\(\int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [206]

Optimal result

Integrand size = 24, antiderivative size = 115 \[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{3/4} d}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{3/4} d} \] Output:

-1/2*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/(a^(1/2)-b^(1/2)) 
^(1/2)/b^(3/4)/d+1/2*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/ 
(a^(1/2)+b^(1/2))^(1/2)/b^(3/4)/d
 

Mathematica [C] (verified)

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

Time = 6.58 (sec) , antiderivative size = 365, normalized size of antiderivative = 3.17 \[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\frac {\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+3 c \text {$\#$1}^2+3 d x \text {$\#$1}^2+6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-3 c \text {$\#$1}^4-3 d x \text {$\#$1}^4-6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+c \text {$\#$1}^6+d x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{8 d} \] Input:

Integrate[Sinh[c + d*x]^3/(a - b*Sinh[c + d*x]^4),x]
 

Output:

-1/8*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , ( 
-c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2 
]*#1 - Sinh[(c + d*x)/2]*#1] + 3*c*#1^2 + 3*d*x*#1^2 + 6*Log[-Cosh[(c + d* 
x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*# 
1^2 - 3*c*#1^4 - 3*d*x*#1^4 - 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] 
 + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + c*#1^6 + d*x*#1^6 + 
 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sin 
h[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^ 
7) & ]/d
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 26, 3694, 1480, 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.

\(\displaystyle \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {i \sin (i c+i d x)^3}{a-b \sin (i c+i d x)^4}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {\sin (i c+i d x)^3}{a-b \sin (i c+i d x)^4}dx\)

\(\Big \downarrow \) 3694

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 218

\(\displaystyle -\frac {\frac {\arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {1}{2} \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {\arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}+\sqrt {b}}}}{d}\)

Input:

Int[Sinh[c + d*x]^3/(a - b*Sinh[c + d*x]^4),x]
 

Output:

-((ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*Sqrt[Sqrt[a] 
 - Sqrt[b]]*b^(3/4)) - ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt 
[b]]]/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/4)))/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 1480

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]
 

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 = 0.84 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77

method	result	size
risch	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (-1+\left (256 a \,b^{3} d^{4}-256 b^{4} d^{4}\right ) \textit {\_Z}^{4}+32 d^{2} \textit {\_Z}^{2} b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (128 a \,b^{2} d^{3}-128 b^{3} d^{3}\right ) \textit {\_R}^{3}+16 b d \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\)	\(89\)
derivativedivides	\(\frac {8 a \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b -\sqrt {a b}\, a}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b +\sqrt {a b}\, a}}\right )}{d}\)	\(148\)
default	\(\frac {8 a \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b -\sqrt {a b}\, a}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{16 a b \sqrt {-a b +\sqrt {a b}\, a}}\right )}{d}\)	\(148\)

Input:

int(sinh(d*x+c)^3/(a-b*sinh(d*x+c)^4),x,method=_RETURNVERBOSE)
 

Output:

sum(_R*ln(exp(2*d*x+2*c)+((128*a*b^2*d^3-128*b^3*d^3)*_R^3+16*b*d*_R)*exp( 
d*x+c)+1),_R=RootOf(-1+(256*a*b^3*d^4-256*b^4*d^4)*_Z^4+32*d^2*_Z^2*b^2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 975 vs. \(2 (79) = 158\).

Time = 0.13 (sec) , antiderivative size = 975, normalized size of antiderivative = 8.48 \[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(sinh(d*x+c)^3/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")
 

Output:

1/4*sqrt(-((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/(( 
a*b - b^2)*d^2))*log(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sin 
h(d*x + c)^2 + 2*(b*d*cosh(d*x + c) + b*d*sinh(d*x + c) - ((a*b^2 - b^3)*d 
^3*cosh(d*x + c) + (a*b^2 - b^3)*d^3*sinh(d*x + c))*sqrt(a/((a^2*b^3 - 2*a 
*b^4 + b^5)*d^4)))*sqrt(-((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5 
)*d^4)) + 1)/((a*b - b^2)*d^2)) + 1) - 1/4*sqrt(-((a*b - b^2)*d^2*sqrt(a/( 
(a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2))*log(cosh(d*x + c)^ 
2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 2*(b*d*cosh(d*x + c) 
 + b*d*sinh(d*x + c) - ((a*b^2 - b^3)*d^3*cosh(d*x + c) + (a*b^2 - b^3)*d^ 
3*sinh(d*x + c))*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)))*sqrt(-((a*b - b^ 
2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2)) + 1 
) + 1/4*sqrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1) 
/((a*b - b^2)*d^2))*log(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + 
sinh(d*x + c)^2 + 2*(b*d*cosh(d*x + c) + b*d*sinh(d*x + c) + ((a*b^2 - b^3 
)*d^3*cosh(d*x + c) + (a*b^2 - b^3)*d^3*sinh(d*x + c))*sqrt(a/((a^2*b^3 - 
2*a*b^4 + b^5)*d^4)))*sqrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b 
^5)*d^4)) - 1)/((a*b - b^2)*d^2)) + 1) - 1/4*sqrt(((a*b - b^2)*d^2*sqrt(a/ 
((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1)/((a*b - b^2)*d^2))*log(cosh(d*x + c) 
^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 2*(b*d*cosh(d*x + c 
) + b*d*sinh(d*x + c) + ((a*b^2 - b^3)*d^3*cosh(d*x + c) + (a*b^2 - b^3...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\text {Timed out} \] Input:

integrate(sinh(d*x+c)**3/(a-b*sinh(d*x+c)**4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{4} - a} \,d x } \] Input:

integrate(sinh(d*x+c)^3/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

-integrate(sinh(d*x + c)^3/(b*sinh(d*x + c)^4 - a), x)
 

Giac [F]

\[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{4} - a} \,d x } \] Input:

integrate(sinh(d*x+c)^3/(a-b*sinh(d*x+c)^4),x, algorithm="giac")
 

Output:

sage0*x
 

Mupad [B] (verification not implemented)

Time = 7.01 (sec) , antiderivative size = 975, normalized size of antiderivative = 8.48 \[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx =\text {Too large to display} \] Input:

int(sinh(c + d*x)^3/(a - b*sinh(c + d*x)^4),x)
 

Output:

log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^7*(a - b)^2) 
 - (8388608*a^4*d^3*exp(c + d*x)*(a + b)*(-(b^2 - (a*b^3)^(1/2))/(b^3*d^2* 
(a - b)))^(1/2))/(b^7*(a - b)))*(-(b^2 - (a*b^3)^(1/2))/(b^3*d^2*(a - b))) 
^(1/2))/4 + (2097152*a^4*d*exp(c + d*x))/(b^8*(a - b)))*(-(b^2 - (a*b^3)^( 
1/2))/(b^3*d^2*(a - b)))^(1/2))/4 - (262144*a^4*(exp(2*c + 2*d*x) + 1)*(a 
+ b))/(b^9*(a - b)^2))*((b^2 - (a*b^3)^(1/2))/(16*(b^4*d^2 - a*b^3*d^2)))^ 
(1/2) - log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^7*(a 
 - b)^2) + (8388608*a^4*d^3*exp(c + d*x)*(a + b)*(-(b^2 - (a*b^3)^(1/2))/( 
b^3*d^2*(a - b)))^(1/2))/(b^7*(a - b)))*(-(b^2 - (a*b^3)^(1/2))/(b^3*d^2*( 
a - b)))^(1/2))/4 - (2097152*a^4*d*exp(c + d*x))/(b^8*(a - b)))*(-(b^2 - ( 
a*b^3)^(1/2))/(b^3*d^2*(a - b)))^(1/2))/4 - (262144*a^4*(exp(2*c + 2*d*x) 
+ 1)*(a + b))/(b^9*(a - b)^2))*((b^2 - (a*b^3)^(1/2))/(16*(b^4*d^2 - a*b^3 
*d^2)))^(1/2) + log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b)) 
/(b^7*(a - b)^2) - (8388608*a^4*d^3*exp(c + d*x)*(a + b)*(-(b^2 + (a*b^3)^ 
(1/2))/(b^3*d^2*(a - b)))^(1/2))/(b^7*(a - b)))*(-(b^2 + (a*b^3)^(1/2))/(b 
^3*d^2*(a - b)))^(1/2))/4 + (2097152*a^4*d*exp(c + d*x))/(b^8*(a - b)))*(- 
(b^2 + (a*b^3)^(1/2))/(b^3*d^2*(a - b)))^(1/2))/4 - (262144*a^4*(exp(2*c + 
 2*d*x) + 1)*(a + b))/(b^9*(a - b)^2))*((b^2 + (a*b^3)^(1/2))/(16*(b^4*d^2 
 - a*b^3*d^2)))^(1/2) - log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3 
*a + b))/(b^7*(a - b)^2) + (8388608*a^4*d^3*exp(c + d*x)*(a + b)*(-(b^2...
 

Reduce [F]

\[ \int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx=-\left (\int \frac {\sinh \left (d x +c \right )^{3}}{\sinh \left (d x +c \right )^{4} b -a}d x \right ) \] Input:

int(sinh(d*x+c)^3/(a-b*sinh(d*x+c)^4),x)
 

Output:

 - int(sinh(c + d*x)**3/(sinh(c + d*x)**4*b - a),x)