\(\int \frac {x \cosh (c+d x)}{(a+b x^2)^3} \, dx\) [74]

Optimal result

Integrand size = 17, antiderivative size = 512 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {\cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \sinh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \sinh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.88 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.63 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {-d e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (-i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )-d e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (-i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )+\frac {4 \sqrt {a} b \cosh (d x) \left (-2 a \cosh (c)+d x \left (a+b x^2\right ) \sinh (c)\right )}{\left (a+b x^2\right )^2}+\frac {4 \sqrt {a} b \left (d x \left (a+b x^2\right ) \cosh (c)-2 a \sinh (c)\right ) \sinh (d x)}{\left (a+b x^2\right )^2}}{32 a^{3/2} b^2} \] Input:

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

Output:

(-(d*E^(c - (I*Sqrt[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)])) - 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)/Sqrt[b] - d*x]) + (4*Sqrt[a]*b*Cosh[d*x]* 
(-2*a*Cosh[c] + d*x*(a + b*x^2)*Sinh[c]))/(a + b*x^2)^2 + (4*Sqrt[a]*b*(d* 
x*(a + b*x^2)*Cosh[c] - 2*a*Sinh[c])*Sinh[d*x])/(a + b*x^2)^2)/(32*a^(3/2) 
*b^2)
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 507, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {5812, 5803, 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.

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

\(\Big \downarrow \) 5812

\(\displaystyle \frac {d \int \frac {\sinh (c+d x)}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {\cosh (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 5803

\(\displaystyle \frac {d \int \left (-\frac {b \sinh (c+d x)}{2 a \left (-b^2 x^2-a b\right )}-\frac {b \sinh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \sinh (c+d x)}{4 a \left (b x+\sqrt {-a} \sqrt {b}\right )^2}\right )dx}{4 b}-\frac {\cosh (c+d x)}{4 b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/4*Cosh[c + d*x]/(b*(a + b*x^2)^2) + (d*(-1/4*(d*Cosh[c + (Sqrt[-a]*d)/S 
qrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(a*b) - (d*Cosh[c - (Sqr 
t[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a*b) + (Cos 
hIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*( 
-a)^(3/2)*Sqrt[b]) - (CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (S 
qrt[-a]*d)/Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) - Sinh[c + d*x]/(4*a*Sqrt[b]*( 
Sqrt[-a] - Sqrt[b]*x)) + Sinh[c + d*x]/(4*a*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x) 
) + (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d* 
x])/(4*(-a)^(3/2)*Sqrt[b]) + (d*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegra 
l[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a*b) + (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*S 
inhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (d*Sinh[ 
c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a*b 
)))/(4*b)
 

Defintions of rubi rules used

rule 2009

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

rule 5803

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> In 
t[ExpandIntegrand[Sinh[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])
 

rule 5812

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p 
_), x_Symbol] :> Simp[e^m*(a + b*x^n)^(p + 1)*(Cosh[c + d*x]/(b*n*(p + 1))) 
, x] - Simp[d*(e^m/(b*n*(p + 1)))   Int[(a + b*x^n)^(p + 1)*Sinh[c + d*x], 
x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && EqQ[m - n + 1, 
0] && LtQ[p, -1] && (IntegerQ[n] || GtQ[e, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1502\) vs. \(2(398)=796\).

Time = 0.97 (sec) , antiderivative size = 1503, normalized size of antiderivative = 2.94

method	result	size
risch	\(\text {Expression too large to display}\)	\(1503\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1607 vs. \(2 (399) = 798\).

Time = 0.14 (sec) , antiderivative size = 1607, normalized size of antiderivative = 3.14 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/32*(8*a^2*b*cosh(d*x + c) + (((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^ 
2)*cosh(d*x + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x 
+ c)^2 + ((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*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - (a*b^2* 
d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^2 - ((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)) 
 + (((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - (a*b^2* 
d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^2 - ((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*b^2*d^2*x^4 + 2*a^2* 
b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + 
a^3*d^2)*sinh(d*x + c)^2 + ((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)) - 4*(a*b^2*d*x^3 + a^2*b*d 
*x)*sinh(d*x + c) + (((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x 
 + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^2 + (( 
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*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/2*(x*e^(d*x + 2*c) - x*e^(-d*x))/(b^3*d*x^6*e^c + 3*a*b^2*d*x^4*e^c + 3* 
a^2*b*d*x^2*e^c + a^3*d*e^c) + 1/2*integrate((5*b*x^2*e^c - a*e^c)*e^(d*x) 
/(b^4*d*x^8 + 4*a*b^3*d*x^6 + 6*a^2*b^2*d*x^4 + 4*a^3*b*d*x^2 + a^4*d), x) 
 - 1/2*integrate((5*b*x^2 - a)*e^(-d*x)/(b^4*d*x^8*e^c + 4*a*b^3*d*x^6*e^c 
 + 6*a^2*b^2*d*x^4*e^c + 4*a^3*b*d*x^2*e^c + a^4*d*e^c), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {-e^{2 d x +2 c} a +e^{d x +2 c} \left (\int \frac {e^{d x}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a^{3} d +2 e^{d x +2 c} \left (\int \frac {e^{d x}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a^{2} b d \,x^{2}+e^{d x +2 c} \left (\int \frac {e^{d x}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a \,b^{2} d \,x^{4}+e^{d x} \left (\int \frac {x^{2}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) a^{2} b d +2 e^{d x} \left (\int \frac {x^{2}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) a \,b^{2} d \,x^{2}+e^{d x} \left (\int \frac {x^{2}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) b^{3} d \,x^{4}+2 e^{d x} \left (\int \frac {x}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) a^{2} b +4 e^{d x} \left (\int \frac {x}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) a \,b^{2} x^{2}+2 e^{d x} \left (\int \frac {x}{e^{d x} a^{2}+2 e^{d x} a b \,x^{2}+e^{d x} b^{2} x^{4}}d x \right ) b^{3} x^{4}+b \,x^{2}}{8 e^{d x +c} a b \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

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

Output:

( - e**(2*c + 2*d*x)*a + e**(2*c + d*x)*int(e**(d*x)/(a**2 + 2*a*b*x**2 + 
b**2*x**4),x)*a**3*d + 2*e**(2*c + d*x)*int(e**(d*x)/(a**2 + 2*a*b*x**2 + 
b**2*x**4),x)*a**2*b*d*x**2 + e**(2*c + d*x)*int(e**(d*x)/(a**2 + 2*a*b*x* 
*2 + b**2*x**4),x)*a*b**2*d*x**4 + 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**2*b*d + 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**2*d*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)*b**3*d*x**4 + 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*b + 4*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**2*x**2 + 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)*b**3*x**4 + b* 
x**2)/(8*e**(c + d*x)*a*b*(a**2 + 2*a*b*x**2 + b**2*x**4))