\(\int \frac {x^2 \cosh (c+d x)}{(a+b x^3)^2} \, dx\) [103]

Optimal result

Integrand size = 19, antiderivative size = 373 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=-\frac {\cosh (c+d x)}{3 b \left (a+b x^3\right )}+\frac {d \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}-\frac {\sqrt [3]{-1} d \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}+\frac {(-1)^{2/3} d \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}+\frac {\sqrt [3]{-1} d \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}+\frac {d \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}+\frac {(-1)^{2/3} d \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}} \]

[Out]

-1/3*cosh(d*x+c)/b/(b*x^3+a)-1/9*(-1)^(1/3)*d*cosh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))*Shi(-(-1)^(1/3)*a^(1/3)*d/b
^(1/3)+d*x)/a^(2/3)/b^(4/3)+1/9*d*cosh(c-a^(1/3)*d/b^(1/3))*Shi(a^(1/3)*d/b^(1/3)+d*x)/a^(2/3)/b^(4/3)+1/9*(-1
)^(2/3)*d*cosh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Shi((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(2/3)/b^(4/3)+1/9*d*Chi
(a^(1/3)*d/b^(1/3)+d*x)*sinh(c-a^(1/3)*d/b^(1/3))/a^(2/3)/b^(4/3)-1/9*(-1)^(1/3)*d*Chi((-1)^(1/3)*a^(1/3)*d/b^
(1/3)-d*x)*sinh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(2/3)/b^(4/3)+1/9*(-1)^(2/3)*d*Chi(-(-1)^(2/3)*a^(1/3)*d/b^(
1/3)-d*x)*sinh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(2/3)/b^(4/3)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5397, 5388, 3384, 3379, 3382} \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\frac {d \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}-\frac {\sqrt [3]{-1} d \sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}+\frac {(-1)^{2/3} d \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}+\frac {\sqrt [3]{-1} d \cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}+\frac {d \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}+\frac {(-1)^{2/3} d \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}-\frac {\cosh (c+d x)}{3 b \left (a+b x^3\right )} \]

[In]

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

[Out]

-1/3*Cosh[c + d*x]/(b*(a + b*x^3)) + (d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])
/(9*a^(2/3)*b^(4/3)) - ((-1)^(1/3)*d*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a
^(1/3)*d)/b^(1/3)])/(9*a^(2/3)*b^(4/3)) + ((-1)^(2/3)*d*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*
Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(2/3)*b^(4/3)) + ((-1)^(1/3)*d*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/
b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(2/3)*b^(4/3)) + (d*Cosh[c - (a^(1/3)*d)/b^(
1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) + ((-1)^(2/3)*d*Cosh[c - ((-1)^(2/3)*a^(1/3
)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3))

Rule 3379

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 3382

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 3384

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 5388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sinh[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 5397

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[e^m*(a + b*x^
n)^(p + 1)*(Cosh[c + d*x]/(b*n*(p + 1))), x] - Dist[d*(e^m/(b*n*(p + 1))), Int[(a + b*x^n)^(p + 1)*Sinh[c + d*
x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && EqQ[m - n + 1, 0] && LtQ[p, -1] && (IntegerQ[n
] || GtQ[e, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh (c+d x)}{3 b \left (a+b x^3\right )}+\frac {d \int \frac {\sinh (c+d x)}{a+b x^3} \, dx}{3 b} \\ & = -\frac {\cosh (c+d x)}{3 b \left (a+b x^3\right )}+\frac {d \int \left (-\frac {\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{3 b} \\ & = -\frac {\cosh (c+d x)}{3 b \left (a+b x^3\right )}-\frac {d \int \frac {\sinh (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac {d \int \frac {\sinh (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac {d \int \frac {\sinh (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b} \\ & = -\frac {\cosh (c+d x)}{3 b \left (a+b x^3\right )}-\frac {\left (d \cosh \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}{9 a^{2/3} b}-\frac {\left (i d \cosh \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}{9 a^{2/3} b}-\frac {\left (i d \cosh \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}{9 a^{2/3} b}-\frac {\left (d \sinh \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}{9 a^{2/3} b}-\frac {\left (d \sinh \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}{9 a^{2/3} b}-\frac {\left (d \sinh \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}{9 a^{2/3} b} \\ & = -\frac {\cosh (c+d x)}{3 b \left (a+b x^3\right )}+\frac {d \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}-\frac {\sqrt [3]{-1} d \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}+\frac {(-1)^{2/3} d \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}+\frac {\sqrt [3]{-1} d \cosh \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 )}{9 a^{2/3} b^{4/3}}+\frac {d \cosh \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 )}{9 a^{2/3} b^{4/3}}+\frac {(-1)^{2/3} d \cosh \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 )}{9 a^{2/3} b^{4/3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 0.13 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.54 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\frac {-\frac {6 b \cosh (c+d x)}{a+b x^3}-d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]+d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]}{18 b^2} \]

[In]

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

[Out]

((-6*b*Cosh[c + d*x])/(a + b*x^3) - d*RootSum[a + b*#1^3 & , (Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] - CoshIn
tegral[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 & ] + 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 &
])/(18*b^2)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.30 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.59

method	result	size
risch	\(-\frac {d^{3} {\mathrm e}^{-d x -c}}{6 b \left (b \,d^{3} x^{3}+d^{3} a \right )}-\frac {\munderset {\textit {\_R2} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 c^{2} b \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (2 \textit {\_R2}^{2} b c -3 \textit {\_R2} b \,c^{2}-d^{3} a +b \,c^{3}+2 \textit {\_R2} b c \right ) {\mathrm e}^{-\textit {\_R2}} \operatorname {Ei}_{1}\left (d x -\textit {\_R2} +c \right )}{\textit {\_R2}^{2}-2 \textit {\_R2} c +c^{2}}}{18 a \,b^{2}}-\frac {c^{2} \left (\munderset {\textit {\_R2} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 c^{2} b \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R2} -c +2\right ) {\mathrm e}^{-\textit {\_R2}} \operatorname {Ei}_{1}\left (d x -\textit {\_R2} +c \right )}{\textit {\_R2}^{2}-2 \textit {\_R2} c +c^{2}}\right )}{18 a b}+\frac {c \left (\munderset {\textit {\_R2} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 c^{2} b \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R2}^{2}-\textit {\_R2} c +\textit {\_R2} +c \right ) {\mathrm e}^{-\textit {\_R2}} \operatorname {Ei}_{1}\left (d x -\textit {\_R2} +c \right )}{\textit {\_R2}^{2}-2 \textit {\_R2} c +c^{2}}\right )}{9 a b}-\frac {d^{3} {\mathrm e}^{d x +c}}{6 b \left (b \,d^{3} x^{3}+d^{3} a \right )}+\frac {\munderset {\textit {\_R2} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 c^{2} b \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (2 \textit {\_R2}^{2} b c -3 \textit {\_R2} b \,c^{2}-d^{3} a +b \,c^{3}-2 \textit {\_R2} b c \right ) {\mathrm e}^{\textit {\_R2}} \operatorname {Ei}_{1}\left (-d x +\textit {\_R2} -c \right )}{\textit {\_R2}^{2}-2 \textit {\_R2} c +c^{2}}}{18 a \,b^{2}}+\frac {c^{2} \left (\munderset {\textit {\_R2} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 c^{2} b \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R2} -c -2\right ) {\mathrm e}^{\textit {\_R2}} \operatorname {Ei}_{1}\left (-d x +\textit {\_R2} -c \right )}{\textit {\_R2}^{2}-2 \textit {\_R2} c +c^{2}}\right )}{18 a b}-\frac {c \left (\munderset {\textit {\_R2} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 c^{2} b \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R2}^{2}-\textit {\_R2} c -\textit {\_R2} -c \right ) {\mathrm e}^{\textit {\_R2}} \operatorname {Ei}_{1}\left (-d x +\textit {\_R2} -c \right )}{\textit {\_R2}^{2}-2 \textit {\_R2} c +c^{2}}\right )}{9 a b}\)	\(594\)

[In]

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

[Out]

-1/6*d^3*exp(-d*x-c)/b/(b*d^3*x^3+a*d^3)-1/18/a/b^2*sum((2*_R2^2*b*c-3*_R2*b*c^2-a*d^3+b*c^3+2*_R2*b*c)/(_R2^2
-2*_R2*c+c^2)*exp(-_R2)*Ei(1,d*x-_R2+c),_R2=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/18*c^2/a/b*sum
((_R2-c+2)/(_R2^2-2*_R2*c+c^2)*exp(-_R2)*Ei(1,d*x-_R2+c),_R2=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))
+1/9*c/a/b*sum((_R2^2-_R2*c+_R2+c)/(_R2^2-2*_R2*c+c^2)*exp(-_R2)*Ei(1,d*x-_R2+c),_R2=RootOf(_Z^3*b-3*_Z^2*b*c+
3*_Z*b*c^2+a*d^3-b*c^3))-1/6*d^3*exp(d*x+c)/b/(b*d^3*x^3+a*d^3)+1/18/a/b^2*sum((2*_R2^2*b*c-3*_R2*b*c^2-a*d^3+
b*c^3-2*_R2*b*c)/(_R2^2-2*_R2*c+c^2)*exp(_R2)*Ei(1,-d*x+_R2-c),_R2=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b
*c^3))+1/18*c^2/a/b*sum((_R2-c-2)/(_R2^2-2*_R2*c+c^2)*exp(_R2)*Ei(1,-d*x+_R2-c),_R2=RootOf(_Z^3*b-3*_Z^2*b*c+3
*_Z*b*c^2+a*d^3-b*c^3))-1/9*c/a/b*sum((_R2^2-_R2*c-_R2-c)/(_R2^2-2*_R2*c+c^2)*exp(_R2)*Ei(1,-d*x+_R2-c),_R2=Ro
otOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1276 vs. \(2 (263) = 526\).

Time = 0.29 (sec) , antiderivative size = 1276, normalized size of antiderivative = 3.42 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/36*((a*d^3/b)^(1/3)*((b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 + sqrt(-3)*(b*x^3 + a) + a
)*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)*((b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*sin
h(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)*((b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*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)*((b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*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)*((b*x^3 + a)*cosh(d*x + c)^2 - (b*x^3 + a)*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)*((b*x^3 + a)*cosh(d*x + c)^2 - (b*x^3 + a)*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)*((b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 -
(b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*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)*((b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^
3 + sqrt(-3)*(b*x^3 + a) + a)*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)*((b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 -
sqrt(-3)*(b*x^3 + a) + a)*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)*((b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 - sqrt
(-3)*(b*x^3 + a) + a)*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)*((b*x^3 + a)*cosh(d*x + c)^2 - (b*x^3 + a)*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)*((b*x^3 + a)*cosh(d*x + c)^2 - (b*x^3 + a)
*sinh(d*x + c)^2)*Ei(d*x + (a*d^3/b)^(1/3))*sinh(-c + (a*d^3/b)^(1/3)) + 12*a*cosh(d*x + c))/((a*b^2*x^3 + a^2
*b)*cosh(d*x + c)^2 - (a*b^2*x^3 + a^2*b)*sinh(d*x + c)^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\int { \frac {x^{2} \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

1/2*((d*x^2*e^(2*c) + 4*x*e^(2*c))*e^(d*x) - (d*x^2 - 4*x)*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) + 1/2*integrate(2*(10*b*x^3*e^c - 3*a*d*x*e^c - 2*a*e^c)*e^(d*x)/(b^3*d^2*x^9 + 3*a*b^2*d^2*x^6 + 3
*a^2*b*d^2*x^3 + a^3*d^2), x) + 1/2*integrate(2*(10*b*x^3 + 3*a*d*x - 2*a)*e^(-d*x)/(b^3*d^2*x^9*e^c + 3*a*b^2
*d^2*x^6*e^c + 3*a^2*b*d^2*x^3*e^c + a^3*d^2*e^c), x)

Giac [F]

\[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\int { \frac {x^{2} \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^2} \,d x \]

[In]

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

[Out]

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