\(\int \frac {x^4 \cosh (c+d x)}{a+b x^3} \, dx\) [94]

Optimal result

Integrand size = 19, antiderivative size = 373 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^3} \, dx=-\frac {\cosh (c+d x)}{b d^2}+\frac {(-1)^{2/3} a^{2/3} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{5/3}}-\frac {\sqrt [3]{-1} a^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{5/3}}+\frac {a^{2/3} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{5/3}}+\frac {x \sinh (c+d x)}{b d}-\frac {(-1)^{2/3} a^{2/3} \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{5/3}}+\frac {a^{2/3} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{5/3}}-\frac {\sqrt [3]{-1} a^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{5/3}} \] Output:

-cosh(d*x+c)/b/d^2+1/3*(-1)^(2/3)*a^(2/3)*cosh(c+(-1)^(1/3)*a^(1/3)*d/b^(1 
/3))*Chi((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)/b^(5/3)-1/3*(-1)^(1/3)*a^(2/3)* 
cosh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Chi(-(-1)^(2/3)*a^(1/3)*d/b^(1/3)-d*x 
)/b^(5/3)+1/3*a^(2/3)*cosh(c-a^(1/3)*d/b^(1/3))*Chi(a^(1/3)*d/b^(1/3)+d*x) 
/b^(5/3)+x*sinh(d*x+c)/b/d+1/3*(-1)^(2/3)*a^(2/3)*sinh(c+(-1)^(1/3)*a^(1/3 
)*d/b^(1/3))*Shi(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)/b^(5/3)+1/3*a^(2/3)*si 
nh(c-a^(1/3)*d/b^(1/3))*Shi(a^(1/3)*d/b^(1/3)+d*x)/b^(5/3)-1/3*(-1)^(1/3)* 
a^(2/3)*sinh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Shi((-1)^(2/3)*a^(1/3)*d/b^(1 
/3)+d*x)/b^(5/3)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 0.14 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.57 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^3} \, dx=-\frac {a d^2 \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}}\&\right ]+a d^2 \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}}\&\right ]+6 b (\cosh (c+d x)-d x \sinh (c+d x))}{6 b^2 d^2} \] Input:

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

Output:

-1/6*(a*d^2*RootSum[a + b*#1^3 & , (Cosh[c + d*#1]*CoshIntegral[d*(x - #1) 
] - CoshIntegral[d*(x - #1)]*Sinh[c + d*#1] - Cosh[c + d*#1]*SinhIntegral[ 
d*(x - #1)] + Sinh[c + d*#1]*SinhIntegral[d*(x - #1)])/#1 & ] + a*d^2*Root 
Sum[a + b*#1^3 & , (Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] + CoshIntegral 
[d*(x - #1)]*Sinh[c + d*#1] + Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] + Si 
nh[c + d*#1]*SinhIntegral[d*(x - #1)])/#1 & ] + 6*b*(Cosh[c + d*x] - d*x*S 
inh[c + d*x]))/(b^2*d^2)
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5816, 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.

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

\(\Big \downarrow \) 5816

\(\displaystyle \int \left (\frac {x \cosh (c+d x)}{b}-\frac {a x \cosh (c+d x)}{b \left (a+b x^3\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(-1)^{2/3} a^{2/3} \cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{5/3}}-\frac {\sqrt [3]{-1} a^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}+\frac {a^{2/3} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}-\frac {(-1)^{2/3} a^{2/3} \sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{5/3}}+\frac {a^{2/3} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}-\frac {\sqrt [3]{-1} a^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{5/3}}-\frac {\cosh (c+d x)}{b d^2}+\frac {x \sinh (c+d x)}{b d}\)

Input:

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

Output:

-(Cosh[c + d*x]/(b*d^2)) + ((-1)^(2/3)*a^(2/3)*Cosh[c + ((-1)^(1/3)*a^(1/3 
)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b^(5/ 
3)) - ((-1)^(1/3)*a^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshInt 
egral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*b^(5/3)) + (a^(2/3)*Cos 
h[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b^( 
5/3)) + (x*Sinh[c + d*x])/(b*d) - ((-1)^(2/3)*a^(2/3)*Sinh[c + ((-1)^(1/3) 
*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/( 
3*b^(5/3)) + (a^(2/3)*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)* 
d)/b^(1/3) + d*x])/(3*b^(5/3)) - ((-1)^(1/3)*a^(2/3)*Sinh[c - ((-1)^(2/3)* 
a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3 
*b^(5/3))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5816

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sy 
mbol] :> Int[ExpandIntegrand[Cosh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Fr 
eeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 
2] || EqQ[p, -1])
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.69 (sec) , antiderivative size = 925, normalized size of antiderivative = 2.48

method	result	size
risch	\(\text {Expression too large to display}\)	\(925\)

Input:

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

Output:

-1/6/d^2/b*c^4*sum(1/(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=Roo 
tOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/6/d^2/b*c^4*sum(1/(_R1^2- 
2*_R1*c+c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b 
*c^2+a*d^3-b*c^3))+2/3/d^2/b*c^3*sum(_R1/(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1 
,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+2/3/d^2 
/b*c^3*sum(_R1/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z 
^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/2/d/b*exp(-d*x-c)*x-1/d^2/b*c^2 
*sum(_R1^2/(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b 
-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/2/d/b*exp(d*x+c)*x-1/d^2/b*c^2*sum( 
_R1^2/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z 
^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/2/d^2/b*exp(-d*x-c)+2/3/d^2/b^2*sum((3*_ 
R1^2*b*c-3*_R1*b*c^2-a*d^3+b*c^3)/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1,d*x-_ 
R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*c+2/3/d^2/b^2* 
sum((3*_R1^2*b*c-3*_R1*b*c^2-a*d^3+b*c^3)/(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei( 
1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*c-1/2/ 
d^2/b*exp(d*x+c)-1/6/d^2/b^2*sum((6*_R1^2*b*c^2-_R1*a*d^3-8*_R1*b*c^3-3*a* 
c*d^3+3*b*c^4)/(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z 
^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/6/d^2/b^2*sum((6*_R1^2*b*c^2-_R 
1*a*d^3-8*_R1*b*c^3-3*a*c*d^3+3*b*c^4)/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1, 
d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 989 vs. \(2 (265) = 530\).

Time = 0.14 (sec) , antiderivative size = 989, normalized size of antiderivative = 2.65 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*((a*d^3/b)^(2/3)*((sqrt(-3) - 1)*cosh(d*x + c)^2 - (sqrt(-3) - 1)*sin 
h(d*x + c)^2)*Ei(d*x - 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1))*cosh(1/2*(a*d^3 
/b)^(1/3)*(sqrt(-3) + 1) + c) + (-a*d^3/b)^(2/3)*((sqrt(-3) - 1)*cosh(d*x 
+ c)^2 - (sqrt(-3) - 1)*sinh(d*x + c)^2)*Ei(-d*x - 1/2*(-a*d^3/b)^(1/3)*(s 
qrt(-3) + 1))*cosh(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1) - c) - (a*d^3/b)^(2 
/3)*((sqrt(-3) + 1)*cosh(d*x + c)^2 - (sqrt(-3) + 1)*sinh(d*x + c)^2)*Ei(d 
*x + 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1))*cosh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3 
) - 1) - c) - (-a*d^3/b)^(2/3)*((sqrt(-3) + 1)*cosh(d*x + c)^2 - (sqrt(-3) 
 + 1)*sinh(d*x + c)^2)*Ei(-d*x + 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1))*cosh 
(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1) + c) + 2*(-a*d^3/b)^(2/3)*(cosh(d*x + 
 c)^2 - sinh(d*x + c)^2)*Ei(-d*x + (-a*d^3/b)^(1/3))*cosh(c + (-a*d^3/b)^( 
1/3)) + 2*(a*d^3/b)^(2/3)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(d*x + (a* 
d^3/b)^(1/3))*cosh(-c + (a*d^3/b)^(1/3)) + (a*d^3/b)^(2/3)*((sqrt(-3) - 1) 
*cosh(d*x + c)^2 - (sqrt(-3) - 1)*sinh(d*x + c)^2)*Ei(d*x - 1/2*(a*d^3/b)^ 
(1/3)*(sqrt(-3) + 1))*sinh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1) + c) + (-a*d 
^3/b)^(2/3)*((sqrt(-3) - 1)*cosh(d*x + c)^2 - (sqrt(-3) - 1)*sinh(d*x + c) 
^2)*Ei(-d*x - 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1))*sinh(1/2*(-a*d^3/b)^(1/ 
3)*(sqrt(-3) + 1) - c) + (a*d^3/b)^(2/3)*((sqrt(-3) + 1)*cosh(d*x + c)^2 - 
 (sqrt(-3) + 1)*sinh(d*x + c)^2)*Ei(d*x + 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 
1))*sinh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1) - c) + (-a*d^3/b)^(2/3)*((s...
 

Sympy [F]

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

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

Output:

Integral(x**4*cosh(c + d*x)/(a + b*x**3), x)
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^3} \, dx=\int \frac {x^4\,\mathrm {cosh}\left (c+d\,x\right )}{b\,x^3+a} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^4 \cosh (c+d x)}{a+b x^3} \, dx=\frac {-\cosh \left (d x +c \right )-\left (\int \frac {\cosh \left (d x +c \right ) x}{b \,x^{3}+a}d x \right ) a \,d^{2}+\sinh \left (d x +c \right ) d x}{b \,d^{2}} \] Input:

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

Output:

( - cosh(c + d*x) - int((cosh(c + d*x)*x)/(a + b*x**3),x)*a*d**2 + sinh(c 
+ d*x)*d*x)/(b*d**2)