Integrand size = 21, antiderivative size = 56 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {a \coth (c+d x)}{d}-\frac {a \coth ^3(c+d x)}{3 d}+\frac {b \log (\tanh (c+d x))}{d}-\frac {b \tanh ^2(c+d x)}{2 d} \] Output:
a*coth(d*x+c)/d-1/3*a*coth(d*x+c)^3/d+b*ln(tanh(d*x+c))/d-1/2*b*tanh(d*x+c )^2/d
Time = 0.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.32 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {2 a \coth (c+d x)}{3 d}-\frac {a \coth (c+d x) \text {csch}^2(c+d x)}{3 d}-\frac {b \left (2 \log (\cosh (c+d x))-2 \log (\sinh (c+d x))-\text {sech}^2(c+d x)\right )}{2 d} \] Input:
Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3),x]
Output:
(2*a*Coth[c + d*x])/(3*d) - (a*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (b*( 2*Log[Cosh[c + d*x]] - 2*Log[Sinh[c + d*x]] - Sech[c + d*x]^2))/(2*d)
Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 4146, 2333, 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.
\(\displaystyle \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+i b \tan (i c+i d x)^3}{\sin (i c+i d x)^4}dx\) |
\(\Big \downarrow \) 4146 |
\(\displaystyle \frac {\int \coth ^4(c+d x) \left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^3(c+d x)+a\right )d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 2333 |
\(\displaystyle \frac {\int \left (a \coth ^4(c+d x)-a \coth ^2(c+d x)+b \coth (c+d x)-b \tanh (c+d x)\right )d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{3} a \coth ^3(c+d x)+a \coth (c+d x)-\frac {1}{2} b \tanh ^2(c+d x)+b \log (\tanh (c+d x))}{d}\) |
Input:
Int[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3),x]
Output:
(a*Coth[c + d*x] - (a*Coth[c + d*x]^3)/3 + b*Log[Tanh[c + d*x]] - (b*Tanh[ c + d*x]^2)/2)/d
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Time = 8.78 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {a \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+b \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\) | \(46\) |
default | \(\frac {a \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+b \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\) | \(46\) |
risch | \(-\frac {2 \left (-3 \,{\mathrm e}^{8 d x +8 c} b +6 \,{\mathrm e}^{6 d x +6 c} a +9 \,{\mathrm e}^{6 d x +6 c} b +10 \,{\mathrm e}^{4 d x +4 c} a -9 b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +3 \,{\mathrm e}^{2 d x +2 c} b -2 a \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d}\) | \(156\) |
Input:
int(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)+b*(1/2/cosh(d*x+c)^2+ln(tanh(d* x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 1739 vs. \(2 (52) = 104\).
Time = 0.10 (sec) , antiderivative size = 1739, normalized size of antiderivative = 31.05 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3),x, algorithm="fricas")
Output:
1/3*(6*b*cosh(d*x + c)^8 + 48*b*cosh(d*x + c)*sinh(d*x + c)^7 + 6*b*sinh(d *x + c)^8 - 6*(2*a + 3*b)*cosh(d*x + c)^6 + 6*(28*b*cosh(d*x + c)^2 - 2*a - 3*b)*sinh(d*x + c)^6 + 12*(28*b*cosh(d*x + c)^3 - 3*(2*a + 3*b)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(10*a - 9*b)*cosh(d*x + c)^4 + 2*(210*b*cosh(d* x + c)^4 - 45*(2*a + 3*b)*cosh(d*x + c)^2 - 10*a + 9*b)*sinh(d*x + c)^4 + 8*(42*b*cosh(d*x + c)^5 - 15*(2*a + 3*b)*cosh(d*x + c)^3 - (10*a - 9*b)*co sh(d*x + c))*sinh(d*x + c)^3 - 2*(2*a + 3*b)*cosh(d*x + c)^2 + 2*(84*b*cos h(d*x + c)^6 - 45*(2*a + 3*b)*cosh(d*x + c)^4 - 6*(10*a - 9*b)*cosh(d*x + c)^2 - 2*a - 3*b)*sinh(d*x + c)^2 - 3*(b*cosh(d*x + c)^10 + 10*b*cosh(d*x + c)*sinh(d*x + c)^9 + b*sinh(d*x + c)^10 - b*cosh(d*x + c)^8 + (45*b*cosh (d*x + c)^2 - b)*sinh(d*x + c)^8 + 8*(15*b*cosh(d*x + c)^3 - b*cosh(d*x + c))*sinh(d*x + c)^7 - 2*b*cosh(d*x + c)^6 + 2*(105*b*cosh(d*x + c)^4 - 14* b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^6 + 4*(63*b*cosh(d*x + c)^5 - 14*b*co sh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^5 + 2*b*cosh(d*x + c)^4 + 2*(105*b*cosh(d*x + c)^6 - 35*b*cosh(d*x + c)^4 - 15*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^4 + 8*(15*b*cosh(d*x + c)^7 - 7*b*cosh(d*x + c)^5 - 5*b*c osh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c)^3 + b*cosh(d*x + c)^2 + (4 5*b*cosh(d*x + c)^8 - 28*b*cosh(d*x + c)^6 - 30*b*cosh(d*x + c)^4 + 12*b*c osh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 2*(5*b*cosh(d*x + c)^9 - 4*b*cosh(d* x + c)^7 - 6*b*cosh(d*x + c)^5 + 4*b*cosh(d*x + c)^3 + b*cosh(d*x + c))...
\[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right ) \operatorname {csch}^{4}{\left (c + d x \right )}\, dx \] Input:
integrate(csch(d*x+c)**4*(a+b*tanh(d*x+c)**3),x)
Output:
Integral((a + b*tanh(c + d*x)**3)*csch(c + d*x)**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (52) = 104\).
Time = 0.13 (sec) , antiderivative size = 184, normalized size of antiderivative = 3.29 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {4}{3} \, a {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \] Input:
integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3),x, algorithm="maxima")
Output:
b*(log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d - log(e^(-2*d*x - 2*c ) + 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) + 4/3*a*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4* c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c ) + e^(-6*d*x - 6*c) - 1)))
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (52) = 104\).
Time = 0.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.66 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=-\frac {6 \, b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 6 \, b \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) - \frac {3 \, {\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 10 \, b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}} + \frac {11 \, b e^{\left (6 \, d x + 6 \, c\right )} - 33 \, b e^{\left (4 \, d x + 4 \, c\right )} + 24 \, a e^{\left (2 \, d x + 2 \, c\right )} + 33 \, b e^{\left (2 \, d x + 2 \, c\right )} - 8 \, a - 11 \, b}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \] Input:
integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3),x, algorithm="giac")
Output:
-1/6*(6*b*log(e^(2*d*x + 2*c) + 1) - 6*b*log(abs(e^(2*d*x + 2*c) - 1)) - 3 *(3*b*e^(4*d*x + 4*c) + 10*b*e^(2*d*x + 2*c) + 3*b)/(e^(2*d*x + 2*c) + 1)^ 2 + (11*b*e^(6*d*x + 6*c) - 33*b*e^(4*d*x + 4*c) + 24*a*e^(2*d*x + 2*c) + 33*b*e^(2*d*x + 2*c) - 8*a - 11*b)/(e^(2*d*x + 2*c) - 1)^3)/d
Time = 2.31 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.89 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {2\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {4\,a}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,b}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {8\,a}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {2\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {-d^2}} \] Input:
int((a + b*tanh(c + d*x)^3)/sinh(c + d*x)^4,x)
Output:
(2*b)/(d*(exp(2*c + 2*d*x) + 1)) - (4*a)/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (2*b)/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - ( 8*a)/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) ) - (2*atan((b*exp(2*c)*exp(2*d*x)*(-d^2)^(1/2))/(d*(b^2)^(1/2)))*(b^2)^(1 /2))/(-d^2)^(1/2)
Time = 0.22 (sec) , antiderivative size = 552, normalized size of antiderivative = 9.86 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {4 a -6 b +3 \,\mathrm {log}\left (e^{2 d x +2 c}+1\right ) b -3 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b +6 e^{10 d x +10 c} b +30 e^{4 d x +4 c} b -20 e^{4 d x +4 c} a -30 e^{6 d x +6 c} b -3 \,\mathrm {log}\left (e^{d x +c}-1\right ) b -3 \,\mathrm {log}\left (e^{d x +c}+1\right ) b -3 e^{10 d x +10 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b +3 e^{10 d x +10 c} \mathrm {log}\left (e^{d x +c}-1\right ) b +3 e^{10 d x +10 c} \mathrm {log}\left (e^{d x +c}+1\right ) b -3 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}-1\right ) b -3 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}+1\right ) b -4 e^{2 d x +2 c} a -12 e^{6 d x +6 c} a -6 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) b -6 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) b +6 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) b +6 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) b +3 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) b +3 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) b +3 e^{8 d x +8 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b +6 e^{6 d x +6 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b -6 e^{4 d x +4 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b}{3 d \left (e^{10 d x +10 c}-e^{8 d x +8 c}-2 e^{6 d x +6 c}+2 e^{4 d x +4 c}+e^{2 d x +2 c}-1\right )} \] Input:
int(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3),x)
Output:
( - 3*e**(10*c + 10*d*x)*log(e**(2*c + 2*d*x) + 1)*b + 3*e**(10*c + 10*d*x )*log(e**(c + d*x) - 1)*b + 3*e**(10*c + 10*d*x)*log(e**(c + d*x) + 1)*b + 6*e**(10*c + 10*d*x)*b + 3*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x) + 1)*b - 3*e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*b - 3*e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*b + 6*e**(6*c + 6*d*x)*log(e**(2*c + 2*d*x) + 1)*b - 6*e**(6* c + 6*d*x)*log(e**(c + d*x) - 1)*b - 6*e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*b - 12*e**(6*c + 6*d*x)*a - 30*e**(6*c + 6*d*x)*b - 6*e**(4*c + 4*d*x) *log(e**(2*c + 2*d*x) + 1)*b + 6*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*b + 6*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*b - 20*e**(4*c + 4*d*x)*a + 30* e**(4*c + 4*d*x)*b - 3*e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x) + 1)*b + 3*e* *(2*c + 2*d*x)*log(e**(c + d*x) - 1)*b + 3*e**(2*c + 2*d*x)*log(e**(c + d* x) + 1)*b - 4*e**(2*c + 2*d*x)*a + 3*log(e**(2*c + 2*d*x) + 1)*b - 3*log(e **(c + d*x) - 1)*b - 3*log(e**(c + d*x) + 1)*b + 4*a - 6*b)/(3*d*(e**(10*c + 10*d*x) - e**(8*c + 8*d*x) - 2*e**(6*c + 6*d*x) + 2*e**(4*c + 4*d*x) + e**(2*c + 2*d*x) - 1))