Integrand size = 21, antiderivative size = 71 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {b \arctan (\sinh (c+d x))}{2 d}+\frac {a \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {b \text {sech}(c+d x) \tanh (c+d x)}{2 d} \] Output:
1/2*b*arctan(sinh(d*x+c))/d+1/2*a*arctanh(cosh(d*x+c))/d-1/2*a*coth(d*x+c) *csch(d*x+c)/d+1/2*b*sech(d*x+c)*tanh(d*x+c)/d
Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.61 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {b \arctan (\sinh (c+d x))}{2 d}-\frac {a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {b \text {sech}(c+d x) \tanh (c+d x)}{2 d} \] Input:
Integrate[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^3),x]
Output:
(b*ArcTan[Sinh[c + d*x]])/(2*d) - (a*Csch[(c + d*x)/2]^2)/(8*d) + (a*Log[C osh[(c + d*x)/2]])/(2*d) - (a*Log[Sinh[(c + d*x)/2]])/(2*d) - (a*Sech[(c + d*x)/2]^2)/(8*d) + (b*Sech[c + d*x]*Tanh[c + d*x])/(2*d)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 26, 4149, 2009}
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.
\(\displaystyle \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \left (a+i b \tan (i c+i d x)^3\right )}{\sin (i c+i d x)^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {i b \tan (i c+i d x)^3+a}{\sin (i c+i d x)^3}dx\) |
\(\Big \downarrow \) 4149 |
\(\displaystyle -i \int \left (i a \text {csch}^3(c+d x)+i b \text {sech}^3(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -i \left (\frac {i a \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {i a \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {i b \arctan (\sinh (c+d x))}{2 d}+\frac {i b \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )\) |
Input:
Int[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^3),x]
Output:
(-I)*(((I/2)*b*ArcTan[Sinh[c + d*x]])/d + ((I/2)*a*ArcTanh[Cosh[c + d*x]]) /d - ((I/2)*a*Coth[c + d*x]*Csch[c + d*x])/d + ((I/2)*b*Sech[c + d*x]*Tanh [c + d*x])/d)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
Time = 4.77 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(54\) |
default | \(\frac {a \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(54\) |
risch | \(-\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{6 d x +6 c} a -{\mathrm e}^{6 d x +6 c} b +3 \,{\mathrm e}^{4 d x +4 c} a +3 b \,{\mathrm e}^{4 d x +4 c}+3 \,{\mathrm e}^{2 d x +2 c} a -3 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {a \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {a \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}+\frac {i b \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}-\frac {i b \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}\) | \(177\) |
Input:
int(csch(d*x+c)^3*(a+b*tanh(d*x+c)^3),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+b*(1/2*sech(d*x+ c)*tanh(d*x+c)+arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 1188 vs. \(2 (63) = 126\).
Time = 0.11 (sec) , antiderivative size = 1188, normalized size of antiderivative = 16.73 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^3),x, algorithm="fricas")
Output:
-1/2*(2*(a - b)*cosh(d*x + c)^7 + 14*(a - b)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(a - b)*sinh(d*x + c)^7 + 6*(a + b)*cosh(d*x + c)^5 + 6*(7*(a - b)*co sh(d*x + c)^2 + a + b)*sinh(d*x + c)^5 + 10*(7*(a - b)*cosh(d*x + c)^3 + 3 *(a + b)*cosh(d*x + c))*sinh(d*x + c)^4 + 6*(a - b)*cosh(d*x + c)^3 + 2*(3 5*(a - b)*cosh(d*x + c)^4 + 30*(a + b)*cosh(d*x + c)^2 + 3*a - 3*b)*sinh(d *x + c)^3 + 6*(7*(a - b)*cosh(d*x + c)^5 + 10*(a + b)*cosh(d*x + c)^3 + 3* (a - b)*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(b*cosh(d*x + c)^8 + 56*b*cosh( d*x + c)^3*sinh(d*x + c)^5 + 28*b*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*b*co sh(d*x + c)*sinh(d*x + c)^7 + b*sinh(d*x + c)^8 - 2*b*cosh(d*x + c)^4 + 2* (35*b*cosh(d*x + c)^4 - b)*sinh(d*x + c)^4 + 8*(7*b*cosh(d*x + c)^5 - b*co sh(d*x + c))*sinh(d*x + c)^3 + 4*(7*b*cosh(d*x + c)^6 - 3*b*cosh(d*x + c)^ 2)*sinh(d*x + c)^2 + 8*(b*cosh(d*x + c)^7 - b*cosh(d*x + c)^3)*sinh(d*x + c) + b)*arctan(cosh(d*x + c) + sinh(d*x + c)) + 2*(a + b)*cosh(d*x + c) - (a*cosh(d*x + c)^8 + 56*a*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*a*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*a*cosh(d*x + c)*sinh(d*x + c)^7 + a*sinh(d*x + c)^8 - 2*a*cosh(d*x + c)^4 + 2*(35*a*cosh(d*x + c)^4 - a)*sinh(d*x + c)^4 + 8*(7*a*cosh(d*x + c)^5 - a*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a*cosh( d*x + c)^6 - 3*a*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(a*cosh(d*x + c)^7 - a*cosh(d*x + c)^3)*sinh(d*x + c) + a)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + (a*cosh(d*x + c)^8 + 56*a*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*a*...
\[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right ) \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(csch(d*x+c)**3*(a+b*tanh(d*x+c)**3),x)
Output:
Integral((a + b*tanh(c + d*x)**3)*csch(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (63) = 126\).
Time = 0.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.20 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=-b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {1}{2} \, a {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^3),x, algorithm="maxima")
Output:
-b*(arctan(e^(-d*x - c))/d - (e^(-d*x - c) - e^(-3*d*x - 3*c))/(d*(2*e^(-2 *d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) + 1/2*a*(log(e^(-d*x - c) + 1)/d - l og(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d *x - 2*c) - e^(-4*d*x - 4*c) - 1)))
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (63) = 126\).
Time = 0.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.01 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {2 \, b \arctan \left (e^{\left (d x + c\right )}\right ) + a \log \left (e^{\left (d x + c\right )} + 1\right ) - a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, {\left (a e^{\left (7 \, d x + 7 \, c\right )} - b e^{\left (7 \, d x + 7 \, c\right )} + 3 \, a e^{\left (5 \, d x + 5 \, c\right )} + 3 \, b e^{\left (5 \, d x + 5 \, c\right )} + 3 \, a e^{\left (3 \, d x + 3 \, c\right )} - 3 \, b e^{\left (3 \, d x + 3 \, c\right )} + a e^{\left (d x + c\right )} + b e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}^{2}}}{2 \, d} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^3),x, algorithm="giac")
Output:
1/2*(2*b*arctan(e^(d*x + c)) + a*log(e^(d*x + c) + 1) - a*log(abs(e^(d*x + c) - 1)) - 2*(a*e^(7*d*x + 7*c) - b*e^(7*d*x + 7*c) + 3*a*e^(5*d*x + 5*c) + 3*b*e^(5*d*x + 5*c) + 3*a*e^(3*d*x + 3*c) - 3*b*e^(3*d*x + 3*c) + a*e^( d*x + c) + b*e^(d*x + c))/(e^(4*d*x + 4*c) - 1)^2)/d
Time = 3.62 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.44 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {a\,\ln \left ({\mathrm {e}}^{c+d\,x}+1\right )}{2\,d}-\frac {\frac {4\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a-b\right )}{d}+\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )}{d}}{{\mathrm {e}}^{8\,c+8\,d\,x}-2\,{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {a\,\ln \left ({\mathrm {e}}^{c+d\,x}-1\right )}{2\,d}-\frac {\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a-b\right )}{d}+\frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1}-\frac {b\,\ln \left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d}+\frac {b\,\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d} \] Input:
int((a + b*tanh(c + d*x)^3)/sinh(c + d*x)^3,x)
Output:
(a*log(exp(c + d*x) + 1))/(2*d) - ((4*exp(3*c + 3*d*x)*(a - b))/d + (4*exp (c + d*x)*(a + b))/d)/(exp(8*c + 8*d*x) - 2*exp(4*c + 4*d*x) + 1) - (a*log (exp(c + d*x) - 1))/(2*d) - ((exp(3*c + 3*d*x)*(a - b))/d + (3*exp(c + d*x )*(a + b))/d)/(exp(4*c + 4*d*x) - 1) - (b*log(exp(c + d*x) - 1i)*1i)/(2*d) + (b*log(exp(c + d*x) + 1i)*1i)/(2*d)
Time = 0.23 (sec) , antiderivative size = 299, normalized size of antiderivative = 4.21 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {2 e^{8 d x +8 c} \mathit {atan} \left (e^{d x +c}\right ) b -4 e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) b +2 \mathit {atan} \left (e^{d x +c}\right ) b -e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}-1\right ) a +e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}+1\right ) a -2 e^{7 d x +7 c} a +2 e^{7 d x +7 c} b -6 e^{5 d x +5 c} a -6 e^{5 d x +5 c} b +2 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a -2 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a -6 e^{3 d x +3 c} a +6 e^{3 d x +3 c} b -2 e^{d x +c} a -2 e^{d x +c} b -\mathrm {log}\left (e^{d x +c}-1\right ) a +\mathrm {log}\left (e^{d x +c}+1\right ) a}{2 d \left (e^{8 d x +8 c}-2 e^{4 d x +4 c}+1\right )} \] Input:
int(csch(d*x+c)^3*(a+b*tanh(d*x+c)^3),x)
Output:
(2*e**(8*c + 8*d*x)*atan(e**(c + d*x))*b - 4*e**(4*c + 4*d*x)*atan(e**(c + d*x))*b + 2*atan(e**(c + d*x))*b - e**(8*c + 8*d*x)*log(e**(c + d*x) - 1) *a + e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*a - 2*e**(7*c + 7*d*x)*a + 2*e **(7*c + 7*d*x)*b - 6*e**(5*c + 5*d*x)*a - 6*e**(5*c + 5*d*x)*b + 2*e**(4* c + 4*d*x)*log(e**(c + d*x) - 1)*a - 2*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a - 6*e**(3*c + 3*d*x)*a + 6*e**(3*c + 3*d*x)*b - 2*e**(c + d*x)*a - 2 *e**(c + d*x)*b - log(e**(c + d*x) - 1)*a + log(e**(c + d*x) + 1)*a)/(2*d* (e**(8*c + 8*d*x) - 2*e**(4*c + 4*d*x) + 1))