Integrand size = 24, antiderivative size = 127 \[ \int \frac {\sinh ^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\frac {x}{b}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b d}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b d} \]
-x/b+1/2*a^(1/4)*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/b/d/ (a^(1/2)-b^(1/2))^(1/2)+1/2*a^(1/4)*arctanh((a^(1/2)+b^(1/2))^(1/2)*tanh(d *x+c)/a^(1/4))/b/d/(a^(1/2)+b^(1/2))^(1/2)
Time = 2.82 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13 \[ \int \frac {\sinh ^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\frac {-2 (c+d x)-\frac {\sqrt {a} \arctan \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {\sqrt {a} \text {arctanh}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}}{2 b d} \]
(-2*(c + d*x) - (Sqrt[a]*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[- a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (Sqrt[a]*ArcTanh[((Sqr t[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a ]*Sqrt[b]])/(2*b*d)
Time = 0.38 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3696, 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.
| \(\displaystyle \int \frac {\sinh ^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (i c+i d x)^4}{a-b \sin (i c+i d x)^4}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1610 |
| \(\displaystyle \frac {\int \left (\frac {a \left (1-\tanh ^2(c+d x)\right )}{b \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}+\frac {1}{b \left (\tanh ^2(c+d x)-1\right )}\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\text {arctanh}(\tanh (c+d x))}{b}}{d}\) |
(-(ArcTanh[Tanh[c + d*x]]/b) + (a^(1/4)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*T anh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b) + (a^(1/4)*ArcTanh[( Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b] ]*b))/d
3.3.36.3.1 Defintions of rubi rules used
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[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 = 0.34 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(-\frac {x}{b}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (256 a \,b^{4} d^{4}-256 b^{5} d^{4}\right ) \textit {\_Z}^{4}-32 a \,b^{2} d^{2} \textit {\_Z}^{2}+a \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (-128 a \,b^{2} d^{3}+128 b^{3} d^{3}\right ) \textit {\_R}^{3}+\left (32 a \,d^{2} b -32 b^{2} d^{2}\right ) \textit {\_R}^{2}+\left (8 a d +8 b d \right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )\) | \(121\) |
| derivativedivides | \(\frac {-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+3 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{4 b}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}}{d}\) | \(139\) |
| default | \(\frac {-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+3 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{4 b}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}}{d}\) | \(139\) |
-x/b+sum(_R*ln(exp(2*d*x+2*c)+(-128*a*b^2*d^3+128*b^3*d^3)*_R^3+(32*a*b*d^ 2-32*b^2*d^2)*_R^2+(8*a*d+8*b*d)*_R-2*a/b-1),_R=RootOf((256*a*b^4*d^4-256* b^5*d^4)*_Z^4-32*a*b^2*d^2*_Z^2+a))
Leaf count of result is larger than twice the leaf count of optimal. 1009 vs. \(2 (91) = 182\).
Time = 0.33 (sec) , antiderivative size = 1009, normalized size of antiderivative = 7.94 \[ \int \frac {\sinh ^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\frac {b \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (2 \, {\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 2 \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - b d\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - 1\right ) - b \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (2 \, {\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 2 \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - b d\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - 1\right ) - b \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (-2 \, {\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 2 \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + b d\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - 1\right ) + b \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (-2 \, {\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 2 \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + b d\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - 1\right ) + 4 \, x}{4 \, b} \]
-1/4*(b*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d^2))*log(2*(a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - b*d )*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a *b^2 - b^3)*d^2)) - 1) - b*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a* b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d^2))*log(2*(a*b - b^2)*d^2*sqrt(a/(( a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d* x + c) + sinh(d*x + c)^2 - 2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - b*d)*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d^2)) - 1) - b*sqrt(-((a*b^2 - b^3)*d^2*sqr t(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2))*log(-2*(a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + cosh(d*x + c)^2 + 2* cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*((a*b^2 - b^3)*d^3*sqrt( a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + b*d)*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/ ((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2)) - 1) + b*sqrt(- ((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2))*log(-2*(a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 2*(( a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + b*d)*sqrt(-(...
Timed out. \[ \int \frac {\sinh ^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sinh ^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{4}}{b \sinh \left (d x + c\right )^{4} - a} \,d x } \]
-16*a*integrate(e^(4*d*x + 4*c)/(b^2*e^(8*d*x + 8*c) - 4*b^2*e^(6*d*x + 6* c) - 4*b^2*e^(2*d*x + 2*c) + b^2 - 2*(8*a*b*e^(4*c) - 3*b^2*e^(4*c))*e^(4* d*x)), x) - x/b
\[ \int \frac {\sinh ^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{4}}{b \sinh \left (d x + c\right )^{4} - a} \,d x } \]
Time = 11.49 (sec) , antiderivative size = 1861, normalized size of antiderivative = 14.65 \[ \int \frac {\sinh ^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \]
log((((((524288*a^3*d^2*(31*a*b^2 - 128*a^2*b + 128*a^3 - b^3 + 256*a^3*ex p(2*c + 2*d*x) + b^3*exp(2*c + 2*d*x) + 21*a*b^2*exp(2*c + 2*d*x) - 240*a^ 2*b*exp(2*c + 2*d*x)))/(b^8*(a - b)) + (1048576*a^3*d^3*((a*b^2 - (a*b^5)^ (1/2))/(b^4*d^2*(a - b)))^(1/2)*(45*a*b^2 - 104*a^2*b + 64*a^3 - 3*b^3 + 4 *b^3*exp(2*c + 2*d*x) - 50*a*b^2*exp(2*c + 2*d*x) + 48*a^2*b*exp(2*c + 2*d *x)))/(b^7*(a - b)))*((a*b^2 - (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2))/4 + (262144*a^4*d*(72*a*b - 64*a^2 - 9*b^2 + 256*a^2*exp(2*c + 2*d*x) + 31*b ^2*exp(2*c + 2*d*x) - 288*a*b*exp(2*c + 2*d*x)))/(b^9*(a - b)))*((a*b^2 - (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2))/4 + (32768*a^4*(128*a*b - 128*a^2 - 15*b^2 + 256*a^2*exp(2*c + 2*d*x) + 29*b^2*exp(2*c + 2*d*x) - 304*a*b*e xp(2*c + 2*d*x)))/(b^10*(a - b)))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(b^5*d^2 - a*b^4*d^2)))^(1/2) - log((((((524288*a^3*d^2*(31*a*b^2 - 128*a^2*b + 128* a^3 - b^3 + 256*a^3*exp(2*c + 2*d*x) + b^3*exp(2*c + 2*d*x) + 21*a*b^2*exp (2*c + 2*d*x) - 240*a^2*b*exp(2*c + 2*d*x)))/(b^8*(a - b)) - (1048576*a^3* d^3*((a*b^2 - (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2)*(45*a*b^2 - 104*a^2* b + 64*a^3 - 3*b^3 + 4*b^3*exp(2*c + 2*d*x) - 50*a*b^2*exp(2*c + 2*d*x) + 48*a^2*b*exp(2*c + 2*d*x)))/(b^7*(a - b)))*((a*b^2 - (a*b^5)^(1/2))/(b^4*d ^2*(a - b)))^(1/2))/4 - (262144*a^4*d*(72*a*b - 64*a^2 - 9*b^2 + 256*a^2*e xp(2*c + 2*d*x) + 31*b^2*exp(2*c + 2*d*x) - 288*a*b*exp(2*c + 2*d*x)))/(b^ 9*(a - b)))*((a*b^2 - (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2))/4 + (327...