\(\int \frac {\cosh (c+d x)}{x^3 (a+b x^2)} \, dx\) [64]

Optimal result

Integrand size = 19, antiderivative size = 270 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=-\frac {\cosh (c+d x)}{2 a x^2}-\frac {b \cosh (c) \text {Chi}(d x)}{a^2}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}+\frac {b \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2}-\frac {d \sinh (c+d x)}{2 a x}-\frac {b \sinh (c) \text {Shi}(d x)}{a^2}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {b \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.71 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.86 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\frac {b e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+b e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )-\frac {2 a \cosh (d x) (\cosh (c)+d x \sinh (c))}{x^2}-\frac {2 a (d x \cosh (c)+\sinh (c)) \sinh (d x)}{x^2}+2 \left (-2 b+a d^2\right ) (\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x))}{4 a^2} \] Input:

Integrate[Cosh[c + d*x]/(x^3*(a + b*x^2)),x]
 

Output:

(b*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegra 
lEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] 
 + x)]) + b*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]) - (2*a*Cosh[d*x]*(Cosh[c] + d*x*Sinh[c]))/x^2 - (2*a*(d* 
x*Cosh[c] + Sinh[c])*Sinh[d*x])/x^2 + 2*(-2*b + a*d^2)*(Cosh[c]*CoshIntegr 
al[d*x] + Sinh[c]*SinhIntegral[d*x]))/(4*a^2)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 270, 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.

\(\displaystyle \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx\)

\(\Big \downarrow \) 5816

\(\displaystyle \int \left (\frac {b^2 x \cosh (c+d x)}{a^2 \left (a+b x^2\right )}-\frac {b \cosh (c+d x)}{a^2 x}+\frac {\cosh (c+d x)}{a x^3}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[Cosh[c + d*x]/(x^3*(a + b*x^2)),x]
 

Output:

-1/2*Cosh[c + d*x]/(a*x^2) - (b*Cosh[c]*CoshIntegral[d*x])/a^2 + (d^2*Cosh 
[c]*CoshIntegral[d*x])/(2*a) + (b*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshInteg 
ral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2) + (b*Cosh[c - (Sqrt[-a]*d)/Sqrt[b 
]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2) - (d*Sinh[c + d*x])/( 
2*a*x) - (b*Sinh[c]*SinhIntegral[d*x])/a^2 + (d^2*Sinh[c]*SinhIntegral[d*x 
])/(2*a) - (b*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqr 
t[b] - d*x])/(2*a^2) + (b*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqr 
t[-a]*d)/Sqrt[b] + d*x])/(2*a^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 = 330, normalized size of antiderivative = 1.22

method	result	size
risch	\(\frac {d \,{\mathrm e}^{-d x -c}}{4 a x}-\frac {{\mathrm e}^{-d x -c}}{4 a \,x^{2}}-\frac {d^{2} {\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right )}{4 a}+\frac {{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) b}{2 a^{2}}-\frac {b \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right )}{4 a^{2}}-\frac {b \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right )}{4 a^{2}}-\frac {d \,{\mathrm e}^{d x +c}}{4 a x}-\frac {{\mathrm e}^{d x +c}}{4 a \,x^{2}}-\frac {d^{2} {\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right )}{4 a}+\frac {{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right ) b}{2 a^{2}}-\frac {b \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right )}{4 a^{2}}-\frac {b \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right )}{4 a^{2}}\)	\(330\)

Input:

int(cosh(d*x+c)/x^3/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (222) = 444\).

Time = 0.11 (sec) , antiderivative size = 583, normalized size of antiderivative = 2.16 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=-\frac {2 \, a d x \sinh \left (d x + c\right ) + 2 \, a \cosh \left (d x + c\right ) - {\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{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 ({\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (d x\right ) + {\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{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 ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{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 ({\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (d x\right ) - {\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) + {\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{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 \, {\left (a^{2} x^{2} \cosh \left (d x + c\right )^{2} - a^{2} x^{2} \sinh \left (d x + c\right )^{2}\right )}} \] Input:

integrate(cosh(d*x+c)/x^3/(b*x^2+a),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(cosh(d*x+c)/x**3/(b*x**2+a),x)
 

Output:

Integral(cosh(c + d*x)/(x**3*(a + b*x**2)), x)
 

Maxima [F]

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

integrate(cosh(d*x+c)/x^3/(b*x^2+a),x, algorithm="maxima")
 

Output:

integrate(cosh(d*x + c)/((b*x^2 + a)*x^3), x)
 

Giac [F]

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

integrate(cosh(d*x+c)/x^3/(b*x^2+a),x, algorithm="giac")
 

Output:

integrate(cosh(d*x + c)/((b*x^2 + a)*x^3), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^3\,\left (b\,x^2+a\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\frac {-e^{2 d x +2 c} d x -e^{2 d x +2 c}+e^{d x +2 c} \left (\int \frac {e^{d x}}{b \,x^{3}+a x}d x \right ) a \,d^{2} x^{2}-2 e^{d x +2 c} \left (\int \frac {e^{d x}}{b \,x^{3}+a x}d x \right ) b \,x^{2}+e^{d x +2 c} \left (\int \frac {e^{d x} x}{b \,x^{2}+a}d x \right ) b \,d^{2} x^{2}+e^{d x} \left (\int \frac {x}{e^{d x} a +e^{d x} b \,x^{2}}d x \right ) b \,d^{2} x^{2}+e^{d x} \left (\int \frac {1}{e^{d x} a x +e^{d x} b \,x^{3}}d x \right ) a \,d^{2} x^{2}-2 e^{d x} \left (\int \frac {1}{e^{d x} a x +e^{d x} b \,x^{3}}d x \right ) b \,x^{2}+d x -1}{4 e^{d x +c} a \,x^{2}} \] Input:

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

Output:

( - e**(2*c + 2*d*x)*d*x - e**(2*c + 2*d*x) + e**(2*c + d*x)*int(e**(d*x)/ 
(a*x + b*x**3),x)*a*d**2*x**2 - 2*e**(2*c + d*x)*int(e**(d*x)/(a*x + b*x** 
3),x)*b*x**2 + e**(2*c + d*x)*int((e**(d*x)*x)/(a + b*x**2),x)*b*d**2*x**2 
 + e**(d*x)*int(x/(e**(d*x)*a + e**(d*x)*b*x**2),x)*b*d**2*x**2 + e**(d*x) 
*int(1/(e**(d*x)*a*x + e**(d*x)*b*x**3),x)*a*d**2*x**2 - 2*e**(d*x)*int(1/ 
(e**(d*x)*a*x + e**(d*x)*b*x**3),x)*b*x**2 + d*x - 1)/(4*e**(c + d*x)*a*x* 
*2)