Integrand size = 21, antiderivative size = 51 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {(a-2 b) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {b \text {sech}(c+d x)}{d} \] Output:
1/2*(a-2*b)*arctanh(cosh(d*x+c))/d-1/2*a*coth(d*x+c)*csch(d*x+c)/d+b*sech( d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(123\) vs. \(2(51)=102\).
Time = 0.18 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.41 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {a \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {b \text {sech}(c+d x)}{d} \] Input:
Integrate[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^2),x]
Output:
-1/8*(a*Csch[(c + d*x)/2]^2)/d + (a*Log[Cosh[(c + d*x)/2]])/(2*d) - (b*Log [Cosh[(c + d*x)/2]])/d - (a*Log[Sinh[(c + d*x)/2]])/(2*d) + (b*Log[Sinh[(c + d*x)/2]])/d - (a*Sech[(c + d*x)/2]^2)/(8*d) + (b*Sech[c + d*x])/d
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 26, 4147, 360, 299, 219}
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.
\(\displaystyle \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \left (a-b \tan (i c+i d x)^2\right )}{\sin (i c+i d x)^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {a-b \tan (i c+i d x)^2}{\sin (i c+i d x)^3}dx\) |
\(\Big \downarrow \) 4147 |
\(\displaystyle -\frac {\int \frac {\text {sech}^2(c+d x) \left (-b \text {sech}^2(c+d x)+a+b\right )}{\left (1-\text {sech}^2(c+d x)\right )^2}d\text {sech}(c+d x)}{d}\) |
\(\Big \downarrow \) 360 |
\(\displaystyle -\frac {\frac {a \text {sech}(c+d x)}{2 \left (1-\text {sech}^2(c+d x)\right )}-\frac {1}{2} \int \frac {a-2 b \text {sech}^2(c+d x)}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)}{d}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle -\frac {\frac {1}{2} \left (-(a-2 b) \int \frac {1}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)-2 b \text {sech}(c+d x)\right )+\frac {a \text {sech}(c+d x)}{2 \left (1-\text {sech}^2(c+d x)\right )}}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {1}{2} (-((a-2 b) \text {arctanh}(\text {sech}(c+d x)))-2 b \text {sech}(c+d x))+\frac {a \text {sech}(c+d x)}{2 \left (1-\text {sech}^2(c+d x)\right )}}{d}\) |
Input:
Int[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^2),x]
Output:
-(((-((a - 2*b)*ArcTanh[Sech[c + d*x]]) - 2*b*Sech[c + d*x])/2 + (a*Sech[c + d*x])/(2*(1 - Sech[c + d*x]^2)))/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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Simp[1/(f*ff^ m) Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m + 1 )), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[( m - 1)/2]
Time = 2.94 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b \left (\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(50\) |
default | \(\frac {a \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b \left (\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(50\) |
risch | \(-\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{4 d x +4 c} a -2 b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 \,{\mathrm e}^{2 d x +2 c} b +a -2 b \right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {a \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d}+\frac {a \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d}\) | \(150\) |
Input:
int(csch(d*x+c)^3*(a+tanh(d*x+c)^2*b),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+b*(1/cosh(d*x+c) -2*arctanh(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (47) = 94\).
Time = 0.10 (sec) , antiderivative size = 924, normalized size of antiderivative = 18.12 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^2),x, algorithm="fricas")
Output:
-1/2*(2*(a - 2*b)*cosh(d*x + c)^5 + 10*(a - 2*b)*cosh(d*x + c)*sinh(d*x + c)^4 + 2*(a - 2*b)*sinh(d*x + c)^5 + 4*(a + 2*b)*cosh(d*x + c)^3 + 4*(5*(a - 2*b)*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^3 + 4*(5*(a - 2*b)*cosh(d *x + c)^3 + 3*(a + 2*b)*cosh(d*x + c))*sinh(d*x + c)^2 + 2*(a - 2*b)*cosh( d*x + c) - ((a - 2*b)*cosh(d*x + c)^6 + 6*(a - 2*b)*cosh(d*x + c)*sinh(d*x + c)^5 + (a - 2*b)*sinh(d*x + c)^6 - (a - 2*b)*cosh(d*x + c)^4 + (15*(a - 2*b)*cosh(d*x + c)^2 - a + 2*b)*sinh(d*x + c)^4 + 4*(5*(a - 2*b)*cosh(d*x + c)^3 - (a - 2*b)*cosh(d*x + c))*sinh(d*x + c)^3 - (a - 2*b)*cosh(d*x + c)^2 + (15*(a - 2*b)*cosh(d*x + c)^4 - 6*(a - 2*b)*cosh(d*x + c)^2 - a + 2 *b)*sinh(d*x + c)^2 + 2*(3*(a - 2*b)*cosh(d*x + c)^5 - 2*(a - 2*b)*cosh(d* x + c)^3 - (a - 2*b)*cosh(d*x + c))*sinh(d*x + c) + a - 2*b)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a - 2*b)*cosh(d*x + c)^6 + 6*(a - 2*b)*cosh( d*x + c)*sinh(d*x + c)^5 + (a - 2*b)*sinh(d*x + c)^6 - (a - 2*b)*cosh(d*x + c)^4 + (15*(a - 2*b)*cosh(d*x + c)^2 - a + 2*b)*sinh(d*x + c)^4 + 4*(5*( a - 2*b)*cosh(d*x + c)^3 - (a - 2*b)*cosh(d*x + c))*sinh(d*x + c)^3 - (a - 2*b)*cosh(d*x + c)^2 + (15*(a - 2*b)*cosh(d*x + c)^4 - 6*(a - 2*b)*cosh(d *x + c)^2 - a + 2*b)*sinh(d*x + c)^2 + 2*(3*(a - 2*b)*cosh(d*x + c)^5 - 2* (a - 2*b)*cosh(d*x + c)^3 - (a - 2*b)*cosh(d*x + c))*sinh(d*x + c) + a - 2 *b)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(5*(a - 2*b)*cosh(d*x + c)^ 4 + 6*(a + 2*b)*cosh(d*x + c)^2 + a - 2*b)*sinh(d*x + c))/(d*cosh(d*x +...
\[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(csch(d*x+c)**3*(a+b*tanh(d*x+c)**2),x)
Output:
Integral((a + b*tanh(c + d*x)**2)*csch(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (47) = 94\).
Time = 0.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.98 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {1}{2} \, a {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {2 \, e^{\left (-d x - c\right )}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^2),x, algorithm="maxima")
Output:
1/2*a*(log(e^(-d*x - c) + 1)/d - log(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))) - b* (log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d - 2*e^(-d*x - c)/(d*(e^ (-2*d*x - 2*c) + 1)))
Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (47) = 94\).
Time = 0.13 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.78 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {{\left (a - 2 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - {\left (a - 2 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 2 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 8 \, b\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 4 \, e^{\left (d x + c\right )} - 4 \, e^{\left (-d x - c\right )}}}{4 \, d} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^2),x, algorithm="giac")
Output:
1/4*((a - 2*b)*log(e^(d*x + c) + e^(-d*x - c) + 2) - (a - 2*b)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 4*(a*(e^(d*x + c) + e^(-d*x - c))^2 - 2*b*(e^(d *x + c) + e^(-d*x - c))^2 + 8*b)/((e^(d*x + c) + e^(-d*x - c))^3 - 4*e^(d* x + c) - 4*e^(-d*x - c)))/d
Time = 2.38 (sec) , antiderivative size = 156, normalized size of antiderivative = 3.06 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a\,\sqrt {-d^2}-2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^2-4\,a\,b+4\,b^2}}\right )\,\sqrt {a^2-4\,a\,b+4\,b^2}}{\sqrt {-d^2}}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:
int((a + b*tanh(c + d*x)^2)/sinh(c + d*x)^3,x)
Output:
(atan((exp(d*x)*exp(c)*(a*(-d^2)^(1/2) - 2*b*(-d^2)^(1/2)))/(d*(a^2 - 4*a* b + 4*b^2)^(1/2)))*(a^2 - 4*a*b + 4*b^2)^(1/2))/(-d^2)^(1/2) - (a*exp(c + d*x))/(d*(exp(2*c + 2*d*x) - 1)) + (2*b*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1)) - (2*a*exp(c + d*x))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1) )
Time = 0.26 (sec) , antiderivative size = 440, normalized size of antiderivative = 8.63 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {-e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) a +2 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) b +e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) a -2 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) b -2 e^{5 d x +5 c} a +4 e^{5 d x +5 c} b +e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a -2 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) b -e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a +2 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) b -4 e^{3 d x +3 c} a -8 e^{3 d x +3 c} b +e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a -2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) b -e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a +2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) b -2 e^{d x +c} a +4 e^{d x +c} b -\mathrm {log}\left (e^{d x +c}-1\right ) a +2 \,\mathrm {log}\left (e^{d x +c}-1\right ) b +\mathrm {log}\left (e^{d x +c}+1\right ) a -2 \,\mathrm {log}\left (e^{d x +c}+1\right ) b}{2 d \left (e^{6 d x +6 c}-e^{4 d x +4 c}-e^{2 d x +2 c}+1\right )} \] Input:
int(csch(d*x+c)^3*(a+b*tanh(d*x+c)^2),x)
Output:
( - e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*a + 2*e**(6*c + 6*d*x)*log(e**( c + d*x) - 1)*b + e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a - 2*e**(6*c + 6 *d*x)*log(e**(c + d*x) + 1)*b - 2*e**(5*c + 5*d*x)*a + 4*e**(5*c + 5*d*x)* b + 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)*b - 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)*b - 4*e**(3*c + 3*d*x)*a - 8*e**(3*c + 3*d*x)* b + e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a - 2*e**(2*c + 2*d*x)*log(e**( c + d*x) - 1)*b - e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a + 2*e**(2*c + 2 *d*x)*log(e**(c + d*x) + 1)*b - 2*e**(c + d*x)*a + 4*e**(c + d*x)*b - log( e**(c + d*x) - 1)*a + 2*log(e**(c + d*x) - 1)*b + log(e**(c + d*x) + 1)*a - 2*log(e**(c + d*x) + 1)*b)/(2*d*(e**(6*c + 6*d*x) - e**(4*c + 4*d*x) - e **(2*c + 2*d*x) + 1))