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

Optimal. Leaf size=283 \[ \frac{\cosh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac{\cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-x d-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}+\frac{\cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}-\frac{\sinh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac{\sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}+\frac{\sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b} \]

[Out]

(Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b) + (Cosh[c
- ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*b) + (Cosh[c - (a^
(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b) - (Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*S
inhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b) + (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3
)*d)/b^(1/3) + d*x])/(3*b) + (Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(
1/3) + d*x])/(3*b)

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Rubi [A]  time = 0.455137, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {5293, 3303, 3298, 3301} \[ \frac{\cosh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac{\cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-x d-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}+\frac{\cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}-\frac{\sinh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac{\sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}+\frac{\sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b) + (Cosh[c
- ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*b) + (Cosh[c - (a^
(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b) - (Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*S
inhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b) + (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3
)*d)/b^(1/3) + d*x])/(3*b) + (Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(
1/3) + d*x])/(3*b)

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 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^2 \cosh (c+d x)}{a+b x^3} \, dx &=\int \left (\frac{\cosh (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\cosh (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\cosh (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx\\ &=\frac{\int \frac{\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac{\int \frac{\cosh (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac{\int \frac{\cosh (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}\\ &=\frac{\cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{\cosh \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac{\cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{\cos \left (\frac{(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac{\cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{\cos \left (\frac{\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac{\sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{\sinh \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac{\left (i \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac{\left (i \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}\\ &=\frac{\cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac{\cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac{\cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b}-\frac{\sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac{\sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b}+\frac{\sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b}\\ \end{align*}

Mathematica [C]  time = 0.225213, size = 170, normalized size = 0.6 \[ \frac{\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,-\sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-\cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))\& \right ]+\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))\& \right ]}{6 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(RootSum[a + b*#1^3 & , Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] - CoshIntegral[d*(x - #1)]*Sinh[c + d*#1] - Co
sh[c + d*#1]*SinhIntegral[d*(x - #1)] + Sinh[c + d*#1]*SinhIntegral[d*(x - #1)] & ] + RootSum[a + b*#1^3 & , C
osh[c + d*#1]*CoshIntegral[d*(x - #1)] + CoshIntegral[d*(x - #1)]*Sinh[c + d*#1] + Cosh[c + d*#1]*SinhIntegral
[d*(x - #1)] + Sinh[c + d*#1]*SinhIntegral[d*(x - #1)] & ])/(6*b)

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Maple [C]  time = 0.032, size = 423, normalized size = 1.5 \begin{align*} -{\frac{1}{6\,b}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }{\frac{{{\it \_R1}}^{2}{{\rm e}^{-{\it \_R1}}}{\it Ei} \left ( 1,dx-{\it \_R1}+c \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}-{\frac{{c}^{2}}{6\,b}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }{\frac{{{\rm e}^{-{\it \_R1}}}{\it Ei} \left ( 1,dx-{\it \_R1}+c \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}+{\frac{c}{3\,b}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }{\frac{{\it \_R1}\,{{\rm e}^{-{\it \_R1}}}{\it Ei} \left ( 1,dx-{\it \_R1}+c \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}-{\frac{1}{6\,b}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }{\frac{{{\it \_R1}}^{2}{{\rm e}^{{\it \_R1}}}{\it Ei} \left ( 1,-dx+{\it \_R1}-c \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}-{\frac{{c}^{2}}{6\,b}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }{\frac{{{\rm e}^{{\it \_R1}}}{\it Ei} \left ( 1,-dx+{\it \_R1}-c \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}+{\frac{c}{3\,b}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }{\frac{{\it \_R1}\,{{\rm e}^{{\it \_R1}}}{\it Ei} \left ( 1,-dx+{\it \_R1}-c \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/b*sum(_R1^2/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b
*c^3))-1/6*c^2/b*sum(1/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a
*d^3-b*c^3))+1/3*c/b*sum(_R1/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b
*c^2+a*d^3-b*c^3))-1/6/b*sum(_R1^2/(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+
3*_Z*b*c^2+a*d^3-b*c^3))-1/6*c^2/b*sum(1/(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^
2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/3*c/b*sum(_R1/(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b
-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (d x^{2} e^{\left (2 \, c\right )} + x e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} -{\left (d x^{2} - x\right )} e^{\left (-d x\right )}}{2 \,{\left (b d^{2} x^{3} e^{c} + a d^{2} e^{c}\right )}} + \frac{1}{2} \, \int \frac{{\left (2 \, b x^{3} e^{c} - 3 \, a d x e^{c} - a e^{c}\right )} e^{\left (d x\right )}}{b^{2} d^{2} x^{6} + 2 \, a b d^{2} x^{3} + a^{2} d^{2}}\,{d x} + \frac{1}{2} \, \int \frac{{\left (2 \, b x^{3} + 3 \, a d x - a\right )} e^{\left (-d x\right )}}{b^{2} d^{2} x^{6} e^{c} + 2 \, a b d^{2} x^{3} 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^2*cosh(d*x+c)/(b*x^3+a),x, algorithm="maxima")

[Out]

1/2*((d*x^2*e^(2*c) + x*e^(2*c))*e^(d*x) - (d*x^2 - x)*e^(-d*x))/(b*d^2*x^3*e^c + a*d^2*e^c) + 1/2*integrate((
2*b*x^3*e^c - 3*a*d*x*e^c - a*e^c)*e^(d*x)/(b^2*d^2*x^6 + 2*a*b*d^2*x^3 + a^2*d^2), x) + 1/2*integrate((2*b*x^
3 + 3*a*d*x - a)*e^(-d*x)/(b^2*d^2*x^6*e^c + 2*a*b*d^2*x^3*e^c + a^2*d^2*e^c), x)

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Fricas [B]  time = 1.95249, size = 1322, normalized size = 4.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(Ei(d*x - 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1))*cosh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1) + c) + Ei(-d*x - 1/2
*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1))*cosh(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1) - c) + Ei(d*x + 1/2*(a*d^3/b)^(1/3)
*(sqrt(-3) - 1))*cosh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1) - c) + Ei(-d*x + 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1))
*cosh(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1) + c) + Ei(-d*x + (-a*d^3/b)^(1/3))*cosh(c + (-a*d^3/b)^(1/3)) + Ei(d
*x + (a*d^3/b)^(1/3))*cosh(-c + (a*d^3/b)^(1/3)) + Ei(d*x - 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1))*sinh(1/2*(a*d^
3/b)^(1/3)*(sqrt(-3) + 1) + c) + Ei(-d*x - 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1))*sinh(1/2*(-a*d^3/b)^(1/3)*(sqr
t(-3) + 1) - c) - Ei(d*x + 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1))*sinh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1) - c) -
Ei(-d*x + 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1))*sinh(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1) + c) - Ei(-d*x + (-a*d
^3/b)^(1/3))*sinh(c + (-a*d^3/b)^(1/3)) - Ei(d*x + (a*d^3/b)^(1/3))*sinh(-c + (a*d^3/b)^(1/3)))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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