Integrand size = 24, antiderivative size = 221 \[ \int \frac {\sinh ^8(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\frac {x}{b^2}-\frac {\sqrt [4]{a} \left (4 \sqrt {a}-5 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^2 d}-\frac {\sqrt [4]{a} \left (4 \sqrt {a}+5 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^2 d}-\frac {a \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 (a-b) b d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \] Output:
x/b^2-1/8*a^(1/4)*(4*a^(1/2)-5*b^(1/2))*arctanh((a^(1/2)-b^(1/2))^(1/2)*ta nh(d*x+c)/a^(1/4))/(a^(1/2)-b^(1/2))^(3/2)/b^2/d-1/8*a^(1/4)*(4*a^(1/2)+5* b^(1/2))*arctanh((a^(1/2)+b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/(a^(1/2)+b^( 1/2))^(3/2)/b^2/d-1/4*a*tanh(d*x+c)*(1-2*tanh(d*x+c)^2)/(a-b)/b/d/(a-2*a*t anh(d*x+c)^2+(a-b)*tanh(d*x+c)^4)
Time = 9.02 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.19 \[ \int \frac {\sinh ^8(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\frac {8 (c+d x)+\frac {\sqrt {a} \left (4 \sqrt {a}-5 \sqrt {b}\right ) \arctan \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {-a+\sqrt {a} \sqrt {b}}}-\frac {\sqrt {a} \left (4 \sqrt {a}+5 \sqrt {b}\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {2 a b (-6 \sinh (2 (c+d x))+\sinh (4 (c+d x)))}{(a-b) (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}}{8 b^2 d} \] Input:
Integrate[Sinh[c + d*x]^8/(a - b*Sinh[c + d*x]^4)^2,x]
Output:
(8*(c + d*x) + (Sqrt[a]*(4*Sqrt[a] - 5*Sqrt[b])*ArcTan[((Sqrt[a] - Sqrt[b] )*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])*Sqrt[-a + Sqrt[a]*Sqrt[b]]) - (Sqrt[a]*(4*Sqrt[a] + 5*Sqrt[b])*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b]) *Sqrt[a + Sqrt[a]*Sqrt[b]]) + (2*a*b*(-6*Sinh[2*(c + d*x)] + Sinh[4*(c + d *x)]))/((a - b)*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) )/(8*b^2*d)
Time = 0.83 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.76, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3696, 1650, 27, 1598, 27, 1442, 27, 1480, 221, 1610, 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.
| \(\displaystyle \int \frac {\sinh ^8(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (i c+i d x)^8}{\left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\tanh ^8(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1650 |
| \(\displaystyle \frac {\frac {\int \frac {a \tanh ^4(c+d x) \left (1-\tanh ^2(c+d x)\right )}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{b}-\frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a \int \frac {\tanh ^4(c+d x) \left (1-\tanh ^2(c+d x)\right )}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{b}-\frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 1598 |
| \(\displaystyle \frac {\frac {a \left (\frac {\int -\frac {2 b \tanh ^4(c+d x)}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{8 a b}+\frac {\tanh ^5(c+d x)}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}\right )}{b}-\frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tanh ^5(c+d x)}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\int \frac {\tanh ^4(c+d x)}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{4 a}\right )}{b}-\frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tanh ^5(c+d x)}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\frac {\tanh (c+d x)}{a-b}-\frac {\int \frac {a \left (1-2 \tanh ^2(c+d x)\right )}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a-b}}{4 a}\right )}{b}-\frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tanh ^5(c+d x)}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\frac {\tanh (c+d x)}{a-b}-\frac {a \int \frac {1-2 \tanh ^2(c+d x)}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a-b}}{4 a}\right )}{b}-\frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tanh ^5(c+d x)}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\frac {\tanh (c+d x)}{a-b}-\frac {a \left (-\frac {1}{2} \left (2-\frac {a+b}{\sqrt {a} \sqrt {b}}\right ) \int \frac {1}{(a-b) \tanh ^2(c+d x)-\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}d\tanh (c+d x)-\frac {1}{2} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right ) \int \frac {1}{(a-b) \tanh ^2(c+d x)-\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}d\tanh (c+d x)\right )}{a-b}}{4 a}\right )}{b}-\frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tanh ^5(c+d x)}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\frac {\tanh (c+d x)}{a-b}-\frac {a \left (\frac {\left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} \left (\sqrt {a}+\sqrt {b}\right )}+\frac {\left (2-\frac {a+b}{\sqrt {a} \sqrt {b}}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{a-b}}{4 a}\right )}{b}-\frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 1610 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tanh ^5(c+d x)}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\frac {\tanh (c+d x)}{a-b}-\frac {a \left (\frac {\left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} \left (\sqrt {a}+\sqrt {b}\right )}+\frac {\left (2-\frac {a+b}{\sqrt {a} \sqrt {b}}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{a-b}}{4 a}\right )}{b}-\frac {\int \left (\frac {a \left (1-\tanh ^2(c+d x)\right )}{b \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}+\frac {1}{b \left (\tanh ^2(c+d x)-1\right )}\right )d\tanh (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tanh ^5(c+d x)}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\frac {\tanh (c+d x)}{a-b}-\frac {a \left (\frac {\left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} \left (\sqrt {a}+\sqrt {b}\right )}+\frac {\left (2-\frac {a+b}{\sqrt {a} \sqrt {b}}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{a-b}}{4 a}\right )}{b}-\frac {\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\text {arctanh}(\tanh (c+d x))}{b}}{b}}{d}\) |
Input:
Int[Sinh[c + d*x]^8/(a - b*Sinh[c + d*x]^4)^2,x]
Output:
(-((-(ArcTanh[Tanh[c + d*x]]/b) + (a^(1/4)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b] ]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b) + (a^(1/4)*ArcTan h[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt [b]]*b))/b) + (a*(-1/4*(-((a*(((2 + (a + b)/(Sqrt[a]*Sqrt[b]))*ArcTanh[(Sq rt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(1/4)*Sqrt[Sqrt[a] - S qrt[b]]*(Sqrt[a] + Sqrt[b])) + ((2 - (a + b)/(Sqrt[a]*Sqrt[b]))*ArcTanh[(S qrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(1/4)*(Sqrt[a] - Sqrt [b])*Sqrt[Sqrt[a] + Sqrt[b]])))/(a - b)) + Tanh[c + d*x]/(a - b))/a + Tanh [c + d*x]^5/(4*a*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))))/b) /d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[d^4/(c*(m + 4*p + 1)) Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x ] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* p] && (IntegerQ[p] || IntegerQ[m])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_.), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1) *((b*d - 2*a*e - (b*e - 2*c*d)*x^2)/(2*(p + 1)*(b^2 - 4*a*c))), x] - Simp[f ^2/(2*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1 )*Simp[(m - 1)*(b*d - 2*a*e) - (4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4 *a*c, 0] && IntegerQ[q] && IntegerQ[m]
Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-f^4/(c*d^2 - b*d*e + a*e^2) Int[(f*x)^ (m - 4)*(a*d + (b*d - a*e)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] + Simp[d^2*(f ^4/(c*d^2 - b*d*e + a*e^2)) Int[(f*x)^(m - 4)*((a + b*x^2 + c*x^4)^(p + 1 )/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 2]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2) ^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] & & IntegerQ[m/2] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.10 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.50
| method | result | size |
| derivativedivides | \(\frac {\frac {2 a \left (\frac {-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 \left (a -b \right )}+\frac {5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 \left (a -b \right )}+\frac {5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 \left (a -b \right )}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (4 a -5 b \right ) \textit {\_R}^{6}+\left (-12 a +19 b \right ) \textit {\_R}^{4}+\left (12 a -19 b \right ) \textit {\_R}^{2}-4 a +5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{32 a -32 b}\right )}{b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}}{d}\) | \(331\) |
| default | \(\frac {\frac {2 a \left (\frac {-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 \left (a -b \right )}+\frac {5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 \left (a -b \right )}+\frac {5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 \left (a -b \right )}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (4 a -5 b \right ) \textit {\_R}^{6}+\left (-12 a +19 b \right ) \textit {\_R}^{4}+\left (12 a -19 b \right ) \textit {\_R}^{2}-4 a +5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{32 a -32 b}\right )}{b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}}{d}\) | \(331\) |
| risch | \(\frac {x}{b^{2}}+\frac {a \left (-{\mathrm e}^{6 d x +6 c} b +8 \,{\mathrm e}^{4 d x +4 c} a -3 b \,{\mathrm e}^{4 d x +4 c}+5 \,{\mathrm e}^{2 d x +2 c} b -b \right )}{2 b^{2} \left (a -b \right ) d \left (-{\mathrm e}^{8 d x +8 c} b +4 \,{\mathrm e}^{6 d x +6 c} b +16 \,{\mathrm e}^{4 d x +4 c} a -6 b \,{\mathrm e}^{4 d x +4 c}+4 \,{\mathrm e}^{2 d x +2 c} b -b \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (65536 a^{3} b^{8} d^{4}-196608 a^{2} b^{9} d^{4}+196608 a \,b^{10} d^{4}-65536 b^{11} d^{4}\right ) \textit {\_Z}^{4}+\left (-8192 a^{3} b^{4} d^{2}+24064 a^{2} b^{5} d^{2}-17920 a \,b^{6} d^{2}\right ) \textit {\_Z}^{2}+256 a^{3}-800 a^{2} b +625 b^{2} a \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\frac {32768 a^{4} b^{6} d^{3}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}-\frac {147456 a^{3} b^{7} d^{3}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}+\frac {245760 a^{2} b^{8} d^{3}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}-\frac {180224 a \,b^{9} d^{3}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}+\frac {49152 b^{10} d^{3}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}\right ) \textit {\_R}^{3}+\left (\frac {8192 a^{4} b^{4} d^{2}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}-\frac {37376 a^{3} b^{5} d^{2}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}+\frac {62976 a^{2} b^{6} d^{2}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}-\frac {46592 a \,b^{7} d^{2}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}+\frac {12800 d^{2} b^{8}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}\right ) \textit {\_R}^{2}+\left (-\frac {2048 a^{4} b^{2} d}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}+\frac {8320 a^{3} b^{3} d}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}-\frac {9440 a^{2} b^{4} d}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}-\frac {320 a \,b^{5} d}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}+\frac {4000 d \,b^{6}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}\right ) \textit {\_R} -\frac {512 a^{4}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}+\frac {2176 a^{3} b}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}-\frac {2806 a^{2} b^{2}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}+\frac {625 a \,b^{3}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}+\frac {625 b^{4}}{128 a^{3} b -664 a^{2} b^{2}+1125 a \,b^{3}-625 b^{4}}\right )\right )\) | \(999\) |
Input:
int(sinh(d*x+c)^8/(a-b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(2*a/b^2*((-1/4*b/(a-b)*tanh(1/2*d*x+1/2*c)^7+5/4*b/(a-b)*tanh(1/2*d*x +1/2*c)^5+5/4*b/(a-b)*tanh(1/2*d*x+1/2*c)^3-1/4*b/(a-b)*tanh(1/2*d*x+1/2*c ))/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c )^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)+1/32/(a-b)*s um(((4*a-5*b)*_R^6+(-12*a+19*b)*_R^4+(12*a-19*b)*_R^2-4*a+5*b)/(_R^7*a-3*_ R^5*a+3*_R^3*a-8*_R^3*b-_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(a*_Z^8- 4*a*_Z^6+(6*a-16*b)*_Z^4-4*a*_Z^2+a)))-1/b^2*ln(tanh(1/2*d*x+1/2*c)-1)+1/b ^2*ln(tanh(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 6944 vs. \(2 (175) = 350\).
Time = 0.53 (sec) , antiderivative size = 6944, normalized size of antiderivative = 31.42 \[ \int \frac {\sinh ^8(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^8/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sinh ^8(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)**8/(a-b*sinh(d*x+c)**4)**2,x)
Output:
Timed out
\[ \int \frac {\sinh ^8(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{8}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(sinh(d*x+c)^8/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")
Output:
1/2*(2*(a*b*d*e^(8*c) - b^2*d*e^(8*c))*x*e^(8*d*x) + a*b + 2*(a*b*d - b^2* d)*x + (a*b*e^(6*c) - 8*(a*b*d*e^(6*c) - b^2*d*e^(6*c))*x)*e^(6*d*x) - (8* a^2*e^(4*c) - 3*a*b*e^(4*c) + 4*(8*a^2*d*e^(4*c) - 11*a*b*d*e^(4*c) + 3*b^ 2*d*e^(4*c))*x)*e^(4*d*x) - (5*a*b*e^(2*c) + 8*(a*b*d*e^(2*c) - b^2*d*e^(2 *c))*x)*e^(2*d*x))/(a*b^3*d - b^4*d + (a*b^3*d*e^(8*c) - b^4*d*e^(8*c))*e^ (8*d*x) - 4*(a*b^3*d*e^(6*c) - b^4*d*e^(6*c))*e^(6*d*x) - 2*(8*a^2*b^2*d*e ^(4*c) - 11*a*b^3*d*e^(4*c) + 3*b^4*d*e^(4*c))*e^(4*d*x) - 4*(a*b^3*d*e^(2 *c) - b^4*d*e^(2*c))*e^(2*d*x)) + 1/256*integrate(256*(a*b*e^(6*d*x + 6*c) + a*b*e^(2*d*x + 2*c) + 2*(8*a^2*e^(4*c) - 11*a*b*e^(4*c))*e^(4*d*x))/(a* b^3 - b^4 + (a*b^3*e^(8*c) - b^4*e^(8*c))*e^(8*d*x) - 4*(a*b^3*e^(6*c) - b ^4*e^(6*c))*e^(6*d*x) - 2*(8*a^2*b^2*e^(4*c) - 11*a*b^3*e^(4*c) + 3*b^4*e^ (4*c))*e^(4*d*x) - 4*(a*b^3*e^(2*c) - b^4*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\sinh ^8(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{8}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(sinh(d*x+c)^8/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\sinh ^8(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^8}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \] Input:
int(sinh(c + d*x)^8/(a - b*sinh(c + d*x)^4)^2,x)
Output:
int(sinh(c + d*x)^8/(a - b*sinh(c + d*x)^4)^2, x)
\[ \int \frac {\sinh ^8(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {too large to display} \] Input:
int(sinh(d*x+c)^8/(a-b*sinh(d*x+c)^4)^2,x)
Output:
( - 1006632960*e**(12*c + 8*d*x)*int(e**(4*d*x)/(1920*e**(16*c + 16*d*x)*a **2*b**2 + 80*e**(16*c + 16*d*x)*a*b**3 - 3*e**(16*c + 16*d*x)*b**4 - 1536 0*e**(14*c + 14*d*x)*a**2*b**2 - 640*e**(14*c + 14*d*x)*a*b**3 + 24*e**(14 *c + 14*d*x)*b**4 - 61440*e**(12*c + 12*d*x)*a**3*b + 51200*e**(12*c + 12* d*x)*a**2*b**2 + 2336*e**(12*c + 12*d*x)*a*b**3 - 84*e**(12*c + 12*d*x)*b* *4 + 245760*e**(10*c + 10*d*x)*a**3*b - 97280*e**(10*c + 10*d*x)*a**2*b**2 - 4864*e**(10*c + 10*d*x)*a*b**3 + 168*e**(10*c + 10*d*x)*b**4 + 491520*e **(8*c + 8*d*x)*a**4 - 348160*e**(8*c + 8*d*x)*a**3*b + 118272*e**(8*c + 8 *d*x)*a**2*b**2 + 6176*e**(8*c + 8*d*x)*a*b**3 - 210*e**(8*c + 8*d*x)*b**4 + 245760*e**(6*c + 6*d*x)*a**3*b - 97280*e**(6*c + 6*d*x)*a**2*b**2 - 486 4*e**(6*c + 6*d*x)*a*b**3 + 168*e**(6*c + 6*d*x)*b**4 - 61440*e**(4*c + 4* d*x)*a**3*b + 51200*e**(4*c + 4*d*x)*a**2*b**2 + 2336*e**(4*c + 4*d*x)*a*b **3 - 84*e**(4*c + 4*d*x)*b**4 - 15360*e**(2*c + 2*d*x)*a**2*b**2 - 640*e* *(2*c + 2*d*x)*a*b**3 + 24*e**(2*c + 2*d*x)*b**4 + 1920*a**2*b**2 + 80*a*b **3 - 3*b**4),x)*a**7*b*d + 2285895680*e**(12*c + 8*d*x)*int(e**(4*d*x)/(1 920*e**(16*c + 16*d*x)*a**2*b**2 + 80*e**(16*c + 16*d*x)*a*b**3 - 3*e**(16 *c + 16*d*x)*b**4 - 15360*e**(14*c + 14*d*x)*a**2*b**2 - 640*e**(14*c + 14 *d*x)*a*b**3 + 24*e**(14*c + 14*d*x)*b**4 - 61440*e**(12*c + 12*d*x)*a**3* b + 51200*e**(12*c + 12*d*x)*a**2*b**2 + 2336*e**(12*c + 12*d*x)*a*b**3 - 84*e**(12*c + 12*d*x)*b**4 + 245760*e**(10*c + 10*d*x)*a**3*b - 97280*e...