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(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))*sinh(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)

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Rubi [A]  time = 0.73, antiderivative size = 273, normalized size of antiderivative = 1.00, 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 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 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]

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 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 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])

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*}

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Mathematica [C]  time = 0.45, size = 274, normalized size = 1.00 \[ \frac {i a^{3/2} d^3 \cosh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right )-i a^{3/2} d^3 \cosh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-a^{3/2} d^3 \sinh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-a^{3/2} d^3 \sinh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-2 a \sqrt {b} d^2 \sinh (c+d x)+2 b^{3/2} d^2 x^2 \sinh (c+d x)+4 b^{3/2} \sinh (c+d x)-4 b^{3/2} d x \cosh (c+d x)}{2 b^{5/2} d^3} \]

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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fricas [B]  time = 0.80, size = 605, normalized size = 2.22 \[ -\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 )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \cosh \left (d x + c\right )}{b x^{2} + a}\,{d x} \]

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)

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maple [A]  time = 0.29, size = 369, normalized size = 1.35 \[ \frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}-\frac {{\mathrm e}^{-d x -c} x^{2}}{2 d b}+\frac {{\mathrm e}^{-d x -c} a}{2 d \,b^{2}}-\frac {{\mathrm e}^{-d x -c} x}{d^{2} b}-\frac {{\mathrm e}^{-d x -c}}{d^{3} b}-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}+\frac {{\mathrm e}^{d x +c} x^{2}}{2 d b}-\frac {a \,{\mathrm e}^{d x +c}}{2 d \,b^{2}}-\frac {{\mathrm e}^{d x +c} x}{d^{2} b}+\frac {{\mathrm e}^{d x +c}}{d^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/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*exp(-d*x-c)/b*x^2+1/2/d*ex
p(-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+1/2/d*exp(d*x+c)/b*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]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b d^{2} x^{4} e^{\left (2 \, c\right )} - 2 \, b d x^{3} e^{\left (2 \, c\right )} - 2 \, a d x e^{\left (2 \, c\right )} + 2 \, b x^{2} e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - {\left (b d^{2} x^{4} + 2 \, b d x^{3} + 2 \, a d x + 2 \, b x^{2}\right )} e^{\left (-d x\right )}}{2 \, {\left (b^{2} d^{3} x^{2} e^{c} + a b d^{3} e^{c}\right )}} + \frac {1}{2} \, \int \frac {2 \, {\left (a b d x^{2} e^{c} + a^{2} d e^{c} + {\left (a^{2} d^{2} e^{c} - 2 \, a b e^{c}\right )} x\right )} e^{\left (d x\right )}}{b^{3} d^{3} x^{4} + 2 \, a b^{2} d^{3} x^{2} + a^{2} b d^{3}}\,{d x} + \frac {1}{2} \, \int \frac {2 \, {\left (a b d x^{2} + a^{2} d - {\left (a^{2} d^{2} - 2 \, a b\right )} x\right )} e^{\left (-d x\right )}}{b^{3} d^{3} x^{4} e^{c} + 2 \, a b^{2} d^{3} x^{2} e^{c} + a^{2} b d^{3} e^{c}}\,{d x} \]

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]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\mathrm {cosh}\left (c+d\,x\right )}{b\,x^2+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \cosh {\left (c + d x \right )}}{a + b x^{2}}\, dx \]

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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