\(\int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx\) [59]

Optimal result

Integrand size = 19, antiderivative size = 226 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx=\frac {\sqrt {-a} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{3/2}}+\frac {\sinh (c+d x)}{b d}-\frac {\sqrt {-a} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{3/2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx=\frac {i \sqrt {a} e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{4 b^{3/2}}-\frac {i \sqrt {a} e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{4 b^{3/2}}+\frac {\cosh (d x) \sinh (c)}{b d}+\frac {\cosh (c) \sinh (d x)}{b d} \] Input:

Integrate[(x^2*Cosh[c + d*x])/(a + b*x^2),x]
 

Output:

((I/4)*Sqrt[a]*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(((2*I)*Sqrt[a]*d)/Sqrt[b] 
)*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] - ExpIntegralEi[d*((I*Sqrt 
[a])/Sqrt[b] + x)]))/b^(3/2) - ((I/4)*Sqrt[a]*E^(-c - (I*Sqrt[a]*d)/Sqrt[b 
])*(E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - 
 d*x] - ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]))/b^(3/2) + (Cosh[d*x]* 
Sinh[c])/(b*d) + (Cosh[c]*Sinh[d*x])/(b*d)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 226, 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.

\(\displaystyle \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx\)

\(\Big \downarrow \) 5816

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^2*Cosh[c + d*x])/(a + b*x^2),x]
 

Output:

(Sqrt[-a]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] 
 - d*x])/(2*b^(3/2)) - (Sqrt[-a]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegr 
al[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(3/2)) + Sinh[c + d*x]/(b*d) - (Sqrt[ 
-a]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x 
])/(2*b^(3/2)) - (Sqrt[-a]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sq 
rt[-a]*d)/Sqrt[b] + d*x])/(2*b^(3/2))
 

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 [A] (verified)

Time = 0.57 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.15

method	result	size
risch	\(-\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right ) a}{4 b \sqrt {-a b}}+\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right ) a}{4 b \sqrt {-a b}}+\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right ) a}{4 b \sqrt {-a b}}-\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right ) a}{4 b \sqrt {-a b}}-\frac {{\mathrm e}^{-d x -c}}{2 d b}+\frac {{\mathrm e}^{d x +c}}{2 b d}\)	\(259\)

Input:

int(x^2*cosh(d*x+c)/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-1/4/b/(-a*b)^(1/2)*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+b*( 
d*x+c)-c*b)/b)*a+1/4/b/(-a*b)^(1/2)*exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d* 
(-a*b)^(1/2)-b*(d*x+c)+c*b)/b)*a+1/4/b/(-a*b)^(1/2)*exp((d*(-a*b)^(1/2)+c* 
b)/b)*Ei(1,(d*(-a*b)^(1/2)-b*(d*x+c)+c*b)/b)*a-1/4/b/(-a*b)^(1/2)*exp((-d* 
(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+b*(d*x+c)-c*b)/b)*a-1/2/d/b*exp 
(-d*x-c)+1/2/b/d*exp(d*x+c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (170) = 340\).

Time = 0.09 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.19 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx=\frac {{\left (\sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + \sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (\sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + \sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (\sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - \sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (\sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - \sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) + 4 \, \sinh \left (d x + c\right )}{4 \, {\left (b d \cosh \left (d x + c\right )^{2} - b d \sinh \left (d x + c\right )^{2}\right )}} \] Input:

integrate(x^2*cosh(d*x+c)/(b*x^2+a),x, algorithm="fricas")
 

Output:

1/4*((sqrt(-a*d^2/b)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(d*x - sqrt(-a* 
d^2/b)) + sqrt(-a*d^2/b)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(-d*x + sqr 
t(-a*d^2/b)))*cosh(c + sqrt(-a*d^2/b)) - (sqrt(-a*d^2/b)*(cosh(d*x + c)^2 
- sinh(d*x + c)^2)*Ei(d*x + sqrt(-a*d^2/b)) + sqrt(-a*d^2/b)*(cosh(d*x + c 
)^2 - sinh(d*x + c)^2)*Ei(-d*x - sqrt(-a*d^2/b)))*cosh(-c + sqrt(-a*d^2/b) 
) + (sqrt(-a*d^2/b)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(d*x - sqrt(-a*d 
^2/b)) - sqrt(-a*d^2/b)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(-d*x + sqrt 
(-a*d^2/b)))*sinh(c + sqrt(-a*d^2/b)) + (sqrt(-a*d^2/b)*(cosh(d*x + c)^2 - 
 sinh(d*x + c)^2)*Ei(d*x + sqrt(-a*d^2/b)) - sqrt(-a*d^2/b)*(cosh(d*x + c) 
^2 - sinh(d*x + c)^2)*Ei(-d*x - sqrt(-a*d^2/b)))*sinh(-c + sqrt(-a*d^2/b)) 
 + 4*sinh(d*x + c))/(b*d*cosh(d*x + c)^2 - b*d*sinh(d*x + c)^2)
 

Sympy [F]

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

integrate(x**2*cosh(d*x+c)/(b*x**2+a),x)
 

Output:

Integral(x**2*cosh(c + d*x)/(a + b*x**2), x)
 

Maxima [F]

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

integrate(x^2*cosh(d*x+c)/(b*x^2+a),x, algorithm="maxima")
 

Output:

-a*integrate(x*e^(d*x + c)/(b^2*d*x^4 + 2*a*b*d*x^2 + a^2*d), x) + a*integ 
rate(x*e^(-d*x)/(b^2*d*x^4*e^c + 2*a*b*d*x^2*e^c + a^2*d*e^c), x) + 1/2*(x 
^2*e^(d*x + 2*c) - x^2*e^(-d*x))/(b*d*x^2*e^c + a*d*e^c)
 

Giac [F]

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

integrate(x^2*cosh(d*x+c)/(b*x^2+a),x, algorithm="giac")
 

Output:

integrate(x^2*cosh(d*x + c)/(b*x^2 + a), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx=\int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{b\,x^2+a} \,d x \] Input:

int((x^2*cosh(c + d*x))/(a + b*x^2),x)
 

Output:

int((x^2*cosh(c + d*x))/(a + b*x^2), x)
 

Reduce [F]

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

int(x^2*cosh(d*x+c)/(b*x^2+a),x)
 

Output:

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