\(\int \frac {\cosh (c+d x)}{x (a+b x^2)^3} \, dx\) [76]

Optimal result

Integrand size = 19, antiderivative size = 730 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, dx=\frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}-\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^3}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}-\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^3}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^2 b}+\frac {5 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {5 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\frac {5 d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \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^3}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}+\frac {5 d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{5/2} \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^3}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^2 b} \] Output:

1/4*cosh(d*x+c)/a/(b*x^2+a)^2+1/2*cosh(d*x+c)/a^2/(b*x^2+a)+cosh(c)*Chi(d* 
x)/a^3-1/2*cosh(c+(-a)^(1/2)*d/b^(1/2))*Chi((-a)^(1/2)*d/b^(1/2)-d*x)/a^3+ 
1/16*d^2*cosh(c+(-a)^(1/2)*d/b^(1/2))*Chi((-a)^(1/2)*d/b^(1/2)-d*x)/a^2/b- 
1/2*cosh(c-(-a)^(1/2)*d/b^(1/2))*Chi((-a)^(1/2)*d/b^(1/2)+d*x)/a^3+1/16*d^ 
2*cosh(c-(-a)^(1/2)*d/b^(1/2))*Chi((-a)^(1/2)*d/b^(1/2)+d*x)/a^2/b+5/16*d* 
Chi((-a)^(1/2)*d/b^(1/2)+d*x)*sinh(c-(-a)^(1/2)*d/b^(1/2))/(-a)^(5/2)/b^(1 
/2)-5/16*d*Chi((-a)^(1/2)*d/b^(1/2)-d*x)*sinh(c+(-a)^(1/2)*d/b^(1/2))/(-a) 
^(5/2)/b^(1/2)+1/16*d*sinh(d*x+c)/a^2/b^(1/2)/((-a)^(1/2)-b^(1/2)*x)-1/16* 
d*sinh(d*x+c)/a^2/b^(1/2)/((-a)^(1/2)+b^(1/2)*x)+sinh(c)*Shi(d*x)/a^3-5/16 
*d*cosh(c+(-a)^(1/2)*d/b^(1/2))*Shi(-(-a)^(1/2)*d/b^(1/2)+d*x)/(-a)^(5/2)/ 
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^ 
3+1/16*d^2*sinh(c+(-a)^(1/2)*d/b^(1/2))*Shi(-(-a)^(1/2)*d/b^(1/2)+d*x)/a^2 
/b+5/16*d*cosh(c-(-a)^(1/2)*d/b^(1/2))*Shi((-a)^(1/2)*d/b^(1/2)+d*x)/(-a)^ 
(5/2)/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^3+1/16*d^2*sinh(c-(-a)^(1/2)*d/b^(1/2))*Shi((-a)^(1/2)*d/b^(1/2)+d*x) 
/a^2/b
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.44 (sec) , antiderivative size = 674, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, dx=\frac {\frac {8 a \left (3 a+2 b x^2\right ) \cosh (c+d x)}{\left (a+b x^2\right )^2}+32 \cosh (c) \text {Chi}(d x)+\frac {4 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}}-8 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 {\sqrt {a} d e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (-i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{b}+\frac {4 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}}-8 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 {\sqrt {a} d e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (-i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{b}-\frac {4 a d x \sinh (c+d x)}{a+b x^2}+32 \sinh (c) \text {Shi}(d x)}{32 a^3} \] Input:

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

Output:

((8*a*(3*a + 2*b*x^2)*Cosh[c + d*x])/(a + b*x^2)^2 + 32*Cosh[c]*CoshIntegr 
al[d*x] + ((4*I)*Sqrt[a]*d*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(((2*I)*Sqrt[a 
]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] - ExpIntegralE 
i[d*((I*Sqrt[a])/Sqrt[b] + x)]))/Sqrt[b] - 8*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)]) + (Sqrt[a]*d*E^(c - (I*S 
qrt[a]*d)/Sqrt[b])*((I*Sqrt[b] + Sqrt[a]*d)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])* 
ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + ((-I)*Sqrt[b] + Sqrt[a]*d) 
*ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)]))/b + ((4*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] - 8*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[a]*d*E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*((I*Sqrt[b] 
+ Sqrt[a]*d)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/ 
Sqrt[b] - d*x] + ((-I)*Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[(I*Sqrt[a]*d)/Sq 
rt[b] - d*x]))/b - (4*a*d*x*Sinh[c + d*x])/(a + b*x^2) + 32*Sinh[c]*SinhIn 
tegral[d*x])/(32*a^3)
 

Rubi [A] (verified)

Time = 2.51 (sec) , antiderivative size = 730, 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.

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

\(\Big \downarrow \) 5816

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

\(\Big \downarrow \) 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^3}-\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^3}+\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^3}-\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^3}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\frac {d^2 \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^2 b}-\frac {d^2 \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^2 b}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^2 b}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {5 d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {5 d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {5 d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {5 d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2}\)

Input:

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

Output:

Cosh[c + d*x]/(4*a*(a + b*x^2)^2) + Cosh[c + d*x]/(2*a^2*(a + b*x^2)) + (C 
osh[c]*CoshIntegral[d*x])/a^3 - (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegr 
al[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^3) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[ 
b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (Cosh[c - (Sqrt 
[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^3) + (d^2* 
Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/( 
16*a^2*b) + (5*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[- 
a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) - (5*d*CoshIntegral[(Sqrt[-a]*d)/S 
qrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) + (d 
*Sinh[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) - (d*Sinh[c + d*x] 
)/(16*a^2*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) + (Sinh[c]*SinhIntegral[d*x])/a^ 
3 + (5*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] 
- d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhInte 
gral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^3) - (d^2*Sinh[c + (Sqrt[-a]*d)/Sqr 
t[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) + (5*d*Cosh[c - 
 (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^ 
(5/2)*Sqrt[b]) - (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d) 
/Sqrt[b] + d*x])/(2*a^3) + (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegra 
l[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b)
 

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 = 0.96 (sec) , antiderivative size = 1090, normalized size of antiderivative = 1.49

method	result	size
risch	\(\text {Expression too large to display}\)	\(1090\)

Input:

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

Output:

1/16*exp(-d*x-c)*d^2*(b*(d*x+c)^3-3*(d*x+c)^2*b*c+(d*x+c)*a*d^2+3*(d*x+c)* 
b*c^2-d^2*c*a-b*c^3+4*(d*x+c)^2*b-8*b*(d*x+c)*c+6*a*d^2+4*b*c^2)/a^2/((d*x 
+c)^4*b^2-4*(d*x+c)^3*c*b^2+2*(d*x+c)^2*a*b*d^2+6*(d*x+c)^2*c^2*b^2-4*a*b* 
(d*x+c)*c*d^2-4*b^2*(d*x+c)*c^3+a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)-1/2/a^3*exp 
(-c)*Ei(1,d*x)-1/32/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)*d^2-1/32/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)*d^2-5/32/a^2/(-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+5/32/a^2/(-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^3*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^3*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+b*(d*x+c)-c*b 
)/b)-1/16*exp(d*x+c)*d^2*(b*(d*x+c)^3-3*(d*x+c)^2*b*c+(d*x+c)*a*d^2+3*(d*x 
+c)*b*c^2-d^2*c*a-b*c^3-4*(d*x+c)^2*b+8*b*(d*x+c)*c-6*a*d^2-4*b*c^2)/a^2/( 
(d*x+c)^4*b^2-4*(d*x+c)^3*c*b^2+2*(d*x+c)^2*a*b*d^2+6*(d*x+c)^2*c^2*b^2-4* 
a*b*(d*x+c)*c*d^2-4*b^2*(d*x+c)*c^3+a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)-1/2/a^3 
*exp(c)*Ei(1,-d*x)-1/32/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)*d^2-1/32/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)*d^2+5/32/a^2/(-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-5/32/a^2/(-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)*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2076 vs. \(2 (575) = 1150\).

Time = 0.15 (sec) , antiderivative size = 2076, normalized size of antiderivative = 2.84 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/32*(8*(2*a*b^2*x^2 + 3*a^2*b)*cosh(d*x + c) + (((a^3*d^2 + (a*b^2*d^2 - 
8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3 
*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*si 
nh(d*x + c)^2 + 5*((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3* 
x^4 + 2*a*b^2*x^2 + a^2*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt( 
-a*d^2/b)) + ((a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 
- 8*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a 
^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*sinh(d*x + c)^2 - 5*((b^3*x^4 + 2*a*b^ 
2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*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) 
) + 16*((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*Ei(d*x) + (b^3*x^4 + 2*a*b^2*x^2 + 
 a^2*b)*Ei(-d*x))*cosh(c) + (((a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b 
 + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 
8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*sinh(d*x + c)^2 - 5*(( 
b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*b^2*x^2 + 
a^2*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) + ((a^3*d 
^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*cosh 
(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 
- 8*a*b^2)*x^2)*sinh(d*x + c)^2 + 5*((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh( 
d*x + c)^2 - (b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*sinh(d*x + c)^2)*sqrt(-a*d...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^2\right )^3} \, dx=\int \frac {\cosh \left (d x +c \right )}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \] Input:

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

Output:

int(cosh(c + d*x)/(a**3*x + 3*a**2*b*x**3 + 3*a*b**2*x**5 + b**3*x**7),x)