Integrand size = 21, antiderivative size = 125 \[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {15 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{8 (a+b)^{7/2} d}+\frac {\cosh (c+d x)}{(a+b)^3 d}-\frac {b \text {sech}(c+d x)}{4 (a+b)^2 d \left (a+b-b \text {sech}^2(c+d x)\right )^2}-\frac {7 b \text {sech}(c+d x)}{8 (a+b)^3 d \left (a+b-b \text {sech}^2(c+d x)\right )} \] Output:
-15/8*b^(1/2)*arctanh(b^(1/2)*sech(d*x+c)/(a+b)^(1/2))/(a+b)^(7/2)/d+cosh( d*x+c)/(a+b)^3/d-1/4*b*sech(d*x+c)/(a+b)^2/d/(a+b-b*sech(d*x+c)^2)^2-7/8*b *sech(d*x+c)/(a+b)^3/d/(a+b-b*sech(d*x+c)^2)
Result contains complex when optimal does not.
Time = 1.41 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.26 \[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {-\frac {15 i \sqrt {b} \left (\arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+\arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )\right )}{(a+b)^{7/2}}+\frac {2 \cosh (c+d x) \left (4-\frac {4 b^2}{(a-b+(a+b) \cosh (2 (c+d x)))^2}-\frac {9 b}{a-b+(a+b) \cosh (2 (c+d x))}\right )}{(a+b)^3}}{8 d} \] Input:
Integrate[Sinh[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(((-15*I)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/S qrt[b]] + ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])) /(a + b)^(7/2) + (2*Cosh[c + d*x]*(4 - (4*b^2)/(a - b + (a + b)*Cosh[2*(c + d*x)])^2 - (9*b)/(a - b + (a + b)*Cosh[2*(c + d*x)])))/(a + b)^3)/(8*d)
Time = 0.33 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 26, 4147, 253, 253, 264, 221}
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.
| \(\displaystyle \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i c+i d x)}{\left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i c+i d x)}{\left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle -\frac {\int \frac {\cosh ^2(c+d x)}{\left (-b \text {sech}^2(c+d x)+a+b\right )^3}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle -\frac {\frac {5 \int \frac {\cosh ^2(c+d x)}{\left (-b \text {sech}^2(c+d x)+a+b\right )^2}d\text {sech}(c+d x)}{4 (a+b)}+\frac {\cosh (c+d x)}{4 (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle -\frac {\frac {5 \left (\frac {3 \int \frac {\cosh ^2(c+d x)}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{2 (a+b)}+\frac {\cosh (c+d x)}{2 (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}\right )}{4 (a+b)}+\frac {\cosh (c+d x)}{4 (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {\frac {5 \left (\frac {3 \left (\frac {b \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a+b}-\frac {\cosh (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\cosh (c+d x)}{2 (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}\right )}{4 (a+b)}+\frac {\cosh (c+d x)}{4 (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {5 \left (\frac {3 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {\cosh (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\cosh (c+d x)}{2 (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}\right )}{4 (a+b)}+\frac {\cosh (c+d x)}{4 (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{d}\) |
Input:
Int[Sinh[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
-((Cosh[c + d*x]/(4*(a + b)*(a + b - b*Sech[c + d*x]^2)^2) + (5*((3*((Sqrt [b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a + b)^(3/2) - Cosh[c + d*x]/(a + b)))/(2*(a + b)) + Cosh[c + d*x]/(2*(a + b)*(a + b - b*Sech[c + d*x]^2))))/(4*(a + b)))/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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(251\) vs. \(2(111)=222\).
Time = 11.92 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.02
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b \left (\frac {-\frac {\left (9 a^{2}+24 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 a}-\frac {\left (27 a^{3}+78 a^{2} b +88 b^{2} a +16 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 a^{2}}-\frac {\left (27 a^{2}+56 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {9 a}{8}-\frac {b}{4}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}-\frac {15 \,\operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{3}}}{d}\) | \(252\) |
| default | \(\frac {\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b \left (\frac {-\frac {\left (9 a^{2}+24 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 a}-\frac {\left (27 a^{3}+78 a^{2} b +88 b^{2} a +16 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 a^{2}}-\frac {\left (27 a^{2}+56 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {9 a}{8}-\frac {b}{4}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}-\frac {15 \,\operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{3}}}{d}\) | \(252\) |
| risch | \(\frac {{\mathrm e}^{d x +c}}{2 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) d}+\frac {{\mathrm e}^{-d x -c}}{2 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) d}-\frac {\left (9 \,{\mathrm e}^{6 d x +6 c} a +9 \,{\mathrm e}^{6 d x +6 c} b +27 \,{\mathrm e}^{4 d x +4 c} a -b \,{\mathrm e}^{4 d x +4 c}+27 \,{\mathrm e}^{2 d x +2 c} a -{\mathrm e}^{2 d x +2 c} b +9 a +9 b \right ) b \,{\mathrm e}^{d x +c}}{4 \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )^{2} \left (a +b \right ) d \left (a^{2}+2 a b +b^{2}\right )}+\frac {15 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{16 \left (a +b \right )^{4} d}-\frac {15 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{16 \left (a +b \right )^{4} d}\) | \(327\) |
Input:
int(sinh(d*x+c)/(a+tanh(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/(a+b)^3/(tanh(1/2*d*x+1/2*c)+1)-1/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)+2 *b/(a+b)^3*((-1/8*(9*a^2+24*a*b+8*b^2)/a*tanh(1/2*d*x+1/2*c)^6-1/8/a^2*(27 *a^3+78*a^2*b+88*a*b^2+16*b^3)*tanh(1/2*d*x+1/2*c)^4-1/8*(27*a^2+56*a*b+8* b^2)/a*tanh(1/2*d*x+1/2*c)^2-9/8*a-1/4*b)/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh( 1/2*d*x+1/2*c)^2*a+4*b*tanh(1/2*d*x+1/2*c)^2+a)^2-15/16/(a*b+b^2)^(1/2)*ar ctanh(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+2*a+4*b)/(a*b+b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 3792 vs. \(2 (117) = 234\).
Time = 0.20 (sec) , antiderivative size = 7119, normalized size of antiderivative = 56.95 \[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)**2)**3,x)
Output:
Timed out
\[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \] Input:
integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
1/4*(2*a^2 + 4*a*b + 2*b^2 + 2*(a^2*e^(10*c) + 2*a*b*e^(10*c) + b^2*e^(10* c))*e^(10*d*x) + 5*(2*a^2*e^(8*c) - a*b*e^(8*c) - 3*b^2*e^(8*c))*e^(8*d*x) + 5*(4*a^2*e^(6*c) - 7*a*b*e^(6*c) + b^2*e^(6*c))*e^(6*d*x) + 5*(4*a^2*e^ (4*c) - 7*a*b*e^(4*c) + b^2*e^(4*c))*e^(4*d*x) + 5*(2*a^2*e^(2*c) - a*b*e^ (2*c) - 3*b^2*e^(2*c))*e^(2*d*x))/((a^5*d*e^(9*c) + 5*a^4*b*d*e^(9*c) + 10 *a^3*b^2*d*e^(9*c) + 10*a^2*b^3*d*e^(9*c) + 5*a*b^4*d*e^(9*c) + b^5*d*e^(9 *c))*e^(9*d*x) + 4*(a^5*d*e^(7*c) + 3*a^4*b*d*e^(7*c) + 2*a^3*b^2*d*e^(7*c ) - 2*a^2*b^3*d*e^(7*c) - 3*a*b^4*d*e^(7*c) - b^5*d*e^(7*c))*e^(7*d*x) + 2 *(3*a^5*d*e^(5*c) + 7*a^4*b*d*e^(5*c) + 6*a^3*b^2*d*e^(5*c) + 6*a^2*b^3*d* e^(5*c) + 7*a*b^4*d*e^(5*c) + 3*b^5*d*e^(5*c))*e^(5*d*x) + 4*(a^5*d*e^(3*c ) + 3*a^4*b*d*e^(3*c) + 2*a^3*b^2*d*e^(3*c) - 2*a^2*b^3*d*e^(3*c) - 3*a*b^ 4*d*e^(3*c) - b^5*d*e^(3*c))*e^(3*d*x) + (a^5*d*e^c + 5*a^4*b*d*e^c + 10*a ^3*b^2*d*e^c + 10*a^2*b^3*d*e^c + 5*a*b^4*d*e^c + b^5*d*e^c)*e^(d*x)) + 1/ 2*integrate(15/2*(b*e^(3*d*x + 3*c) - b*e^(d*x + c))/(a^4 + 4*a^3*b + 6*a^ 2*b^2 + 4*a*b^3 + b^4 + (a^4*e^(4*c) + 4*a^3*b*e^(4*c) + 6*a^2*b^2*e^(4*c) + 4*a*b^3*e^(4*c) + b^4*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) + 2*a^3*b*e^( 2*c) - 2*a*b^3*e^(2*c) - b^4*e^(2*c))*e^(2*d*x)), x)
Exception generated. \[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \] Input:
int(sinh(c + d*x)/(a + b*tanh(c + d*x)^2)^3,x)
Output:
int(sinh(c + d*x)/(a + b*tanh(c + d*x)^2)^3, x)
Time = 0.45 (sec) , antiderivative size = 18347, normalized size of antiderivative = 146.78 \[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(sinh(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x)
Output:
(32*e**(9*c + 9*d*x)*cosh(c + d*x)*tanh(c + d*x)**4*a**6*b**2 + 192*e**(9* c + 9*d*x)*cosh(c + d*x)*tanh(c + d*x)**4*a**5*b**3 + 480*e**(9*c + 9*d*x) *cosh(c + d*x)*tanh(c + d*x)**4*a**4*b**4 + 640*e**(9*c + 9*d*x)*cosh(c + d*x)*tanh(c + d*x)**4*a**3*b**5 + 480*e**(9*c + 9*d*x)*cosh(c + d*x)*tanh( c + d*x)**4*a**2*b**6 + 192*e**(9*c + 9*d*x)*cosh(c + d*x)*tanh(c + d*x)** 4*a*b**7 + 32*e**(9*c + 9*d*x)*cosh(c + d*x)*tanh(c + d*x)**4*b**8 + 64*e* *(9*c + 9*d*x)*cosh(c + d*x)*tanh(c + d*x)**2*a**7*b + 400*e**(9*c + 9*d*x )*cosh(c + d*x)*tanh(c + d*x)**2*a**6*b**2 + 1056*e**(9*c + 9*d*x)*cosh(c + d*x)*tanh(c + d*x)**2*a**5*b**3 + 1520*e**(9*c + 9*d*x)*cosh(c + d*x)*ta nh(c + d*x)**2*a**4*b**4 + 1280*e**(9*c + 9*d*x)*cosh(c + d*x)*tanh(c + d* x)**2*a**3*b**5 + 624*e**(9*c + 9*d*x)*cosh(c + d*x)*tanh(c + d*x)**2*a**2 *b**6 + 160*e**(9*c + 9*d*x)*cosh(c + d*x)*tanh(c + d*x)**2*a*b**7 + 16*e* *(9*c + 9*d*x)*cosh(c + d*x)*tanh(c + d*x)**2*b**8 + 32*e**(9*c + 9*d*x)*c osh(c + d*x)*a**8 + 208*e**(9*c + 9*d*x)*cosh(c + d*x)*a**7*b + 576*e**(9* c + 9*d*x)*cosh(c + d*x)*a**6*b**2 + 880*e**(9*c + 9*d*x)*cosh(c + d*x)*a* *5*b**3 + 800*e**(9*c + 9*d*x)*cosh(c + d*x)*a**4*b**4 + 432*e**(9*c + 9*d *x)*cosh(c + d*x)*a**3*b**5 + 128*e**(9*c + 9*d*x)*cosh(c + d*x)*a**2*b**6 + 16*e**(9*c + 9*d*x)*cosh(c + d*x)*a*b**7 + 128*e**(7*c + 7*d*x)*cosh(c + d*x)*tanh(c + d*x)**4*a**6*b**2 + 512*e**(7*c + 7*d*x)*cosh(c + d*x)*tan h(c + d*x)**4*a**5*b**3 + 640*e**(7*c + 7*d*x)*cosh(c + d*x)*tanh(c + d...