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

Optimal. Leaf size=121 \[ \frac{3 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d}+\frac{3 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.23984, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3312, 3303, 3298, 3301} \[ \frac{3 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d}+\frac{3 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^3/(c + d*x),x]

[Out]

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

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && 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 ^3(a+b x)}{c+d x} \, dx &=\int \left (\frac{3 \cosh (a+b x)}{4 (c+d x)}+\frac{\cosh (3 a+3 b x)}{4 (c+d x)}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\cosh (3 a+3 b x)}{c+d x} \, dx+\frac{3}{4} \int \frac{\cosh (a+b x)}{c+d x} \, dx\\ &=\frac{1}{4} \cosh \left (3 a-\frac{3 b c}{d}\right ) \int \frac{\cosh \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx+\frac{1}{4} \left (3 \cosh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cosh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx+\frac{1}{4} \sinh \left (3 a-\frac{3 b c}{d}\right ) \int \frac{\sinh \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx+\frac{1}{4} \left (3 \sinh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sinh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx\\ &=\frac{3 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d}+\frac{3 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.227433, size = 102, normalized size = 0.84 \[ \frac{3 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (b \left (\frac{c}{d}+x\right )\right )+\cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b (c+d x)}{d}\right )+3 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (b \left (\frac{c}{d}+x\right )\right )+\sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b (c+d x)}{d}\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Cosh[a - (b*c)/d]*CoshIntegral[b*(c/d + x)] + Cosh[3*a - (3*b*c)/d]*CoshIntegral[(3*b*(c + d*x))/d] + 3*Sin
h[a - (b*c)/d]*SinhIntegral[b*(c/d + x)] + Sinh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*(c + d*x))/d])/(4*d)

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Maple [A]  time = 0.088, size = 166, normalized size = 1.4 \begin{align*} -{\frac{1}{8\,d}{{\rm e}^{-3\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,3\,bx+3\,a-3\,{\frac{da-cb}{d}} \right ) }-{\frac{3}{8\,d}{{\rm e}^{-{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{da-cb}{d}} \right ) }-{\frac{3}{8\,d}{{\rm e}^{{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-bx-a-{\frac{-da+cb}{d}} \right ) }-{\frac{1}{8\,d}{{\rm e}^{3\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-3\,bx-3\,a-3\,{\frac{-da+cb}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/d*exp(-3*(a*d-b*c)/d)*Ei(1,3*b*x+3*a-3*(a*d-b*c)/d)-3/8/d*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)-3/8/d
*exp((a*d-b*c)/d)*Ei(1,-b*x-a-(-a*d+b*c)/d)-1/8/d*exp(3*(a*d-b*c)/d)*Ei(1,-3*b*x-3*a-3*(-a*d+b*c)/d)

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Maxima [A]  time = 1.27836, size = 158, normalized size = 1.31 \begin{align*} -\frac{e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )} E_{1}\left (\frac{3 \,{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac{3 \, e^{\left (-a + \frac{b c}{d}\right )} E_{1}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac{3 \, e^{\left (a - \frac{b c}{d}\right )} E_{1}\left (-\frac{{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac{e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} E_{1}\left (-\frac{3 \,{\left (d x + c\right )} b}{d}\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*e^(-3*a + 3*b*c/d)*exp_integral_e(1, 3*(d*x + c)*b/d)/d - 3/8*e^(-a + b*c/d)*exp_integral_e(1, (d*x + c)*
b/d)/d - 3/8*e^(a - b*c/d)*exp_integral_e(1, -(d*x + c)*b/d)/d - 1/8*e^(3*a - 3*b*c/d)*exp_integral_e(1, -3*(d
*x + c)*b/d)/d

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Fricas [A]  time = 1.98404, size = 398, normalized size = 3.29 \begin{align*} \frac{3 \,{\left ({\rm Ei}\left (\frac{b d x + b c}{d}\right ) +{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \cosh \left (-\frac{b c - a d}{d}\right ) +{\left ({\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) +{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) + 3 \,{\left ({\rm Ei}\left (\frac{b d x + b c}{d}\right ) -{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \sinh \left (-\frac{b c - a d}{d}\right ) +{\left ({\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) -{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*(Ei((b*d*x + b*c)/d) + Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) + (Ei(3*(b*d*x + b*c)/d) + Ei(-3*(b*d
*x + b*c)/d))*cosh(-3*(b*c - a*d)/d) + 3*(Ei((b*d*x + b*c)/d) - Ei(-(b*d*x + b*c)/d))*sinh(-(b*c - a*d)/d) + (
Ei(3*(b*d*x + b*c)/d) - Ei(-3*(b*d*x + b*c)/d))*sinh(-3*(b*c - a*d)/d))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.33214, size = 151, normalized size = 1.25 \begin{align*} \frac{{\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} + 3 \,{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} + 3 \,{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} +{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) + 3*Ei((b*d*x + b*c)/d)*e^(a - b*c/d) + 3*Ei(-(b*d*x + b*c)/d)*e^
(-a + b*c/d) + Ei(-3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d))/d