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

Optimal. Leaf size=209 \[ -\frac{a \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^2}-\frac{a \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}+\frac{a \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^2}-\frac{a \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}-\frac{\cosh (c+d x)}{b d^2}+\frac{x \sinh (c+d x)}{b d} \]

[Out]

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

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Rubi [A]  time = 0.35826, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5293, 3296, 2638, 3303, 3298, 3301} \[ -\frac{a \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^2}-\frac{a \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}+\frac{a \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^2}-\frac{a \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}-\frac{\cosh (c+d x)}{b d^2}+\frac{x \sinh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cosh[c + d*x]/(b*d^2)) - (a*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2)
- (a*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^2) + (x*Sinh[c + d*x])/(b*
d) + (a*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2) - (a*Sinh[c - (Sqrt[-
a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^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 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 2638

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

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

Mathematica [C]  time = 0.348637, size = 210, normalized size = 1. \[ -\frac{a d^2 \cosh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+a d^2 \cosh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+i a d^2 \sinh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )-i a d^2 \sinh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-2 b d x \sinh (c+d x)+2 b \cosh (c+d x)}{2 b^2 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(2*b*Cosh[c + d*x] + a*d^2*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] + a*d^
2*Cosh[c + (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] - 2*b*d*x*Sinh[c + d*x] + I*a*d^2*S
inh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] - I*a*d^2*Sinh[c + (I*Sqrt[a]*d)/Sqrt[
b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])/(2*b^2*d^2)

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Maple [A]  time = 0.06, size = 268, normalized size = 1.3 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}x}{2\,bd}}-{\frac{{{\rm e}^{-dx-c}}}{2\,b{d}^{2}}}+{\frac{a}{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{a}{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{{{\rm e}^{dx+c}}x}{2\,bd}}-{\frac{{{\rm e}^{dx+c}}}{2\,b{d}^{2}}}+{\frac{a}{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{a}{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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (d x^{3} e^{\left (2 \, c\right )} - x^{2} e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} -{\left (d x^{3} + x^{2}\right )} e^{\left (-d x\right )}}{2 \,{\left (b d^{2} x^{2} e^{c} + a d^{2} e^{c}\right )}} - \frac{1}{2} \, \int \frac{2 \,{\left (a d x^{2} e^{c} - a x e^{c}\right )} e^{\left (d x\right )}}{b^{2} d^{2} x^{4} + 2 \, a b d^{2} x^{2} + a^{2} d^{2}}\,{d x} + \frac{1}{2} \, \int \frac{2 \,{\left (a d x^{2} + a x\right )} e^{\left (-d x\right )}}{b^{2} d^{2} x^{4} e^{c} + 2 \, a b d^{2} x^{2} e^{c} + a^{2} d^{2} e^{c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.10293, size = 1100, normalized size = 5.26 \begin{align*} \frac{4 \, b d x \sinh \left (d x + c\right ) - 4 \, b \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 )}{\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 )}{\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 )}{\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 )}{\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 )}{\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 )}{\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 )}{\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 )}{\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^{2} \cosh \left (d x + c\right )^{2} - b^{2} d^{2} \sinh \left (d x + c\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*b*d*x*sinh(d*x + c) - 4*b*cosh(d*x + c) - ((a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c)^2)*Ei(d*x - sqr
t(-a*d^2/b)) + (a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c)^2)*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)*Ei(d*x + sqrt(-a*d^2/b)) + (a*d^2*cosh(d*x + c)^2 - a*
d^2*sinh(d*x + c)^2)*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*si
nh(d*x + c)^2)*Ei(d*x - sqrt(-a*d^2/b)) - (a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c)^2)*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)*Ei(d*x + sqrt(-a*d^2/b)) -
(a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c)^2)*Ei(-d*x - sqrt(-a*d^2/b)))*sinh(-c + sqrt(-a*d^2/b)))/(b^2*d^2
*cosh(d*x + c)^2 - b^2*d^2*sinh(d*x + c)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \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**3*cosh(d*x+c)/(b*x**2+a),x)

[Out]

Integral(x**3*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^{3} \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^3*cosh(d*x+c)/(b*x^2+a),x, algorithm="giac")

[Out]

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