\(\int \frac {\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [207]

Optimal result

Integrand size = 22, antiderivative size = 125 \[ \int \frac {\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a}-\sqrt {b}} \sqrt [4]{b} d}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a}+\sqrt {b}} \sqrt [4]{b} d} \] Output:

1/2*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^(1/2)/(a^(1/2)-b 
^(1/2))^(1/2)/b^(1/4)/d+1/2*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^ 
(1/2))/a^(1/2)/(a^(1/2)+b^(1/2))^(1/2)/b^(1/4)/d
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 4.20 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.77 \[ \int \frac {\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\frac {\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c \text {$\#$1}-d x \text {$\#$1}-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}+c \text {$\#$1}^3+d x \text {$\#$1}^3+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^3}{-b-8 a \text {$\#$1}^2+3 b \text {$\#$1}^2-3 b \text {$\#$1}^4+b \text {$\#$1}^6}\&\right ]}{2 d} \] Input:

Integrate[Sinh[c + d*x]/(a - b*Sinh[c + d*x]^4),x]
 

Output:

-1/2*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , ( 
-(c*#1) - d*x*#1 - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c 
+ d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1 + c*#1^3 + d*x*#1^3 + 2*Log[-Cosh[ 
(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2 
]*#1]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ]/d
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3042, 26, 3694, 1406, 218, 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.

\(\displaystyle \int \frac {\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {i \sin (i c+i d x)}{a-b \sin (i c+i d x)^4}dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \frac {\sin (i c+i d x)}{a-b \sin (i c+i d x)^4}dx\)

\(\Big \downarrow \) 3694

\(\displaystyle \frac {\int \frac {1}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{d}\)

\(\Big \downarrow \) 1406

\(\displaystyle \frac {\frac {\sqrt {b} \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)}{2 \sqrt {a}}-\frac {\sqrt {b} \int \frac {1}{-b \cosh ^2(c+d x)-\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}}d\cosh (c+d x)}{2 \sqrt {a}}}{d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\sqrt {b} \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)}{2 \sqrt {a}}+\frac {\arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}+\sqrt {b}}}}{d}\)

Input:

Int[Sinh[c + d*x]/(a - b*Sinh[c + d*x]^4),x]
 

Output:

(ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*Sqrt[a]*Sqrt[S 
qrt[a] - Sqrt[b]]*b^(1/4)) + ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] 
+ Sqrt[b]]]/(2*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(1/4)))/d
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

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]
 

rule 1406

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 
2 - 4*a*c, 2]}, Simp[c/q   Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q   I 
nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c 
, 0] && PosQ[b^2 - 4*a*c]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3694

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f 
Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, 
 x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 
 1)/2]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.79 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.78

method	result	size
risch	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (-1+\left (256 b \,d^{4} a^{3}-256 a^{2} b^{2} d^{4}\right ) \textit {\_Z}^{4}+32 a \,d^{2} \textit {\_Z}^{2} b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (128 a^{2} b \,d^{3}-128 a \,b^{2} d^{3}\right ) \textit {\_R}^{3}+\left (8 a d +8 b d \right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\)	\(98\)
derivativedivides	\(\frac {2 a \left (\frac {\arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}-\frac {\arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}\right )}{d}\)	\(132\)
default	\(\frac {2 a \left (\frac {\arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}-\frac {\arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}\right )}{d}\)	\(132\)

Input:

int(sinh(d*x+c)/(a-b*sinh(d*x+c)^4),x,method=_RETURNVERBOSE)
 

Output:

sum(_R*ln(exp(2*d*x+2*c)+((128*a^2*b*d^3-128*a*b^2*d^3)*_R^3+(8*a*d+8*b*d) 
*_R)*exp(d*x+c)+1),_R=RootOf(-1+(256*a^3*b*d^4-256*a^2*b^2*d^4)*_Z^4+32*a* 
d^2*_Z^2*b))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 979 vs. \(2 (85) = 170\).

Time = 0.12 (sec) , antiderivative size = 979, normalized size of antiderivative = 7.83 \[ \int \frac {\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(sinh(d*x+c)/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")
 

Output:

1/4*sqrt(-((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1)/ 
((a^2 - a*b)*d^2))*log(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + s 
inh(d*x + c)^2 + 2*(a*d*cosh(d*x + c) + a*d*sinh(d*x + c) - ((a^2*b - a*b^ 
2)*d^3*cosh(d*x + c) + (a^2*b - a*b^2)*d^3*sinh(d*x + c))*sqrt(1/((a^3*b - 
 2*a^2*b^2 + a*b^3)*d^4)))*sqrt(-((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b 
^2 + a*b^3)*d^4)) + 1)/((a^2 - a*b)*d^2)) + 1) - 1/4*sqrt(-((a^2 - a*b)*d^ 
2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1)/((a^2 - a*b)*d^2))*log(co 
sh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 2*(a*d*c 
osh(d*x + c) + a*d*sinh(d*x + c) - ((a^2*b - a*b^2)*d^3*cosh(d*x + c) + (a 
^2*b - a*b^2)*d^3*sinh(d*x + c))*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) 
)*sqrt(-((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1)/(( 
a^2 - a*b)*d^2)) + 1) + 1/4*sqrt(((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b 
^2 + a*b^3)*d^4)) - 1)/((a^2 - a*b)*d^2))*log(cosh(d*x + c)^2 + 2*cosh(d*x 
 + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*(a*d*cosh(d*x + c) + a*d*sinh(d* 
x + c) + ((a^2*b - a*b^2)*d^3*cosh(d*x + c) + (a^2*b - a*b^2)*d^3*sinh(d*x 
 + c))*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)))*sqrt(((a^2 - a*b)*d^2*sq 
rt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1)/((a^2 - a*b)*d^2)) + 1) - 1/4 
*sqrt(((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1)/((a^ 
2 - a*b)*d^2))*log(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh( 
d*x + c)^2 - 2*(a*d*cosh(d*x + c) + a*d*sinh(d*x + c) + ((a^2*b - a*b^2...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Timed out} \] Input:

integrate(sinh(d*x+c)/(a-b*sinh(d*x+c)**4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\int { -\frac {\sinh \left (d x + c\right )}{b \sinh \left (d x + c\right )^{4} - a} \,d x } \] Input:

integrate(sinh(d*x+c)/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

-integrate(sinh(d*x + c)/(b*sinh(d*x + c)^4 - a), x)
 

Giac [F]

\[ \int \frac {\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\int { -\frac {\sinh \left (d x + c\right )}{b \sinh \left (d x + c\right )^{4} - a} \,d x } \] Input:

integrate(sinh(d*x+c)/(a-b*sinh(d*x+c)^4),x, algorithm="giac")
 

Output:

sage0*x
 

Mupad [B] (verification not implemented)

Time = 9.14 (sec) , antiderivative size = 1007, normalized size of antiderivative = 8.06 \[ \int \frac {\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx =\text {Too large to display} \] Input:

int(sinh(c + d*x)/(a - b*sinh(c + d*x)^4),x)
 

Output:

log((((((4194304*a^2*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^5*(a - b)^2) 
 + (16777216*a^3*d^3*exp(c + d*x)*(-(a*b - (a^3*b)^(1/2))/(a^2*b*d^2*(a - 
b)))^(1/2))/(b^5*(a - b)))*(-(a*b - (a^3*b)^(1/2))/(a^2*b*d^2*(a - b)))^(1 
/2))/4 - (2097152*a^2*d*exp(c + d*x))/(b^6*(a - b)))*(-(a*b - (a^3*b)^(1/2 
))/(a^2*b*d^2*(a - b)))^(1/2))/4 - (262144*a*(exp(2*c + 2*d*x) + 1)*(a + b 
))/(b^6*(a - b)^2))*(-(a*b - (a^3*b)^(1/2))/(16*(a^3*b*d^2 - a^2*b^2*d^2)) 
)^(1/2) - log((((((4194304*a^2*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^5* 
(a - b)^2) - (16777216*a^3*d^3*exp(c + d*x)*(-(a*b - (a^3*b)^(1/2))/(a^2*b 
*d^2*(a - b)))^(1/2))/(b^5*(a - b)))*(-(a*b - (a^3*b)^(1/2))/(a^2*b*d^2*(a 
 - b)))^(1/2))/4 + (2097152*a^2*d*exp(c + d*x))/(b^6*(a - b)))*(-(a*b - (a 
^3*b)^(1/2))/(a^2*b*d^2*(a - b)))^(1/2))/4 - (262144*a*(exp(2*c + 2*d*x) + 
 1)*(a + b))/(b^6*(a - b)^2))*(-(a*b - (a^3*b)^(1/2))/(16*(a^3*b*d^2 - a^2 
*b^2*d^2)))^(1/2) - log((((((4194304*a^2*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + 
 b))/(b^5*(a - b)^2) - (16777216*a^3*d^3*exp(c + d*x)*(-(a*b + (a^3*b)^(1/ 
2))/(a^2*b*d^2*(a - b)))^(1/2))/(b^5*(a - b)))*(-(a*b + (a^3*b)^(1/2))/(a^ 
2*b*d^2*(a - b)))^(1/2))/4 + (2097152*a^2*d*exp(c + d*x))/(b^6*(a - b)))*( 
-(a*b + (a^3*b)^(1/2))/(a^2*b*d^2*(a - b)))^(1/2))/4 - (262144*a*(exp(2*c 
+ 2*d*x) + 1)*(a + b))/(b^6*(a - b)^2))*(-(a*b + (a^3*b)^(1/2))/(16*(a^3*b 
*d^2 - a^2*b^2*d^2)))^(1/2) + log((((((4194304*a^2*d^2*(exp(2*c + 2*d*x) + 
 1)*(3*a + b))/(b^5*(a - b)^2) + (16777216*a^3*d^3*exp(c + d*x)*(-(a*b ...
 

Reduce [F]

\[ \int \frac {\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\left (\int \frac {\sinh \left (d x +c \right )}{\sinh \left (d x +c \right )^{4} b -a}d x \right ) \] Input:

int(sinh(d*x+c)/(a-b*sinh(d*x+c)^4),x)
 

Output:

 - int(sinh(c + d*x)/(sinh(c + d*x)**4*b - a),x)