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

Optimal. Leaf size=273 \[ \frac{(-a)^{3/2} \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^{5/2}}-\frac{(-a)^{3/2} \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^{5/2}}-\frac{(-a)^{3/2} \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^{5/2}}-\frac{(-a)^{3/2} \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^{5/2}}-\frac{a \sinh (c+d x)}{b^2 d}+\frac{2 \sinh (c+d x)}{b d^3}-\frac{2 x \cosh (c+d x)}{b d^2}+\frac{x^2 \sinh (c+d x)}{b d} \]

[Out]

(-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))

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Rubi [A]  time = 0.728719, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {5293, 2637, 3296, 5281, 3303, 3298, 3301} \[ \frac{(-a)^{3/2} \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^{5/2}}-\frac{(-a)^{3/2} \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^{5/2}}-\frac{(-a)^{3/2} \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^{5/2}}-\frac{(-a)^{3/2} \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^{5/2}}-\frac{a \sinh (c+d x)}{b^2 d}+\frac{2 \sinh (c+d x)}{b d^3}-\frac{2 x \cosh (c+d x)}{b d^2}+\frac{x^2 \sinh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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))

Rule 5293

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5281

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

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{x^4 \cosh (c+d x)}{a+b x^2} \, dx &=\int \left (-\frac{a \cosh (c+d x)}{b^2}+\frac{x^2 \cosh (c+d x)}{b}+\frac{a^2 \cosh (c+d x)}{b^2 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{a \int \cosh (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{\cosh (c+d x)}{a+b x^2} \, dx}{b^2}+\frac{\int x^2 \cosh (c+d x) \, dx}{b}\\ &=-\frac{a \sinh (c+d x)}{b^2 d}+\frac{x^2 \sinh (c+d x)}{b d}+\frac{a^2 \int \left (\frac{\sqrt{-a} \cosh (c+d x)}{2 a \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{-a} \cosh (c+d x)}{2 a \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{b^2}-\frac{2 \int x \sinh (c+d x) \, dx}{b d}\\ &=-\frac{2 x \cosh (c+d x)}{b d^2}-\frac{a \sinh (c+d x)}{b^2 d}+\frac{x^2 \sinh (c+d x)}{b d}-\frac{(-a)^{3/2} \int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 b^2}-\frac{(-a)^{3/2} \int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 b^2}+\frac{2 \int \cosh (c+d x) \, dx}{b d^2}\\ &=-\frac{2 x \cosh (c+d x)}{b d^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{\left ((-a)^{3/2} \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 b^2}-\frac{\left ((-a)^{3/2} \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 b^2}-\frac{\left ((-a)^{3/2} \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 b^2}+\frac{\left ((-a)^{3/2} \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 b^2}\\ &=-\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}}\\ \end{align*}

Mathematica [C]  time = 0.418781, size = 274, normalized size = 1. \[ \frac{i a^{3/2} d^3 \cosh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )-i a^{3/2} d^3 \cosh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )-a^{3/2} d^3 \sinh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )-a^{3/2} d^3 \sinh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-2 a \sqrt{b} d^2 \sinh (c+d x)+2 b^{3/2} d^2 x^2 \sinh (c+d x)+4 b^{3/2} \sinh (c+d x)-4 b^{3/2} d x \cosh (c+d x)}{2 b^{5/2} d^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*b^(3/2)*d*x*Cosh[c + d*x] + I*a^(3/2)*d^3*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[-((Sqrt[a]*d)/Sqrt[b
]) + I*d*x] - I*a^(3/2)*d^3*Cosh[c + (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] + 4*b^(3/
2)*Sinh[c + d*x] - 2*a*Sqrt[b]*d^2*Sinh[c + d*x] + 2*b^(3/2)*d^2*x^2*Sinh[c + d*x] - a^(3/2)*d^3*Sinh[c - (I*S
qrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] - a^(3/2)*d^3*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinI
ntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])/(2*b^(5/2)*d^3)

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Maple [A]  time = 0.189, size = 369, normalized size = 1.4 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}{x}^{2}}{2\,bd}}+{\frac{{{\rm e}^{-dx-c}}a}{2\,d{b}^{2}}}-{\frac{{{\rm e}^{-dx-c}}x}{b{d}^{2}}}-{\frac{{{\rm e}^{-dx-c}}}{{d}^{3}b}}-{\frac{{a}^{2}}{4\,{b}^{2}}{{\rm e}^{-{\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{b} \left ( d\sqrt{-ab}- \left ( dx+c \right ) b+cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{{a}^{2}}{4\,{b}^{2}}{{\rm e}^{{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) }}}{\it Ei} \left ( 1,{\frac{1}{b} \left ( d\sqrt{-ab}+ \left ( dx+c \right ) b-cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{{a}^{2}}{4\,{b}^{2}}{{\rm e}^{{\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) }}}{\it Ei} \left ( 1,{\frac{1}{b} \left ( d\sqrt{-ab}- \left ( dx+c \right ) b+cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{{a}^{2}}{4\,{b}^{2}}{{\rm e}^{-{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{b} \left ( d\sqrt{-ab}+ \left ( dx+c \right ) b-cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{{{\rm e}^{dx+c}}{x}^{2}}{2\,bd}}-{\frac{a{{\rm e}^{dx+c}}}{2\,d{b}^{2}}}-{\frac{{{\rm e}^{dx+c}}x}{b{d}^{2}}}+{\frac{{{\rm e}^{dx+c}}}{{d}^{3}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.19521, size = 1296, normalized size = 4.75 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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 +
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)))*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*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)) + ((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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \cosh{\left (c + d x \right )}}{a + b x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \cosh \left (d x + c\right )}{b x^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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