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

Optimal. Leaf size=104 \[ \frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 b^3}+\frac{d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 b^3}-\frac{d \sinh (c+d x)}{2 b^2 (a+b x)}-\frac{\cosh (c+d x)}{2 b (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.146297, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3297, 3303, 3298, 3301} \[ \frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 b^3}+\frac{d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 b^3}-\frac{d \sinh (c+d x)}{2 b^2 (a+b x)}-\frac{\cosh (c+d x)}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(a + b*x)^3,x]

[Out]

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

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

Mathematica [A]  time = 0.467275, size = 88, normalized size = 0.85 \[ \frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )+d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-\frac{b (d (a+b x) \sinh (c+d x)+b \cosh (c+d x))}{(a+b x)^2}}{2 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.028, size = 276, normalized size = 2.7 \begin{align*}{\frac{{d}^{3}{{\rm e}^{-dx-c}}x}{4\,b \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}+{\frac{{d}^{3}{{\rm e}^{-dx-c}}a}{4\,{b}^{2} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}-{\frac{{d}^{2}{{\rm e}^{-dx-c}}}{4\,b \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}-{\frac{{d}^{2}}{4\,{b}^{3}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{{d}^{2}{{\rm e}^{dx+c}}}{4\,{b}^{3}} \left ({\frac{da}{b}}+dx \right ) ^{-2}}-{\frac{{d}^{2}{{\rm e}^{dx+c}}}{4\,{b}^{3}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{{d}^{2}}{4\,{b}^{3}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*d^3*exp(-d*x-c)/b/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)*x+1/4*d^3*exp(-d*x-c)/b^2/(b^2*d^2*x^2+2*a*b*d^2*x+a^2
*d^2)*a-1/4*d^2*exp(-d*x-c)/b/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)-1/4*d^2/b^3*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b
*c)/b)-1/4*d^2/b^3*exp(d*x+c)/(1/b*d*a+d*x)^2-1/4*d^2/b^3*exp(d*x+c)/(1/b*d*a+d*x)-1/4*d^2/b^3*exp(-(a*d-b*c)/
b)*Ei(1,-d*x-c-(a*d-b*c)/b)

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Maxima [A]  time = 1.18995, size = 128, normalized size = 1.23 \begin{align*} \frac{d{\left (\frac{e^{\left (-c + \frac{a d}{b}\right )} E_{2}\left (\frac{{\left (b x + a\right )} d}{b}\right )}{{\left (b x + a\right )} b} - \frac{e^{\left (c - \frac{a d}{b}\right )} E_{2}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{{\left (b x + a\right )} b}\right )}}{4 \, b} - \frac{\cosh \left (d x + c\right )}{2 \,{\left (b x + a\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*d*(e^(-c + a*d/b)*exp_integral_e(2, (b*x + a)*d/b)/((b*x + a)*b) - e^(c - a*d/b)*exp_integral_e(2, -(b*x +
a)*d/b)/((b*x + a)*b))/b - 1/2*cosh(d*x + c)/((b*x + a)^2*b)

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Fricas [B]  time = 1.95569, size = 518, normalized size = 4.98 \begin{align*} -\frac{2 \, b^{2} \cosh \left (d x + c\right ) -{\left ({\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) + 2 \,{\left (b^{2} d x + a b d\right )} \sinh \left (d x + c\right ) +{\left ({\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*b^2*cosh(d*x + c) - ((b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*Ei((b*d*x + a*d)/b) + (b^2*d^2*x^2 + 2*a*b*
d^2*x + a^2*d^2)*Ei(-(b*d*x + a*d)/b))*cosh(-(b*c - a*d)/b) + 2*(b^2*d*x + a*b*d)*sinh(d*x + c) + ((b^2*d^2*x^
2 + 2*a*b*d^2*x + a^2*d^2)*Ei((b*d*x + a*d)/b) - (b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*Ei(-(b*d*x + a*d)/b))*s
inh(-(b*c - a*d)/b))/(b^5*x^2 + 2*a*b^4*x + a^2*b^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.18275, size = 402, normalized size = 3.87 \begin{align*} \frac{b^{2} d^{2} x^{2}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} + b^{2} d^{2} x^{2}{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + 2 \, a b d^{2} x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} + 2 \, a b d^{2} x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + a^{2} d^{2}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} + a^{2} d^{2}{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - b^{2} d x e^{\left (d x + c\right )} + b^{2} d x e^{\left (-d x - c\right )} - a b d e^{\left (d x + c\right )} + a b d e^{\left (-d x - c\right )} - b^{2} e^{\left (d x + c\right )} - b^{2} e^{\left (-d x - c\right )}}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^2*d^2*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + b^2*d^2*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 2*a*b*d
^2*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 2*a*b*d^2*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + a^2*d^2*Ei((b*d*x +
a*d)/b)*e^(c - a*d/b) + a^2*d^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - b^2*d*x*e^(d*x + c) + b^2*d*x*e^(-d*x -
c) - a*b*d*e^(d*x + c) + a*b*d*e^(-d*x - c) - b^2*e^(d*x + c) - b^2*e^(-d*x - c))/(b^5*x^2 + 2*a*b^4*x + a^2*
b^3)