp(b*x+a)+exp(a-c))/b*exp(-2*c-2*a)*exp(2*a+2*c)-3/8*ln(exp(b*x +a)+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. 3234 vs. \(2 (85) = 170\).
Time = 0.11 (sec) , antiderivative size = 3234, normalized size of antiderivative = 36.34 \[ \int \text {csch}^3(c+b x) \sinh ^2(a+b x) \, dx=\text {Too large to display} \] 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. 207 vs. \(2 (85) = 170\).
Time = 0.04 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.33 \[ \int \text {csch}^3(c+b x) \sinh ^2(a+b x) \, dx=-\frac {{\left (3 \, e^{\left (4 \, a\right )} + 2 \, e^{\left (2 \, a + 2 \, c\right )} + 3 \, e^{\left (4 \, c\right )}\right )} e^{\left (-2 \, a - 2 \, c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{8 \, b} + \frac {{\left (3 \, e^{\left (4 \, a\right )} + 2 \, e^{\left (2 \, a + 2 \, c\right )} + 3 \, e^{\left (4 \, c\right )}\right )} e^{\left (-2 \, a - 2 \, c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{8 \, b} + \frac {{\left (5 \, e^{\left (4 \, a + 2 \, c\right )} - 2 \, e^{\left (2 \, a + 4 \, c\right )} - 3 \, e^{\left (6 \, c\right )}\right )} e^{\left (-b x - a\right )} - {\left (3 \, e^{\left (6 \, a\right )} + 2 \, e^{\left (4 \, a + 2 \, c\right )} - 5 \, e^{\left (2 \, a + 4 \, c\right )}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{4 \, b {\left (2 \, e^{\left (-2 \, b x + a + 3 \, c\right )} - e^{\left (-4 \, b x + a + c\right )} - e^{\left (a + 5 \, c\right )}\right )}} \] Input:
integrate(csch(b*x+c)^3*sinh(b*x+a)^2,x, algorithm="maxima")
Output:
-1/8*(3*e^(4*a) + 2*e^(2*a + 2*c) + 3*e^(4*c))*e^(-2*a - 2*c)*log(e^(-b*x) + e^c)/b + 1/8*(3*e^(4*a) + 2*e^(2*a + 2*c) + 3*e^(4*c))*e^(-2*a - 2*c)*l og(e^(-b*x) - e^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)))
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (85) = 170\).
Time = 0.16 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.19 \[ \int \text {csch}^3(c+b x) \sinh ^2(a+b x) \, dx=-\frac {{\left (3 \, e^{\left (4 \, a + c\right )} + 2 \, e^{\left (2 \, a + 3 \, c\right )} + 3 \, e^{\left (5 \, c\right )}\right )} e^{\left (-2 \, a - 3 \, c\right )} \log \left (e^{\left (b x + c\right )} + 1\right )}{8 \, b} + \frac {{\left (3 \, e^{\left (4 \, a + c\right )} + 2 \, e^{\left (2 \, a + 3 \, c\right )} + 3 \, e^{\left (5 \, c\right )}\right )} e^{\left (-2 \, a - 3 \, c\right )} \log \left ({\left | e^{\left (b x + c\right )} - 1 \right |}\right )}{8 \, b} - \frac {{\left (5 \, e^{\left (3 \, b x + 4 \, a + 2 \, c\right )} - 2 \, e^{\left (3 \, b x + 2 \, a + 4 \, c\right )} - 3 \, e^{\left (3 \, b x + 6 \, c\right )} - 3 \, e^{\left (b x + 4 \, a\right )} - 2 \, e^{\left (b x + 2 \, a + 2 \, c\right )} + 5 \, e^{\left (b x + 4 \, c\right )}\right )} e^{\left (-2 \, a - c\right )}}{4 \, b {\left (e^{\left (2 \, b x + 2 \, c\right )} - 1\right )}^{2}} \] Input:
integrate(csch(b*x+c)^3*sinh(b*x+a)^2,x, algorithm="giac")
Output:
-1/8*(3*e^(4*a + c) + 2*e^(2*a + 3*c) + 3*e^(5*c))*e^(-2*a - 3*c)*log(e^(b *x + c) + 1)/b + 1/8*(3*e^(4*a + c) + 2*e^(2*a + 3*c) + 3*e^(5*c))*e^(-2*a - 3*c)*log(abs(e^(b*x + c) - 1))/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. \[ \int \text {csch}^3(c+b x) \sinh ^2(a+b x) \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{{\mathrm {sinh}\left (c+b\,x\right )}^3} \,d x \] 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 = 0.23 (sec) , antiderivative size = 520, normalized size of antiderivative = 5.84 \[ \int \text {csch}^3(c+b x) \sinh ^2(a+b x) \, dx=\frac {3 e^{4 b x +4 a +4 c} \mathrm {log}\left (e^{b x +c}-1\right )-3 e^{4 b x +4 a +4 c} \mathrm {log}\left (e^{b x +c}+1\right )+2 e^{4 b x +2 a +6 c} \mathrm {log}\left (e^{b x +c}-1\right )-2 e^{4 b x +2 a +6 c} \mathrm {log}\left (e^{b x +c}+1\right )+3 e^{4 b x +8 c} \mathrm {log}\left (e^{b x +c}-1\right )-3 e^{4 b x +8 c} \mathrm {log}\left (e^{b x +c}+1\right )-10 e^{3 b x +4 a +3 c}+4 e^{3 b x +2 a +5 c}+6 e^{3 b x +7 c}-6 e^{2 b x +4 a +2 c} \mathrm {log}\left (e^{b x +c}-1\right )+6 e^{2 b x +4 a +2 c} \mathrm {log}\left (e^{b x +c}+1\right )-4 e^{2 b x +2 a +4 c} \mathrm {log}\left (e^{b x +c}-1\right )+4 e^{2 b x +2 a +4 c} \mathrm {log}\left (e^{b x +c}+1\right )-6 e^{2 b x +6 c} \mathrm {log}\left (e^{b x +c}-1\right )+6 e^{2 b x +6 c} \mathrm {log}\left (e^{b x +c}+1\right )+6 e^{b x +4 a +c}+4 e^{b x +2 a +3 c}-10 e^{b x +5 c}+3 e^{4 a} \mathrm {log}\left (e^{b x +c}-1\right )-3 e^{4 a} \mathrm {log}\left (e^{b x +c}+1\right )+2 e^{2 a +2 c} \mathrm {log}\left (e^{b x +c}-1\right )-2 e^{2 a +2 c} \mathrm {log}\left (e^{b x +c}+1\right )+3 e^{4 c} \mathrm {log}\left (e^{b x +c}-1\right )-3 e^{4 c} \mathrm {log}\left (e^{b x +c}+1\right )}{8 e^{2 a +2 c} b \left (e^{4 b x +4 c}-2 e^{2 b x +2 c}+1\right )} \] 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 + c) - 1) - 3*e**(4*a + 4*b*x + 4*c) *log(e**(b*x + c) + 1) + 2*e**(2*a + 4*b*x + 6*c)*log(e**(b*x + c) - 1) - 2*e**(2*a + 4*b*x + 6*c)*log(e**(b*x + c) + 1) + 3*e**(4*b*x + 8*c)*log(e* *(b*x + c) - 1) - 3*e**(4*b*x + 8*c)*log(e**(b*x + c) + 1) - 10*e**(4*a + 3*b*x + 3*c) + 4*e**(2*a + 3*b*x + 5*c) + 6*e**(3*b*x + 7*c) - 6*e**(4*a + 2*b*x + 2*c)*log(e**(b*x + c) - 1) + 6*e**(4*a + 2*b*x + 2*c)*log(e**(b*x + c) + 1) - 4*e**(2*a + 2*b*x + 4*c)*log(e**(b*x + c) - 1) + 4*e**(2*a + 2*b*x + 4*c)*log(e**(b*x + c) + 1) - 6*e**(2*b*x + 6*c)*log(e**(b*x + c) - 1) + 6*e**(2*b*x + 6*c)*log(e**(b*x + c) + 1) + 6*e**(4*a + b*x + c) + 4* e**(2*a + b*x + 3*c) - 10*e**(b*x + 5*c) + 3*e**(4*a)*log(e**(b*x + c) - 1 ) - 3*e**(4*a)*log(e**(b*x + c) + 1) + 2*e**(2*a + 2*c)*log(e**(b*x + c) - 1) - 2*e**(2*a + 2*c)*log(e**(b*x + c) + 1) + 3*e**(4*c)*log(e**(b*x + c) - 1) - 3*e**(4*c)*log(e**(b*x + c) + 1))/(8*e**(2*a + 2*c)*b*(e**(4*b*x + 4*c) - 2*e**(2*b*x + 2*c) + 1))