Integrand size = 15, antiderivative size = 36 \[ \int \cosh ^2(a+b x) \text {csch}(c+b x) \, dx=-\frac {\text {arctanh}(\cosh (c+b x)) \cosh ^2(a-c)}{b}+\frac {\cosh (2 a-c+b x)}{b} \] Output:
-arctanh(cosh(b*x+c))*cosh(a-c)^2/b+cosh(b*x+2*a-c)/b
Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \cosh ^2(a+b x) \text {csch}(c+b x) \, dx=\frac {\cosh (2 a-c+b x)+\cosh ^2(a-c) \left (-\log \left (\cosh \left (\frac {1}{2} (c+b x)\right )\right )+\log \left (\sinh \left (\frac {1}{2} (c+b x)\right )\right )\right )}{b} \] Input:
Integrate[Cosh[a + b*x]^2*Csch[c + b*x],x]
Output:
(Cosh[2*a - c + b*x] + Cosh[a - c]^2*(-Log[Cosh[(c + b*x)/2]] + Log[Sinh[( c + b*x)/2]]))/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.
\(\displaystyle \int \cosh ^2(a+b x) \text {csch}(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \cosh ^2(a+b x) \text {csch}(b x+c)dx\) |
Input:
Int[Cosh[a + b*x]^2*Csch[c + b*x],x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(232\) vs. \(2(36)=72\).
Time = 0.43 (sec) , antiderivative size = 233, normalized size of antiderivative = 6.47
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +2 a -c}}{2 b}+\frac {{\mathrm e}^{-b x -2 a +c}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 a}}{4 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{2 a +2 c}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 c}}{4 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 a}}{4 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{2 a +2 c}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-2 c -2 a} {\mathrm e}^{4 c}}{4 b}\) | \(233\) |
Input:
int(cosh(b*x+a)^2*csch(b*x+c),x,method=_RETURNVERBOSE)
Output:
1/2/b*exp(b*x+2*a-c)+1/2/b*exp(-b*x-2*a+c)+1/4*ln(exp(b*x+a)-exp(a-c))/b*e xp(-2*c-2*a)*exp(4*a)+1/2*ln(exp(b*x+a)-exp(a-c))/b*exp(-2*c-2*a)*exp(2*a+ 2*c)+1/4*ln(exp(b*x+a)-exp(a-c))/b*exp(-2*c-2*a)*exp(4*c)-1/4*ln(exp(b*x+a )+exp(a-c))/b*exp(-2*c-2*a)*exp(4*a)-1/2*ln(exp(b*x+a)+exp(a-c))/b*exp(-2* c-2*a)*exp(2*a+2*c)-1/4*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. 872 vs. \(2 (36) = 72\).
Time = 0.11 (sec) , antiderivative size = 872, normalized size of antiderivative = 24.22 \[ \int \cosh ^2(a+b x) \text {csch}(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cosh(b*x+a)^2*csch(b*x+c),x, algorithm="fricas")
Output:
1/4*(2*cosh(b*x + c)^2*cosh(-a + c)^4 - 8*cosh(b*x + c)^2*cosh(-a + c)^3*s inh(-a + c) + 12*cosh(b*x + c)^2*cosh(-a + c)^2*sinh(-a + c)^2 - 8*cosh(b* x + c)^2*cosh(-a + c)*sinh(-a + c)^3 + 2*cosh(b*x + c)^2*sinh(-a + c)^4 + 2*(cosh(-a + c)^4 - 4*cosh(-a + c)^3*sinh(-a + c) + 6*cosh(-a + c)^2*sinh( -a + c)^2 - 4*cosh(-a + c)*sinh(-a + c)^3 + sinh(-a + c)^4)*sinh(b*x + c)^ 2 + (4*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c)^3 - cosh(b*x + c)*sinh(-a + c)^4 - 2*(3*cosh(-a + c)^2 + 1)*cosh(b*x + c)*sinh(-a + c)^2 + 4*(cosh(-a + c)^3 + cosh(-a + c))*cosh(b*x + c)*sinh(-a + c) - (cosh(-a + c)^4 + 2*c osh(-a + c)^2 + 1)*cosh(b*x + c) - (cosh(-a + c)^4 - 4*cosh(-a + c)*sinh(- a + c)^3 + sinh(-a + c)^4 + 2*(3*cosh(-a + c)^2 + 1)*sinh(-a + c)^2 + 2*co sh(-a + c)^2 - 4*(cosh(-a + c)^3 + cosh(-a + c))*sinh(-a + c) + 1)*sinh(b* x + c))*log(cosh(b*x + c) + sinh(b*x + c) + 1) - (4*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c)^3 - cosh(b*x + c)*sinh(-a + c)^4 - 2*(3*cosh(-a + c)^2 + 1)*cosh(b*x + c)*sinh(-a + c)^2 + 4*(cosh(-a + c)^3 + cosh(-a + c))*cosh( b*x + c)*sinh(-a + c) - (cosh(-a + c)^4 + 2*cosh(-a + c)^2 + 1)*cosh(b*x + c) - (cosh(-a + c)^4 - 4*cosh(-a + c)*sinh(-a + c)^3 + sinh(-a + c)^4 + 2 *(3*cosh(-a + c)^2 + 1)*sinh(-a + c)^2 + 2*cosh(-a + c)^2 - 4*(cosh(-a + c )^3 + cosh(-a + c))*sinh(-a + c) + 1)*sinh(b*x + c))*log(cosh(b*x + c) + s inh(b*x + c) - 1) + 4*(cosh(b*x + c)*cosh(-a + c)^4 - 4*cosh(b*x + c)*cosh (-a + c)^3*sinh(-a + c) + 6*cosh(b*x + c)*cosh(-a + c)^2*sinh(-a + c)^2...
\[ \int \cosh ^2(a+b x) \text {csch}(c+b x) \, dx=\int \cosh ^{2}{\left (a + b x \right )} \operatorname {csch}{\left (b x + c \right )}\, dx \] Input:
integrate(cosh(b*x+a)**2*csch(b*x+c),x)
Output:
Integral(cosh(a + b*x)**2*csch(b*x + c), x)
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (36) = 72\).
Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.22 \[ \int \cosh ^2(a+b x) \text {csch}(c+b x) \, dx=-\frac {{\left (e^{\left (4 \, a\right )} + 2 \, e^{\left (2 \, a + 2 \, c\right )} + e^{\left (4 \, c\right )}\right )} e^{\left (-2 \, a - 2 \, c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{4 \, b} + \frac {{\left (e^{\left (4 \, a\right )} + 2 \, e^{\left (2 \, a + 2 \, c\right )} + e^{\left (4 \, c\right )}\right )} e^{\left (-2 \, a - 2 \, c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{4 \, b} + \frac {e^{\left (b x + 2 \, a - c\right )}}{2 \, b} + \frac {e^{\left (-b x - 2 \, a + c\right )}}{2 \, b} \] Input:
integrate(cosh(b*x+a)^2*csch(b*x+c),x, algorithm="maxima")
Output:
-1/4*(e^(4*a) + 2*e^(2*a + 2*c) + e^(4*c))*e^(-2*a - 2*c)*log(e^(-b*x) + e ^c)/b + 1/4*(e^(4*a) + 2*e^(2*a + 2*c) + e^(4*c))*e^(-2*a - 2*c)*log(e^(-b *x) - e^c)/b + 1/2*e^(b*x + 2*a - c)/b + 1/2*e^(-b*x - 2*a + c)/b
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (36) = 72\).
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.31 \[ \int \cosh ^2(a+b x) \text {csch}(c+b x) \, dx=-\frac {{\left (e^{\left (4 \, a + c\right )} + 2 \, e^{\left (2 \, a + 3 \, c\right )} + e^{\left (5 \, c\right )}\right )} e^{\left (-2 \, a - 3 \, c\right )} \log \left (e^{\left (b x + c\right )} + 1\right )}{4 \, b} + \frac {{\left (e^{\left (4 \, a + c\right )} + 2 \, e^{\left (2 \, a + 3 \, c\right )} + e^{\left (5 \, c\right )}\right )} e^{\left (-2 \, a - 3 \, c\right )} \log \left ({\left | e^{\left (b x + c\right )} - 1 \right |}\right )}{4 \, b} + \frac {e^{\left (b x + 2 \, a - c\right )}}{2 \, b} + \frac {e^{\left (-b x - 2 \, a + c\right )}}{2 \, b} \] Input:
integrate(cosh(b*x+a)^2*csch(b*x+c),x, algorithm="giac")
Output:
-1/4*(e^(4*a + c) + 2*e^(2*a + 3*c) + e^(5*c))*e^(-2*a - 3*c)*log(e^(b*x + c) + 1)/b + 1/4*(e^(4*a + c) + 2*e^(2*a + 3*c) + e^(5*c))*e^(-2*a - 3*c)* log(abs(e^(b*x + c) - 1))/b + 1/2*e^(b*x + 2*a - c)/b + 1/2*e^(-b*x - 2*a + c)/b
Time = 0.37 (sec) , antiderivative size = 202, normalized size of antiderivative = 5.61 \[ \int \cosh ^2(a+b x) \text {csch}(c+b x) \, dx=\frac {{\mathrm {e}}^{c-2\,a-b\,x}}{2\,b}+\frac {{\mathrm {e}}^{2\,a-c+b\,x}}{2\,b}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-2\,a}\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{b\,x}\,\left (\sqrt {-b^2}+2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\,\sqrt {-b^2}+{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}\,\sqrt {-b^2}\right )}{b\,\sqrt {{\mathrm {e}}^{-4\,a}\,{\mathrm {e}}^{4\,c}\,\left (4\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+6\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}+4\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,c}+{\mathrm {e}}^{8\,a}\,{\mathrm {e}}^{-8\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{4\,c-4\,a}\,\left (4\,{\mathrm {e}}^{2\,a-2\,c}+6\,{\mathrm {e}}^{4\,a-4\,c}+4\,{\mathrm {e}}^{6\,a-6\,c}+{\mathrm {e}}^{8\,a-8\,c}+1\right )}}{2\,\sqrt {-b^2}} \] Input:
int(cosh(a + b*x)^2/sinh(c + b*x),x)
Output:
exp(c - 2*a - b*x)/(2*b) + exp(2*a - c + b*x)/(2*b) - (atan((exp(-2*a)*exp (3*c)*exp(b*x)*((-b^2)^(1/2) + 2*exp(2*a)*exp(-2*c)*(-b^2)^(1/2) + exp(4*a )*exp(-4*c)*(-b^2)^(1/2)))/(b*(exp(-4*a)*exp(4*c)*(4*exp(2*a)*exp(-2*c) + 6*exp(4*a)*exp(-4*c) + 4*exp(6*a)*exp(-6*c) + exp(8*a)*exp(-8*c) + 1))^(1/ 2)))*(exp(4*c - 4*a)*(4*exp(2*a - 2*c) + 6*exp(4*a - 4*c) + 4*exp(6*a - 6* c) + exp(8*a - 8*c) + 1))^(1/2))/(2*(-b^2)^(1/2))
Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 4.72 \[ \int \cosh ^2(a+b x) \text {csch}(c+b x) \, dx=\frac {2 e^{2 b x +4 a +c}+e^{b x +4 a} \mathrm {log}\left (e^{b x +c}-1\right )-e^{b x +4 a} \mathrm {log}\left (e^{b x +c}+1\right )+2 e^{b x +2 a +2 c} \mathrm {log}\left (e^{b x +c}-1\right )-2 e^{b x +2 a +2 c} \mathrm {log}\left (e^{b x +c}+1\right )+e^{b x +4 c} \mathrm {log}\left (e^{b x +c}-1\right )-e^{b x +4 c} \mathrm {log}\left (e^{b x +c}+1\right )+2 e^{3 c}}{4 e^{b x +2 a +2 c} b} \] Input:
int(cosh(b*x+a)^2*csch(b*x+c),x)
Output:
(2*e**(4*a + 2*b*x + c) + e**(4*a + b*x)*log(e**(b*x + c) - 1) - e**(4*a + b*x)*log(e**(b*x + c) + 1) + 2*e**(2*a + b*x + 2*c)*log(e**(b*x + c) - 1) - 2*e**(2*a + b*x + 2*c)*log(e**(b*x + c) + 1) + e**(b*x + 4*c)*log(e**(b *x + c) - 1) - e**(b*x + 4*c)*log(e**(b*x + c) + 1) + 2*e**(3*c))/(4*e**(2 *a + b*x + 2*c)*b)