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

Optimal. Leaf size=303 \[ -\frac {\cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}+\frac {\sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}+\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.52, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5293, 3303, 3298, 3301} \[ -\frac {\cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}+\frac {\sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}+\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(x*(a + b*x^3)),x]

[Out]

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

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

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Mathematica [C]  time = 0.28, size = 186, normalized size = 0.61 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,-\sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))-\cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))\& \right ]+\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))+\cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))\& \right ]-6 \cosh (c) \text {Chi}(d x)-6 \sinh (c) \text {Shi}(d x)}{6 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(-6*Cosh[c]*CoshIntegral[d*x] + RootSum[a + b*#1^3 & , Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] - CoshInte
gral[d*(x - #1)]*Sinh[c + d*#1] - Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] + Sinh[c + d*#1]*SinhIntegral[d*(x -
#1)] & ] + 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)] & ] - 6*Sinh[c]*Sin
hIntegral[d*x])/a

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fricas [B]  time = 0.52, size = 530, normalized size = 1.75 \[ -\frac {{\rm Ei}\left (d x - \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + c\right ) + {\rm Ei}\left (-d x - \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - c\right ) + {\rm Ei}\left (d x + \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - c\right ) + {\rm Ei}\left (-d x + \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + c\right ) + {\rm Ei}\left (-d x + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \cosh \left (c + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) - 3 \, {\left ({\rm Ei}\left (d x\right ) + {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) + {\rm Ei}\left (d x + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \cosh \left (-c + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) + {\rm Ei}\left (d x - \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + c\right ) + {\rm Ei}\left (-d x - \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - c\right ) - {\rm Ei}\left (d x + \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - c\right ) - {\rm Ei}\left (-d x + \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + c\right ) - {\rm Ei}\left (-d x + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \sinh \left (c + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) - 3 \, {\left ({\rm Ei}\left (d x\right ) - {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c) - {\rm Ei}\left (d x + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \sinh \left (-c + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right )}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/x/(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)) - 3*(
Ei(d*x) + Ei(-d*x))*cosh(c) + 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)*(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 + 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1))*sinh(1/2*(-a*d^3/b)^(1/3)*(sq
rt(-3) - 1) + c) - Ei(-d*x + (-a*d^3/b)^(1/3))*sinh(c + (-a*d^3/b)^(1/3)) - 3*(Ei(d*x) - Ei(-d*x))*sinh(c) - E
i(d*x + (a*d^3/b)^(1/3))*sinh(-c + (a*d^3/b)^(1/3)))/a

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.11, size = 138, normalized size = 0.46 \[ \frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }{\mathrm e}^{-\textit {\_R1}} \Ei \left (1, d x -\textit {\_R1} +c \right )}{6 a}-\frac {{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2 a}+\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }{\mathrm e}^{\textit {\_R1}} \Ei \left (1, -d x +\textit {\_R1} -c \right )}{6 a}-\frac {{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6/a*sum(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/a*exp(-c)*Ei(1,d
*x)+1/6/a*sum(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/a*exp(c)*Ei(
1,-d*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x\,\left (b\,x^3+a\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)/(x*(a + b*x^3)),x)

[Out]

int(cosh(c + d*x)/(x*(a + b*x^3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (c + d x \right )}}{x \left (a + b x^{3}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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