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

Optimal. Leaf size=184 \[ \frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}+\frac {3 b^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2} \]

[Out]

9/8*b^2*Chi(3*b*c/d+3*b*x)*cosh(3*a-3*b*c/d)/d^3+3/8*b^2*Chi(b*c/d+b*x)*cosh(a-b*c/d)/d^3-1/2*cosh(b*x+a)^3/d/
(d*x+c)^2+9/8*b^2*Shi(3*b*c/d+3*b*x)*sinh(3*a-3*b*c/d)/d^3+3/8*b^2*Shi(b*c/d+b*x)*sinh(a-b*c/d)/d^3-3/2*b*cosh
(b*x+a)^2*sinh(b*x+a)/d^2/(d*x+c)

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Rubi [A]  time = 0.34, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3314, 3303, 3298, 3301, 3312} \[ \frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}+\frac {3 b^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^3/(c + d*x)^3,x]

[Out]

-Cosh[a + b*x]^3/(2*d*(c + d*x)^2) + (3*b^2*Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/(8*d^3) + (9*b^2*Co
sh[3*a - (3*b*c)/d]*CoshIntegral[(3*b*c)/d + 3*b*x])/(8*d^3) - (3*b*Cosh[a + b*x]^2*Sinh[a + b*x])/(2*d^2*(c +
d*x)) + (3*b^2*Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(8*d^3) + (9*b^2*Sinh[3*a - (3*b*c)/d]*SinhInte
gral[(3*b*c)/d + 3*b*x])/(8*d^3)

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 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 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rubi steps

\begin {align*} \int \frac {\cosh ^3(a+b x)}{(c+d x)^3} \, dx &=-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}-\frac {3 b \cosh ^2(a+b x) \sinh (a+b x)}{2 d^2 (c+d x)}-\frac {\left (3 b^2\right ) \int \frac {\cosh (a+b x)}{c+d x} \, dx}{d^2}+\frac {\left (9 b^2\right ) \int \frac {\cosh ^3(a+b x)}{c+d x} \, dx}{2 d^2}\\ &=-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}-\frac {3 b \cosh ^2(a+b x) \sinh (a+b x)}{2 d^2 (c+d x)}+\frac {\left (9 b^2\right ) \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}{2 d^2}-\frac {\left (3 b^2 \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d^2}-\frac {\left (3 b^2 \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d^2}\\ &=-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}-\frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d^3}-\frac {3 b \cosh ^2(a+b x) \sinh (a+b x)}{2 d^2 (c+d x)}-\frac {3 b^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d^3}+\frac {\left (9 b^2\right ) \int \frac {\cosh (3 a+3 b x)}{c+d x} \, dx}{8 d^2}+\frac {\left (27 b^2\right ) \int \frac {\cosh (a+b x)}{c+d x} \, dx}{8 d^2}\\ &=-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}-\frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d^3}-\frac {3 b \cosh ^2(a+b x) \sinh (a+b x)}{2 d^2 (c+d x)}-\frac {3 b^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d^3}+\frac {\left (9 b^2 \cosh \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{8 d^2}+\frac {\left (27 b^2 \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{8 d^2}+\frac {\left (9 b^2 \sinh \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{8 d^2}+\frac {\left (27 b^2 \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{8 d^2}\\ &=-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}+\frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}-\frac {3 b \cosh ^2(a+b x) \sinh (a+b x)}{2 d^2 (c+d x)}+\frac {3 b^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}\\ \end {align*}

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Mathematica [A]  time = 0.89, size = 218, normalized size = 1.18 \[ -\frac {-6 b^2 (c+d x)^2 \left (\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right )+3 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b (c+d x)}{d}\right )+\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )+3 \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b (c+d x)}{d}\right )\right )+6 d \cosh (b x) (b \sinh (a) (c+d x)+d \cosh (a))+2 d \cosh (3 b x) (3 b \sinh (3 a) (c+d x)+d \cosh (3 a))+6 d \sinh (b x) (b \cosh (a) (c+d x)+d \sinh (a))+2 d \sinh (3 b x) (3 b \cosh (3 a) (c+d x)+d \sinh (3 a))}{16 d^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(6*d*Cosh[b*x]*(d*Cosh[a] + b*(c + d*x)*Sinh[a]) + 2*d*Cosh[3*b*x]*(d*Cosh[3*a] + 3*b*(c + d*x)*Sinh[3*a
]) + 6*d*(b*(c + d*x)*Cosh[a] + d*Sinh[a])*Sinh[b*x] + 2*d*(3*b*(c + d*x)*Cosh[3*a] + d*Sinh[3*a])*Sinh[3*b*x]
- 6*b^2*(c + d*x)^2*(Cosh[a - (b*c)/d]*CoshIntegral[b*(c/d + x)] + 3*Cosh[3*a - (3*b*c)/d]*CoshIntegral[(3*b*
(c + d*x))/d] + Sinh[a - (b*c)/d]*SinhIntegral[b*(c/d + x)] + 3*Sinh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*(c + d
*x))/d]))/(d^3*(c + d*x)^2)

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fricas [B]  time = 0.55, size = 527, normalized size = 2.86 \[ -\frac {2 \, d^{2} \cosh \left (b x + a\right )^{3} + 6 \, d^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 6 \, {\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right )^{3} + 6 \, d^{2} \cosh \left (b x + a\right ) - 3 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - 9 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 6 \, {\left (b d^{2} x + b c d + 3 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right ) - 3 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - 9 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{16 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(2*d^2*cosh(b*x + a)^3 + 6*d^2*cosh(b*x + a)*sinh(b*x + a)^2 + 6*(b*d^2*x + b*c*d)*sinh(b*x + a)^3 + 6*d
^2*cosh(b*x + a) - 3*((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei((b*d*x + b*c)/d) + (b^2*d^2*x^2 + 2*b^2*c*d*x +
b^2*c^2)*Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) - 9*((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(3*(b*d*x +
b*c)/d) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(-3*(b*d*x + b*c)/d))*cosh(-3*(b*c - a*d)/d) + 6*(b*d^2*x +
b*c*d + 3*(b*d^2*x + b*c*d)*cosh(b*x + a)^2)*sinh(b*x + a) - 3*((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei((b*d*
x + b*c)/d) - (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(-(b*d*x + b*c)/d))*sinh(-(b*c - a*d)/d) - 9*((b^2*d^2*x
^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(3*(b*d*x + b*c)/d) - (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(-3*(b*d*x + b*c)/
d))*sinh(-3*(b*c - a*d)/d))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)

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giac [B]  time = 0.13, size = 602, normalized size = 3.27 \[ \frac {9 \, b^{2} d^{2} x^{2} {\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 \, b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 3 \, b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 9 \, b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} + 18 \, b^{2} c d x {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} + 6 \, b^{2} c d x {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 6 \, b^{2} c d x {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 18 \, b^{2} c d x {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} + 9 \, b^{2} c^{2} {\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 \, b^{2} c^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 3 \, b^{2} c^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 9 \, b^{2} c^{2} {\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 \, b d^{2} x e^{\left (3 \, b x + 3 \, a\right )} - 3 \, b d^{2} x e^{\left (b x + a\right )} + 3 \, b d^{2} x e^{\left (-b x - a\right )} + 3 \, b d^{2} x e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, b c d e^{\left (3 \, b x + 3 \, a\right )} - 3 \, b c d e^{\left (b x + a\right )} + 3 \, b c d e^{\left (-b x - a\right )} + 3 \, b c d e^{\left (-3 \, b x - 3 \, a\right )} - d^{2} e^{\left (3 \, b x + 3 \, a\right )} - 3 \, d^{2} e^{\left (b x + a\right )} - 3 \, d^{2} e^{\left (-b x - a\right )} - d^{2} e^{\left (-3 \, b x - 3 \, a\right )}}{16 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(9*b^2*d^2*x^2*Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) + 3*b^2*d^2*x^2*Ei((b*d*x + b*c)/d)*e^(a - b*c/d)
+ 3*b^2*d^2*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 9*b^2*d^2*x^2*Ei(-3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d)
+ 18*b^2*c*d*x*Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) + 6*b^2*c*d*x*Ei((b*d*x + b*c)/d)*e^(a - b*c/d) + 6*b^2
*c*d*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 18*b^2*c*d*x*Ei(-3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d) + 9*b^2*c^
2*Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) + 3*b^2*c^2*Ei((b*d*x + b*c)/d)*e^(a - b*c/d) + 3*b^2*c^2*Ei(-(b*d*x
+ b*c)/d)*e^(-a + b*c/d) + 9*b^2*c^2*Ei(-3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d) - 3*b*d^2*x*e^(3*b*x + 3*a) -
3*b*d^2*x*e^(b*x + a) + 3*b*d^2*x*e^(-b*x - a) + 3*b*d^2*x*e^(-3*b*x - 3*a) - 3*b*c*d*e^(3*b*x + 3*a) - 3*b*c*
d*e^(b*x + a) + 3*b*c*d*e^(-b*x - a) + 3*b*c*d*e^(-3*b*x - 3*a) - d^2*e^(3*b*x + 3*a) - 3*d^2*e^(b*x + a) - 3*
d^2*e^(-b*x - a) - d^2*e^(-3*b*x - 3*a))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)

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maple [B]  time = 0.32, size = 562, normalized size = 3.05 \[ \frac {3 b^{3} {\mathrm e}^{-3 b x -3 a} x}{16 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +c^{2} b^{2}\right )}+\frac {3 b^{3} {\mathrm e}^{-3 b x -3 a} c}{16 d^{2} \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +c^{2} b^{2}\right )}-\frac {b^{2} {\mathrm e}^{-3 b x -3 a}}{16 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +c^{2} b^{2}\right )}-\frac {9 b^{2} {\mathrm e}^{-\frac {3 \left (d a -c b \right )}{d}} \Ei \left (1, 3 b x +3 a -\frac {3 \left (d a -c b \right )}{d}\right )}{16 d^{3}}+\frac {3 b^{3} {\mathrm e}^{-b x -a} x}{16 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +c^{2} b^{2}\right )}+\frac {3 b^{3} {\mathrm e}^{-b x -a} c}{16 d^{2} \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +c^{2} b^{2}\right )}-\frac {3 b^{2} {\mathrm e}^{-b x -a}}{16 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +c^{2} b^{2}\right )}-\frac {3 b^{2} {\mathrm e}^{-\frac {d a -c b}{d}} \Ei \left (1, b x +a -\frac {d a -c b}{d}\right )}{16 d^{3}}-\frac {3 b^{2} {\mathrm e}^{b x +a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {3 b^{2} {\mathrm e}^{b x +a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )}-\frac {3 b^{2} {\mathrm e}^{\frac {d a -c b}{d}} \Ei \left (1, -b x -a -\frac {-d a +c b}{d}\right )}{16 d^{3}}-\frac {b^{2} {\mathrm e}^{3 b x +3 a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {3 b^{2} {\mathrm e}^{3 b x +3 a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )}-\frac {9 b^{2} {\mathrm e}^{\frac {3 d a -3 c b}{d}} \Ei \left (1, -3 b x -3 a -\frac {3 \left (-d a +c b \right )}{d}\right )}{16 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*b^3*exp(-3*b*x-3*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*x+3/16*b^3*exp(-3*b*x-3*a)/d^2/(b^2*d^2*x^2+2*b^2
*c*d*x+b^2*c^2)*c-1/16*b^2*exp(-3*b*x-3*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)-9/16*b^2/d^3*exp(-3*(a*d-b*c)/d
)*Ei(1,3*b*x+3*a-3*(a*d-b*c)/d)+3/16*b^3*exp(-b*x-a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*x+3/16*b^3*exp(-b*x-a
)/d^2/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*c-3/16*b^2*exp(-b*x-a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)-3/16*b^2/d^
3*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)-3/16*b^2/d^3*exp(b*x+a)/(b*c/d+b*x)^2-3/16*b^2/d^3*exp(b*x+a)/(b*c
/d+b*x)-3/16*b^2/d^3*exp((a*d-b*c)/d)*Ei(1,-b*x-a-(-a*d+b*c)/d)-1/16*b^2/d^3*exp(3*b*x+3*a)/(b*c/d+b*x)^2-3/16
*b^2/d^3*exp(3*b*x+3*a)/(b*c/d+b*x)-9/16*b^2/d^3*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 = 0.45, size = 145, normalized size = 0.79 \[ -\frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{3}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} - \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{3}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*e^(-3*a + 3*b*c/d)*exp_integral_e(3, 3*(d*x + c)*b/d)/((d*x + c)^2*d) - 3/8*e^(-a + b*c/d)*exp_integral_e
(3, (d*x + c)*b/d)/((d*x + c)^2*d) - 3/8*e^(a - b*c/d)*exp_integral_e(3, -(d*x + c)*b/d)/((d*x + c)^2*d) - 1/8
*e^(3*a - 3*b*c/d)*exp_integral_e(3, -3*(d*x + c)*b/d)/((d*x + c)^2*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^3/(c + d*x)^3,x)

[Out]

int(cosh(a + b*x)^3/(c + d*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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