\(\int \frac {\cosh (c+d x)}{(a+b x^2)^3} \, dx\) [75]

Optimal result

Integrand size = 16, antiderivative size = 856 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.89 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.46 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {-e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (3 i b+3 \sqrt {a} \sqrt {b} d-i a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (-3 i b+3 \sqrt {a} \sqrt {b} d+i a d^2\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (3 i b+3 \sqrt {a} \sqrt {b} d-i a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (-3 i b+3 \sqrt {a} \sqrt {b} d+i a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )+\frac {4 \sqrt {a} \sqrt {b} \cosh (d x) \left (b x \left (5 a+3 b x^2\right ) \cosh (c)+a d \left (a+b x^2\right ) \sinh (c)\right )}{\left (a+b x^2\right )^2}+\frac {4 \sqrt {a} \sqrt {b} \left (a d \left (a+b x^2\right ) \cosh (c)+b x \left (5 a+3 b x^2\right ) \sinh (c)\right ) \sinh (d x)}{\left (a+b x^2\right )^2}}{32 a^{5/2} b^{3/2}} \] Input:

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

Output:

(-(E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(((3*I)*b + 3*Sqrt[a]*Sqrt[b]*d - I*a*d^2 
)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + 
x)] + ((-3*I)*b + 3*Sqrt[a]*Sqrt[b]*d + I*a*d^2)*ExpIntegralEi[d*((I*Sqrt[ 
a])/Sqrt[b] + x)])) + E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*(((3*I)*b + 3*Sqrt[a] 
*Sqrt[b]*d - I*a*d^2)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sq 
rt[a]*d)/Sqrt[b] - d*x] + ((-3*I)*b + 3*Sqrt[a]*Sqrt[b]*d + I*a*d^2)*ExpIn 
tegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) + (4*Sqrt[a]*Sqrt[b]*Cosh[d*x]*(b*x 
*(5*a + 3*b*x^2)*Cosh[c] + a*d*(a + b*x^2)*Sinh[c]))/(a + b*x^2)^2 + (4*Sq 
rt[a]*Sqrt[b]*(a*d*(a + b*x^2)*Cosh[c] + b*x*(5*a + 3*b*x^2)*Sinh[c])*Sinh 
[d*x])/(a + b*x^2)^2)/(32*a^(5/2)*b^(3/2))
 

Rubi [A] (verified)

Time = 1.89 (sec) , antiderivative size = 856, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5804, 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.

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

\(\Big \downarrow \) 5804

\(\displaystyle \int \left (-\frac {3 b \cosh (c+d x)}{8 a^2 \left (-a b-b^2 x^2\right )}-\frac {3 b \cosh (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {3 b \cosh (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b^{3/2} \cosh (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}-b x\right )^3}-\frac {b^{3/2} \cosh (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}+b x\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/16*Cosh[c + d*x]/((-a)^(3/2)*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)^2) - (3*Cos 
h[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) + Cosh[c + d*x]/(16*(- 
a)^(3/2)*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)^2) + (3*Cosh[c + d*x])/(16*a^2*Sqr 
t[b]*(Sqrt[-a] + Sqrt[b]*x)) + (3*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshInteg 
ral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (d^2*Cosh[c + ( 
Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3 
/2)*b^(3/2)) - (3*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d) 
/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt 
[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) - ( 
3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b] 
])/(16*a^2*b) - (3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sq 
rt[-a]*d)/Sqrt[b]])/(16*a^2*b) + (d*Sinh[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[ 
-a] - Sqrt[b]*x)) + (d*Sinh[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] + Sqrt[b] 
*x)) + (3*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[ 
b] - d*x])/(16*a^2*b) - (3*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sq 
rt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*Sinh[c + (Sqrt[-a 
]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^( 
3/2)) - (3*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt 
[b] + d*x])/(16*a^2*b) - (3*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(S 
qrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*Sinh[c - (Sqr...
 

Defintions of rubi rules used

rule 2009

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

rule 5804

Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> In 
t[ExpandIntegrand[Cosh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d 
}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
 

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 1064, normalized size of antiderivative = 1.24

method	result	size
risch	\(\text {Expression too large to display}\)	\(1064\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2116 vs. \(2 (655) = 1310\).

Time = 0.14 (sec) , antiderivative size = 2116, normalized size of antiderivative = 2.47 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {e^{d x +2 c} \left (\int \frac {e^{d x}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{3}+e^{d x +2 c} \left (\int \frac {e^{d x}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{2} b \,x^{2}+e^{d x} \left (\int \frac {x^{3}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) a b d +e^{d x} \left (\int \frac {x^{3}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) b^{2} d \,x^{2}-e^{d x} \left (\int \frac {x^{2}}{e^{d x} a^{3}+3 e^{d x} a^{2} b \,x^{2}+3 e^{d x} a \,b^{2} x^{4}+e^{d x} b^{3} x^{6}}d x \right ) a^{2} b -e^{d x} \left (\int \frac {x^{2}}{e^{d x} a^{3}+3 e^{d x} a^{2} b \,x^{2}+3 e^{d x} a \,b^{2} x^{4}+e^{d x} b^{3} x^{6}}d x \right ) a \,b^{2} x^{2}+e^{d x} \left (\int \frac {x^{2}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) a b +e^{d x} \left (\int \frac {x^{2}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) b^{2} x^{2}+e^{d x} \left (\int \frac {x}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) a^{2} d +e^{d x} \left (\int \frac {x}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) a b d \,x^{2}+x}{2 e^{d x +c} a^{2} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

(e**(2*c + d*x)*int(e**(d*x)/(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3* 
x**6),x)*a**3 + e**(2*c + d*x)*int(e**(d*x)/(a**3 + 3*a**2*b*x**2 + 3*a*b* 
*2*x**4 + b**3*x**6),x)*a**2*b*x**2 + e**(d*x)*int(x**3/(e**(d*x)*a**2 + 2 
*e**(d*x)*a*b*x**2 + e**(d*x)*b**2*x**4),x)*a*b*d + e**(d*x)*int(x**3/(e** 
(d*x)*a**2 + 2*e**(d*x)*a*b*x**2 + e**(d*x)*b**2*x**4),x)*b**2*d*x**2 - e* 
*(d*x)*int(x**2/(e**(d*x)*a**3 + 3*e**(d*x)*a**2*b*x**2 + 3*e**(d*x)*a*b** 
2*x**4 + e**(d*x)*b**3*x**6),x)*a**2*b - e**(d*x)*int(x**2/(e**(d*x)*a**3 
+ 3*e**(d*x)*a**2*b*x**2 + 3*e**(d*x)*a*b**2*x**4 + e**(d*x)*b**3*x**6),x) 
*a*b**2*x**2 + e**(d*x)*int(x**2/(e**(d*x)*a**2 + 2*e**(d*x)*a*b*x**2 + e* 
*(d*x)*b**2*x**4),x)*a*b + e**(d*x)*int(x**2/(e**(d*x)*a**2 + 2*e**(d*x)*a 
*b*x**2 + e**(d*x)*b**2*x**4),x)*b**2*x**2 + e**(d*x)*int(x/(e**(d*x)*a**2 
 + 2*e**(d*x)*a*b*x**2 + e**(d*x)*b**2*x**4),x)*a**2*d + e**(d*x)*int(x/(e 
**(d*x)*a**2 + 2*e**(d*x)*a*b*x**2 + e**(d*x)*b**2*x**4),x)*a*b*d*x**2 + x 
)/(2*e**(c + d*x)*a**2*(a + b*x**2))