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

Optimal. Leaf size=358 \[ \frac{\sqrt [3]{-1} \sqrt [3]{a} \cosh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{a} \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-x d-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac{\sqrt [3]{a} \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{a} \sinh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac{\sqrt [3]{a} \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{a} \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}+\frac{\sinh (c+d x)}{b d} \]

[Out]

((-1)^(1/3)*a^(1/3)*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^(4/3)) - ((-1)^(2/3)*a^(1/3)*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^(4/3)) - (a^(1/3)*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) +
d*x])/(3*b^(4/3)) + Sinh[c + d*x]/(b*d) - ((-1)^(1/3)*a^(1/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIn
tegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b^(4/3)) - (a^(1/3)*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegra
l[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3)) - ((-1)^(2/3)*a^(1/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhI
ntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3))

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Rubi [A]  time = 0.648427, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5293, 2637, 5281, 3303, 3298, 3301} \[ \frac{\sqrt [3]{-1} \sqrt [3]{a} \cosh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{a} \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-x d-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac{\sqrt [3]{a} \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{a} \sinh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac{\sqrt [3]{a} \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{a} \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}+\frac{\sinh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1)^(1/3)*a^(1/3)*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^(4/3)) - ((-1)^(2/3)*a^(1/3)*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^(4/3)) - (a^(1/3)*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) +
d*x])/(3*b^(4/3)) + Sinh[c + d*x]/(b*d) - ((-1)^(1/3)*a^(1/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIn
tegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b^(4/3)) - (a^(1/3)*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegra
l[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3)) - ((-1)^(2/3)*a^(1/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhI
ntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3))

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 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^3 \cosh (c+d x)}{a+b x^3} \, dx &=\int \left (\frac{\cosh (c+d x)}{b}-\frac{a \cosh (c+d x)}{b \left (a+b x^3\right )}\right ) \, dx\\ &=\frac{\int \cosh (c+d x) \, dx}{b}-\frac{a \int \frac{\cosh (c+d x)}{a+b x^3} \, dx}{b}\\ &=\frac{\sinh (c+d x)}{b d}-\frac{a \int \left (-\frac{\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac{\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{b}\\ &=\frac{\sinh (c+d x)}{b d}+\frac{\sqrt [3]{a} \int \frac{\cosh (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 b}+\frac{\sqrt [3]{a} \int \frac{\cosh (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 b}+\frac{\sqrt [3]{a} \int \frac{\cosh (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 b}\\ &=\frac{\sinh (c+d x)}{b d}+\frac{\left (\sqrt [3]{a} \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\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}+\frac{\left (\sqrt [3]{a} \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 b}+\frac{\left (\sqrt [3]{a} \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 b}+\frac{\left (\sqrt [3]{a} \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\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}+\frac{\left (i \sqrt [3]{a} \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]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 b}+\frac{\left (i \sqrt [3]{a} \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 )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 b}\\ &=\frac{\sqrt [3]{-1} \sqrt [3]{a} \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^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{a} \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^{4/3}}-\frac{\sqrt [3]{a} \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^{4/3}}+\frac{\sinh (c+d x)}{b d}-\frac{\sqrt [3]{-1} \sqrt [3]{a} \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^{4/3}}-\frac{\sqrt [3]{a} \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^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{a} \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^{4/3}}\\ \end{align*}

Mathematica [C]  time = 0.259845, size = 198, normalized size = 0.55 \[ -\frac{a d \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-\sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-\cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+a d \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{\sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-6 b \sinh (c+d x)}{6 b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*d*RootSum[a + b*#1^3 & , (Cosh[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)])/#1^2 & ] + a*d*RootSum[
a + b*#1^3 & , (Cosh[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)])/#1^2 & ] - 6*b*Sinh[c + d*x])/(6*b^2*
d)

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Maple [C]  time = 0.036, size = 671, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d*exp(-d*x-c)/b-1/6/d/b^2*sum((3*_R1^2*b*c-3*_R1*b*c^2-a*d^3+b*c^3)/(_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/d*c^3/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))-1/2/d*c^2/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))+1/2/d*c/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))+1/2/d/b*exp(d*x+c)-
1/6/d/b^2*sum((3*_R1^2*b*c-3*_R1*b*c^2-a*d^3+b*c^3)/(_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/d*c^3/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))-1/2/d*c^2/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))+1/2/d*c/b*sum(_R1^2/(_R1^2-2*_R1*c+c^2)*exp(_R1)*E
i(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(-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^3*cosh(d*x+c)/(b*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.20258, size = 2591, normalized size = 7.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*((a*d^3/b)^(1/3)*((sqrt(-3) + 1)*cosh(d*x + c)^2 - (sqrt(-3) + 1)*sinh(d*x + c)^2)*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) - (-a*d^3/b)^(1/3)*((sqrt(-3) + 1)*cosh(d*
x + c)^2 - (sqrt(-3) + 1)*sinh(d*x + c)^2)*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) - (a*d^3/b)^(1/3)*((sqrt(-3) - 1)*cosh(d*x + c)^2 - (sqrt(-3) - 1)*sinh(d*x + c)^2)*
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) + (-a*d^3/b)^(1/3)*(
(sqrt(-3) - 1)*cosh(d*x + c)^2 - (sqrt(-3) - 1)*sinh(d*x + c)^2)*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) + 2*(-a*d^3/b)^(1/3)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(-d
*x + (-a*d^3/b)^(1/3))*cosh(c + (-a*d^3/b)^(1/3)) - 2*(a*d^3/b)^(1/3)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(d
*x + (a*d^3/b)^(1/3))*cosh(-c + (a*d^3/b)^(1/3)) + (a*d^3/b)^(1/3)*((sqrt(-3) + 1)*cosh(d*x + c)^2 - (sqrt(-3)
+ 1)*sinh(d*x + c)^2)*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) - (-a*d^3/b)^(1/3)*((sqrt(-3) + 1)*cosh(d*x + c)^2 - (sqrt(-3) + 1)*sinh(d*x + c)^2)*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) + (a*d^3/b)^(1/3)*((sqrt(-3) - 1)*cosh(
d*x + c)^2 - (sqrt(-3) - 1)*sinh(d*x + c)^2)*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) - (-a*d^3/b)^(1/3)*((sqrt(-3) - 1)*cosh(d*x + c)^2 - (sqrt(-3) - 1)*sinh(d*x + c)^2)*
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) - 2*(-a*d^3/b)^(1
/3)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(-d*x + (-a*d^3/b)^(1/3))*sinh(c + (-a*d^3/b)^(1/3)) + 2*(a*d^3/b)^(
1/3)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(d*x + (a*d^3/b)^(1/3))*sinh(-c + (a*d^3/b)^(1/3)) + 12*sinh(d*x +
c))/(b*d*cosh(d*x + c)^2 - b*d*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^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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