3.1.57 \(\int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx\) [57]

3.1.57.1 Optimal result

Integrand size = 19, antiderivative size = 273 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx=-\frac {2 x \cosh (c+d x)}{b d^2}+\frac {(-a)^{3/2} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{5/2}}+\frac {2 \sinh (c+d x)}{b d^3}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {x^2 \sinh (c+d x)}{b d}-\frac {(-a)^{3/2} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{5/2}} \]

output

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

3.1.57.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.08 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.01 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx=\frac {-i a^{3/2} e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+i a^{3/2} e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )+\frac {4 \sqrt {b} \cosh (d x) \left (-2 b d x \cosh (c)+\left (-a d^2+b \left (2+d^2 x^2\right )\right ) \sinh (c)\right )}{d^3}+\frac {4 \sqrt {b} \left (\left (-a d^2+b \left (2+d^2 x^2\right )\right ) \cosh (c)-2 b d x \sinh (c)\right ) \sinh (d x)}{d^3}}{4 b^{5/2}} \]

input

Integrate[(x^4*Cosh[c + d*x])/(a + b*x^2),x]
 

output

((-I)*a^(3/2)*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*a^(3/2)*E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(((2*I)*S 
qrt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] - ExpInte 
gralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) + (4*Sqrt[b]*Cosh[d*x]*(-2*b*d*x*Cosh 
[c] + (-(a*d^2) + b*(2 + d^2*x^2))*Sinh[c]))/d^3 + (4*Sqrt[b]*((-(a*d^2) + 
 b*(2 + d^2*x^2))*Cosh[c] - 2*b*d*x*Sinh[c])*Sinh[d*x])/d^3)/(4*b^(5/2))
 

3.1.57.3 Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 273, 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.

\(\displaystyle \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx\)

\(\Big \downarrow \) 5816

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

\(\Big \downarrow \) 2009

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

input

Int[(x^4*Cosh[c + d*x])/(a + b*x^2),x]
 

output

(-2*x*Cosh[c + d*x])/(b*d^2) + ((-a)^(3/2)*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]* 
CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^(5/2)) - ((-a)^(3/2)*Cosh[c 
 - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(5 
/2)) + (2*Sinh[c + d*x])/(b*d^3) - (a*Sinh[c + d*x])/(b^2*d) + (x^2*Sinh[c 
 + d*x])/(b*d) - ((-a)^(3/2)*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[( 
Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^(5/2)) - ((-a)^(3/2)*Sinh[c - (Sqrt[-a]*d 
)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(5/2))
 

3.1.57.3.1 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])
 

3.1.57.4 Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.35

method	result	size
risch	\(-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}+\frac {{\mathrm e}^{d x +c} x^{2}}{2 d b}-\frac {{\mathrm e}^{-d x -c} x^{2}}{2 d b}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}-\frac {a \,{\mathrm e}^{d x +c}}{2 d \,b^{2}}-\frac {{\mathrm e}^{d x +c} x}{d^{2} b}+\frac {{\mathrm e}^{-d x -c} a}{2 d \,b^{2}}-\frac {{\mathrm e}^{-d x -c} x}{d^{2} b}+\frac {{\mathrm e}^{d x +c}}{d^{3} b}-\frac {{\mathrm e}^{-d x -c}}{d^{3} b}\)	\(369\)

input

int(x^4*cosh(d*x+c)/(b*x^2+a),x,method=_RETURNVERBOSE)
 

output

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

3.1.57.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (217) = 434\).

Time = 0.27 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.22 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx=-\frac {8 \, b d x \cosh \left (d x + c\right ) + {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) - 4 \, {\left (b d^{2} x^{2} - a d^{2} + 2 \, b\right )} \sinh \left (d x + c\right ) + {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{4 \, {\left (b^{2} d^{3} \cosh \left (d x + c\right )^{2} - b^{2} d^{3} \sinh \left (d x + c\right )^{2}\right )}} \]

input

integrate(x^4*cosh(d*x+c)/(b*x^2+a),x, algorithm="fricas")
 

output

-1/4*(8*b*d*x*cosh(d*x + c) + ((a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c 
)^2)*sqrt(-a*d^2/b)*Ei(d*x - sqrt(-a*d^2/b)) + (a*d^2*cosh(d*x + c)^2 - a* 
d^2*sinh(d*x + c)^2)*sqrt(-a*d^2/b)*Ei(-d*x + sqrt(-a*d^2/b)))*cosh(c + sq 
rt(-a*d^2/b)) - ((a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c)^2)*sqrt(-a*d 
^2/b)*Ei(d*x + sqrt(-a*d^2/b)) + (a*d^2*cosh(d*x + c)^2 - a*d^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)) 
 - 4*(b*d^2*x^2 - a*d^2 + 2*b)*sinh(d*x + c) + ((a*d^2*cosh(d*x + c)^2 - a 
*d^2*sinh(d*x + c)^2)*sqrt(-a*d^2/b)*Ei(d*x - sqrt(-a*d^2/b)) - (a*d^2*cos 
h(d*x + c)^2 - a*d^2*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*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x 
+ c)^2)*sqrt(-a*d^2/b)*Ei(d*x + sqrt(-a*d^2/b)) - (a*d^2*cosh(d*x + c)^2 - 
 a*d^2*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)))/(b^2*d^3*cosh(d*x + c)^2 - b^2*d^3*sinh(d*x + c)^2)
 

3.1.57.6 Sympy [F]

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

input

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

output

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

3.1.57.7 Maxima [F]

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

input

integrate(x^4*cosh(d*x+c)/(b*x^2+a),x, algorithm="maxima")
 

output

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

3.1.57.8 Giac [F]

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

input

integrate(x^4*cosh(d*x+c)/(b*x^2+a),x, algorithm="giac")
 

output

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

3.1.57.9 Mupad [F(-1)]

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

input

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

output

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