\(\int \frac {\cosh (c+d x)}{x (a+b x^2)^2} \, dx\) [70]

Optimal result

Integrand size = 19, antiderivative size = 435 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\frac {\cosh (c+d x)}{2 a \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}-\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2} \] Output:

1/2*cosh(d*x+c)/a/(b*x^2+a)+cosh(c)*Chi(d*x)/a^2-1/2*cosh(c+(-a)^(1/2)*d/b 
^(1/2))*Chi((-a)^(1/2)*d/b^(1/2)-d*x)/a^2-1/2*cosh(c-(-a)^(1/2)*d/b^(1/2)) 
*Chi((-a)^(1/2)*d/b^(1/2)+d*x)/a^2-1/4*d*Chi((-a)^(1/2)*d/b^(1/2)+d*x)*sin 
h(c-(-a)^(1/2)*d/b^(1/2))/(-a)^(3/2)/b^(1/2)+1/4*d*Chi((-a)^(1/2)*d/b^(1/2 
)-d*x)*sinh(c+(-a)^(1/2)*d/b^(1/2))/(-a)^(3/2)/b^(1/2)+sinh(c)*Shi(d*x)/a^ 
2+1/4*d*cosh(c+(-a)^(1/2)*d/b^(1/2))*Shi(-(-a)^(1/2)*d/b^(1/2)+d*x)/(-a)^( 
3/2)/b^(1/2)-1/2*sinh(c+(-a)^(1/2)*d/b^(1/2))*Shi(-(-a)^(1/2)*d/b^(1/2)+d* 
x)/a^2-1/4*d*cosh(c-(-a)^(1/2)*d/b^(1/2))*Shi((-a)^(1/2)*d/b^(1/2)+d*x)/(- 
a)^(3/2)/b^(1/2)-1/2*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 = 0.79 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.95 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\frac {\frac {4 a \cosh (c) \cosh (d x)}{a+b x^2}+\frac {i \sqrt {a} d e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{\sqrt {b}}-2 e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+\frac {i \sqrt {a} d e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{\sqrt {b}}-2 e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )+\frac {4 a \sinh (c) \sinh (d x)}{a+b x^2}+8 (\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x))}{8 a^2} \] Input:

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

Output:

((4*a*Cosh[c]*Cosh[d*x])/(a + b*x^2) + (I*Sqrt[a]*d*E^(c - (I*Sqrt[a]*d)/S 
qrt[b])*(E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqr 
t[b] + x)] - ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)]))/Sqrt[b] - 2*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)]) 
 + (I*Sqrt[a]*d*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]))/Sqrt[b] - 2*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] + Exp 
IntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) + (4*a*Sinh[c]*Sinh[d*x])/(a + b* 
x^2) + 8*(Cosh[c]*CoshIntegral[d*x] + Sinh[c]*SinhIntegral[d*x]))/(8*a^2)
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 435, 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.

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

\(\Big \downarrow \) 5816

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\cosh (c+d x)}{2 a \left (a+b x^2\right )}\)

Input:

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

Output:

Cosh[c + d*x]/(2*a*(a + b*x^2)) + (Cosh[c]*CoshIntegral[d*x])/a^2 - (Cosh[ 
c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2 
) - (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d* 
x])/(2*a^2) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[- 
a]*d)/Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt 
[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) + (Sinh[ 
c]*SinhIntegral[d*x])/a^2 - (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral 
[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) + (Sinh[c + (Sqrt[-a] 
*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2) - (d*Cosh[c 
 - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a) 
^(3/2)*Sqrt[b]) - (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-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 = 0.66 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.26

method	result	size
risch	\(\frac {{\mathrm e}^{-d x -c} d^{2}}{4 a \left (\left (d x +c \right )^{2} b -2 b \left (d x +c \right ) c +a \,d^{2}+b \,c^{2}\right )}-\frac {{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right )}{2 a^{2}}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right ) d}{8 a \sqrt {-a b}}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right ) d}{8 a \sqrt {-a b}}+\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right )}{4 a^{2}}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right )}{4 a^{2}}+\frac {{\mathrm e}^{d x +c} d^{2}}{4 a \left (\left (d x +c \right )^{2} b -2 b \left (d x +c \right ) c +a \,d^{2}+b \,c^{2}\right )}-\frac {{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right )}{2 a^{2}}+\frac {\operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right ) {\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} d}{8 a \sqrt {-a b}}-\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right ) d}{8 a \sqrt {-a b}}+\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right )}{4 a^{2}}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right )}{4 a^{2}}\)	\(546\)

Input:

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

Output:

1/4*exp(-d*x-c)*d^2/a/((d*x+c)^2*b-2*b*(d*x+c)*c+a*d^2+b*c^2)-1/2/a^2*exp( 
-c)*Ei(1,d*x)-1/8/a/(-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)*d+1/8/a/(-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)*d+1/4/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/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*exp(d*x+c)*d^2/a/( 
(d*x+c)^2*b-2*b*(d*x+c)*c+a*d^2+b*c^2)-1/2/a^2*exp(c)*Ei(1,-d*x)+1/8/a/(-a 
*b)^(1/2)*Ei(1,(d*(-a*b)^(1/2)-b*(d*x+c)+c*b)/b)*exp((d*(-a*b)^(1/2)+c*b)/ 
b)*d-1/8/a/(-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)*d+1/4/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/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)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 992 vs. \(2 (337) = 674\).

Time = 0.14 (sec) , antiderivative size = 992, normalized size of antiderivative = 2.28 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(4*a*cosh(d*x + c) - ((2*(b*x^2 + a)*cosh(d*x + c)^2 - 2*(b*x^2 + a)*s 
inh(d*x + c)^2 - ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sinh(d*x + c)^ 
2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) + (2*(b*x^2 + a)*cosh(d*x + c) 
^2 - 2*(b*x^2 + a)*sinh(d*x + c)^2 + ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 
 + a)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*x + sqrt(-a*d^2/b)))*cosh(c + 
 sqrt(-a*d^2/b)) + 4*((b*x^2 + a)*Ei(d*x) + (b*x^2 + a)*Ei(-d*x))*cosh(c) 
- ((2*(b*x^2 + a)*cosh(d*x + c)^2 - 2*(b*x^2 + a)*sinh(d*x + c)^2 + ((b*x^ 
2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d 
*x + sqrt(-a*d^2/b)) + (2*(b*x^2 + a)*cosh(d*x + c)^2 - 2*(b*x^2 + a)*sinh 
(d*x + c)^2 - ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sinh(d*x + c)^2)* 
sqrt(-a*d^2/b))*Ei(-d*x - sqrt(-a*d^2/b)))*cosh(-c + sqrt(-a*d^2/b)) - ((2 
*(b*x^2 + a)*cosh(d*x + c)^2 - 2*(b*x^2 + a)*sinh(d*x + c)^2 - ((b*x^2 + a 
)*cosh(d*x + c)^2 - (b*x^2 + a)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - 
sqrt(-a*d^2/b)) - (2*(b*x^2 + a)*cosh(d*x + c)^2 - 2*(b*x^2 + a)*sinh(d*x 
+ c)^2 + ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sinh(d*x + c)^2)*sqrt( 
-a*d^2/b))*Ei(-d*x + sqrt(-a*d^2/b)))*sinh(c + sqrt(-a*d^2/b)) + 4*((b*x^2 
 + a)*Ei(d*x) - (b*x^2 + a)*Ei(-d*x))*sinh(c) + ((2*(b*x^2 + a)*cosh(d*x + 
 c)^2 - 2*(b*x^2 + a)*sinh(d*x + c)^2 + ((b*x^2 + a)*cosh(d*x + c)^2 - (b* 
x^2 + a)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) - (2*(b 
*x^2 + a)*cosh(d*x + c)^2 - 2*(b*x^2 + a)*sinh(d*x + c)^2 - ((b*x^2 + a...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: AttributeError >> type
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\int \frac {\cosh \left (d x +c \right )}{b^{2} x^{5}+2 a b \,x^{3}+a^{2} x}d x \] Input:

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

Output:

int(cosh(c + d*x)/(a**2*x + 2*a*b*x**3 + b**2*x**5),x)