\(\int \frac {\cosh (c+d x)}{(a+b x^2)^2} \, dx\) [69]

Optimal result

Integrand size = 16, antiderivative size = 476 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {\cosh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{4 a \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a b}+\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a b}+\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a b}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.84 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.62 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {4 \sqrt {a} x \cosh (c) \cosh (d x)}{a+b x^2}-\frac {e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (-i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{b}+\frac {e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (-i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{b}+\frac {4 \sqrt {a} x \sinh (c) \sinh (d x)}{a+b x^2}}{8 a^{3/2}} \] Input:

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

Output:

((4*Sqrt[a]*x*Cosh[c]*Cosh[d*x])/(a + b*x^2) - (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)]))/b + (E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*((I*Sqr 
t[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]))/b + (4*Sqrt[a]*x*Sinh[c]*Sinh[d*x])/(a + b*x^2))/(8*a^ 
(3/2))
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 476, 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.

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

\(\Big \downarrow \) 5804

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/4*Cosh[c + d*x]/(a*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) + Cosh[c + d*x]/(4*a 
*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) - (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshInt 
egral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) + (Cosh[c - (Sqr 
t[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)* 
Sqrt[b]) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]* 
d)/Sqrt[b]])/(4*a*b) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c 
+ (Sqrt[-a]*d)/Sqrt[b]])/(4*a*b) + (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhI 
ntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a*b) + (Sinh[c + (Sqrt[-a]*d)/Sqrt 
[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (d 
*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/ 
(4*a*b) + (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b 
] + d*x])/(4*(-a)^(3/2)*Sqrt[b])
 

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 = 0.60 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.06

method	result	size
risch	\(\frac {d^{2} {\mathrm e}^{-d x -c} x}{4 a \left (b \,d^{2} x^{2}+a \,d^{2}\right )}-\frac {d \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right )}{8 b a}-\frac {d \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right )}{8 b a}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right )}{8 \sqrt {-a b}\, a}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right )}{8 \sqrt {-a b}\, a}+\frac {d^{2} {\mathrm e}^{d x +c} x}{4 a \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {d \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right )}{8 b a}+\frac {d \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right )}{8 b a}-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (\frac {d \sqrt {-a b}-b \left (d x +c \right )+c b}{b}\right )}{8 \sqrt {-a b}\, a}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {expIntegral}_{1}\left (-\frac {d \sqrt {-a b}+b \left (d x +c \right )-c b}{b}\right )}{8 \sqrt {-a b}\, a}\)	\(503\)

Input:

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

Output:

1/4*d^2*exp(-d*x-c)*x/a/(b*d^2*x^2+a*d^2)-1/8*d/b/a*exp(-(d*(-a*b)^(1/2)+c 
*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-b*(d*x+c)+c*b)/b)-1/8*d/b/a*exp(-(-d*(-a*b)^( 
1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+b*(d*x+c)-c*b)/b)-1/8/(-a*b)^(1/2)/a*exp 
(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-b*(d*x+c)+c*b)/b)+1/8/(-a* 
b)^(1/2)/a*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^2*exp(d*x+c)*x/a/(b*d^2*x^2+a*d^2)+1/8*d/b/a*exp((d*(-a*b)^(1/ 
2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-b*(d*x+c)+c*b)/b)+1/8*d/b/a*exp((-d*(-a*b) 
^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+b*(d*x+c)-c*b)/b)-1/8/(-a*b)^(1/2)/a* 
exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-b*(d*x+c)+c*b)/b)+1/8/(-a 
*b)^(1/2)/a*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. 1162 vs. \(2 (365) = 730\).

Time = 0.11 (sec) , antiderivative size = 1162, normalized size of antiderivative = 2.44 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(cosh(c + d*x)/(a**2 + 2*a*b*x**2 + b**2*x**4),x)