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

Optimal. Leaf size=270 \[ -\frac{b \cosh (c) \text{Chi}(d x)}{a^2}+\frac{b \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^2}+\frac{b \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^2}-\frac{b \sinh (c) \text{Shi}(d x)}{a^2}-\frac{b \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^2}+\frac{b \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^2}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a}-\frac{\cosh (c+d x)}{2 a x^2}-\frac{d \sinh (c+d x)}{2 a x} \]

[Out]

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

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Rubi [A]  time = 0.504207, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {5293, 3297, 3303, 3298, 3301} \[ -\frac{b \cosh (c) \text{Chi}(d x)}{a^2}+\frac{b \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^2}+\frac{b \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^2}-\frac{b \sinh (c) \text{Shi}(d x)}{a^2}-\frac{b \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^2}+\frac{b \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^2}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a}-\frac{\cosh (c+d x)}{2 a x^2}-\frac{d \sinh (c+d x)}{2 a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cosh[c + d*x]/(2*a*x^2) - (b*Cosh[c]*CoshIntegral[d*x])/a^2 + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a) + (b*Cosh
[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2) + (b*Cosh[c - (Sqrt[-a]*d)/Sqrt[b
]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2) - (d*Sinh[c + d*x])/(2*a*x) - (b*Sinh[c]*SinhIntegral[d*x
])/a^2 + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a) - (b*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/S
qrt[b] - d*x])/(2*a^2) + (b*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^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 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -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{\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx &=\int \left (\frac{\cosh (c+d x)}{a x^3}-\frac{b \cosh (c+d x)}{a^2 x}+\frac{b^2 x \cosh (c+d x)}{a^2 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\cosh (c+d x)}{x^3} \, dx}{a}-\frac{b \int \frac{\cosh (c+d x)}{x} \, dx}{a^2}+\frac{b^2 \int \frac{x \cosh (c+d x)}{a+b x^2} \, dx}{a^2}\\ &=-\frac{\cosh (c+d x)}{2 a x^2}+\frac{b^2 \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}{a^2}+\frac{d \int \frac{\sinh (c+d x)}{x^2} \, dx}{2 a}-\frac{(b \cosh (c)) \int \frac{\cosh (d x)}{x} \, dx}{a^2}-\frac{(b \sinh (c)) \int \frac{\sinh (d x)}{x} \, dx}{a^2}\\ &=-\frac{\cosh (c+d x)}{2 a x^2}-\frac{b \cosh (c) \text{Chi}(d x)}{a^2}-\frac{d \sinh (c+d x)}{2 a x}-\frac{b \sinh (c) \text{Shi}(d x)}{a^2}-\frac{b^{3/2} \int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 a^2}+\frac{b^{3/2} \int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 a^2}+\frac{d^2 \int \frac{\cosh (c+d x)}{x} \, dx}{2 a}\\ &=-\frac{\cosh (c+d x)}{2 a x^2}-\frac{b \cosh (c) \text{Chi}(d x)}{a^2}-\frac{d \sinh (c+d x)}{2 a x}-\frac{b \sinh (c) \text{Shi}(d x)}{a^2}+\frac{\left (d^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{2 a}+\frac{\left (b^{3/2} \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 a^2}-\frac{\left (b^{3/2} \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 a^2}+\frac{\left (d^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{2 a}+\frac{\left (b^{3/2} \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 a^2}+\frac{\left (b^{3/2} \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 a^2}\\ &=-\frac{\cosh (c+d x)}{2 a x^2}-\frac{b \cosh (c) \text{Chi}(d x)}{a^2}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a}+\frac{b \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^2}+\frac{b \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^2}-\frac{d \sinh (c+d x)}{2 a x}-\frac{b \sinh (c) \text{Shi}(d x)}{a^2}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a}-\frac{b \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^2}+\frac{b \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^2}\\ \end{align*}

Mathematica [C]  time = 0.530193, size = 257, normalized size = 0.95 \[ \frac{x^2 \cosh (c) \left (-\left (2 b-a d^2\right )\right ) \text{Chi}(d x)+b x^2 \cosh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+b x^2 \cosh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+i b x^2 \sinh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )-i b x^2 \sinh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )+a d^2 x^2 \sinh (c) \text{Shi}(d x)-a d x \sinh (c+d x)-a \cosh (c+d x)-2 b x^2 \sinh (c) \text{Shi}(d x)}{2 a^2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.063, size = 330, normalized size = 1.2 \begin{align*}{\frac{d{{\rm e}^{-dx-c}}}{4\,ax}}-{\frac{{{\rm e}^{-dx-c}}}{4\,a{x}^{2}}}-{\frac{b}{4\,{a}^{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{b}{4\,{a}^{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{{d}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{4\,a}}+{\frac{b{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{2}}}-{\frac{d{{\rm e}^{dx+c}}}{4\,ax}}-{\frac{{{\rm e}^{dx+c}}}{4\,a{x}^{2}}}-{\frac{b}{4\,{a}^{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{b}{4\,{a}^{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{{d}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{4\,a}}+{\frac{b{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.51016, size = 1283, normalized size = 4.75 \begin{align*} -\frac{2 \, a d x \sinh \left (d x + c\right ) + 2 \, a \cosh \left (d x + c\right ) -{\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )}{\rm Ei}\left (d x - \sqrt{-\frac{a d^{2}}{b}}\right ) +{\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{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} - 2 \, b\right )} x^{2}{\rm Ei}\left (d x\right ) +{\left (a d^{2} - 2 \, b\right )} x^{2}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )}{\rm Ei}\left (d x + \sqrt{-\frac{a d^{2}}{b}}\right ) +{\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{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 (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )}{\rm Ei}\left (d x - \sqrt{-\frac{a d^{2}}{b}}\right ) -{\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{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} - 2 \, b\right )} x^{2}{\rm Ei}\left (d x\right ) -{\left (a d^{2} - 2 \, b\right )} x^{2}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) +{\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )}{\rm Ei}\left (d x + \sqrt{-\frac{a d^{2}}{b}}\right ) -{\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{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 (a^{2} x^{2} \cosh \left (d x + c\right )^{2} - a^{2} x^{2} \sinh \left (d x + c\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a*d*x*sinh(d*x + c) + 2*a*cosh(d*x + c) - ((b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(d*x - sq
rt(-a*d^2/b)) + (b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(-d*x + sqrt(-a*d^2/b)))*cosh(c + sqrt(-a*d^
2/b)) - ((a*d^2 - 2*b)*x^2*Ei(d*x) + (a*d^2 - 2*b)*x^2*Ei(-d*x))*cosh(c) - ((b*x^2*cosh(d*x + c)^2 - b*x^2*sin
h(d*x + c)^2)*Ei(d*x + sqrt(-a*d^2/b)) + (b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(-d*x - sqrt(-a*d^2
/b)))*cosh(-c + sqrt(-a*d^2/b)) - ((b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(d*x - sqrt(-a*d^2/b)) -
(b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(-d*x + sqrt(-a*d^2/b)))*sinh(c + sqrt(-a*d^2/b)) - ((a*d^2
- 2*b)*x^2*Ei(d*x) - (a*d^2 - 2*b)*x^2*Ei(-d*x))*sinh(c) + ((b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei
(d*x + sqrt(-a*d^2/b)) - (b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(-d*x - sqrt(-a*d^2/b)))*sinh(-c +
sqrt(-a*d^2/b)))/(a^2*x^2*cosh(d*x + c)^2 - a^2*x^2*sinh(d*x + c)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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