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

Optimal. Leaf size=145 \[ \frac {3 b \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}+\frac {3 b \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {\cosh ^3(a+b x)}{d (c+d x)} \]

[Out]

-cosh(b*x+a)^3/d/(d*x+c)+3/4*b*cosh(a-b*c/d)*Shi(b*c/d+b*x)/d^2+3/4*b*cosh(3*a-3*b*c/d)*Shi(3*b*c/d+3*b*x)/d^2
+3/4*b*Chi(3*b*c/d+3*b*x)*sinh(3*a-3*b*c/d)/d^2+3/4*b*Chi(b*c/d+b*x)*sinh(a-b*c/d)/d^2

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Rubi [A]  time = 0.24, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3313, 3303, 3298, 3301} \[ \frac {3 b \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}+\frac {3 b \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {\cosh ^3(a+b x)}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^3/(c + d*x)^2,x]

[Out]

-(Cosh[a + b*x]^3/(d*(c + d*x))) + (3*b*CoshIntegral[(3*b*c)/d + 3*b*x]*Sinh[3*a - (3*b*c)/d])/(4*d^2) + (3*b*
CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/(4*d^2) + (3*b*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(
4*d^2) + (3*b*Cosh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*c)/d + 3*b*x])/(4*d^2)

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 3313

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

Rubi steps

\begin {align*} \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx &=-\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {(3 i b) \int \left (-\frac {i \sinh (a+b x)}{4 (c+d x)}-\frac {i \sinh (3 a+3 b x)}{4 (c+d x)}\right ) \, dx}{d}\\ &=-\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {(3 b) \int \frac {\sinh (a+b x)}{c+d x} \, dx}{4 d}+\frac {(3 b) \int \frac {\sinh (3 a+3 b x)}{c+d x} \, dx}{4 d}\\ &=-\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {\left (3 b \cosh \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}{4 d}+\frac {\left (3 b \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{4 d}+\frac {\left (3 b \sinh \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}{4 d}+\frac {\left (3 b \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{4 d}\\ &=-\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {3 b \text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{4 d^2}+\frac {3 b \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{4 d^2}+\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}\\ \end {align*}

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Mathematica [A]  time = 0.53, size = 196, normalized size = 1.35 \[ -\frac {3 b \left (-2 \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )-2 \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )-2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )-2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )\right )}{8 d^2}-\frac {3 \sinh (a) \sinh (b x)}{4 d (c+d x)}-\frac {\sinh (3 a) \sinh (3 b x)}{4 d (c+d x)}-\frac {3 \cosh (a) \cosh (b x)}{4 d (c+d x)}-\frac {\cosh (3 a) \cosh (3 b x)}{4 d (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Cosh[a]*Cosh[b*x])/(4*d*(c + d*x)) - (Cosh[3*a]*Cosh[3*b*x])/(4*d*(c + d*x)) - (3*Sinh[a]*Sinh[b*x])/(4*d*
(c + d*x)) - (Sinh[3*a]*Sinh[3*b*x])/(4*d*(c + d*x)) - (3*b*(-2*CoshIntegral[(3*b*c)/d + 3*b*x]*Sinh[3*a - (3*
b*c)/d] - 2*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d] - 2*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x] -
2*Cosh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*c)/d + 3*b*x]))/(8*d^2)

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fricas [B]  time = 0.53, size = 305, normalized size = 2.10 \[ -\frac {2 \, d \cosh \left (b x + a\right )^{3} + 6 \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 6 \, d \cosh \left (b x + a\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\left (d^{3} x + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(2*d*cosh(b*x + a)^3 + 6*d*cosh(b*x + a)*sinh(b*x + a)^2 + 6*d*cosh(b*x + a) - 3*((b*d*x + b*c)*Ei((b*d*x
+ b*c)/d) - (b*d*x + b*c)*Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) - 3*((b*d*x + b*c)*Ei(3*(b*d*x + b*c)/d)
- (b*d*x + b*c)*Ei(-3*(b*d*x + b*c)/d))*cosh(-3*(b*c - a*d)/d) - 3*((b*d*x + b*c)*Ei((b*d*x + b*c)/d) + (b*d*
x + b*c)*Ei(-(b*d*x + b*c)/d))*sinh(-(b*c - a*d)/d) - 3*((b*d*x + b*c)*Ei(3*(b*d*x + b*c)/d) + (b*d*x + b*c)*E
i(-3*(b*d*x + b*c)/d))*sinh(-3*(b*c - a*d)/d))/(d^3*x + c*d^2)

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giac [B]  time = 0.21, size = 1075, normalized size = 7.41 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(-3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))
+ b*c - a*d)/d)*e^(3*(b*c - a*d)/d) + 3*b^3*c*Ei(-3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*
d)/d)*e^(3*(b*c - a*d)/d) - 3*a*b^2*d*Ei(-3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^(
3*(b*c - a*d)/d) + 3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d
/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 3*b^3*c*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b
*c - a*d)/d)*e^((b*c - a*d)/d) - 3*a*b^2*d*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*
e^((b*c - a*d)/d) - 3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(((d*x + c)*(b - b*c/(d*x + c) + a*d
/(d*x + c)) + b*c - a*d)/d)*e^(-(b*c - a*d)/d) - 3*b^3*c*Ei(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b
*c - a*d)/d)*e^(-(b*c - a*d)/d) + 3*a*b^2*d*Ei(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*
e^(-(b*c - a*d)/d) - 3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(3*((d*x + c)*(b - b*c/(d*x + c) +
a*d/(d*x + c)) + b*c - a*d)/d)*e^(-3*(b*c - a*d)/d) - 3*b^3*c*Ei(3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x +
c)) + b*c - a*d)/d)*e^(-3*(b*c - a*d)/d) + 3*a*b^2*d*Ei(3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c
- a*d)/d)*e^(-3*(b*c - a*d)/d) + b^2*d*e^(3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d) + 3*b^2*d*e^((d*
x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d) + 3*b^2*d*e^(-(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d) +
b^2*d*e^(-3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d))*d^2/(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x +
c))*d^4 + b*c*d^4 - a*d^5)*b)

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maple [A]  time = 0.30, size = 271, normalized size = 1.87 \[ -\frac {b \,{\mathrm e}^{-3 b x -3 a}}{8 \left (b d x +c b \right ) d}+\frac {3 b \,{\mathrm e}^{-\frac {3 \left (d a -c b \right )}{d}} \Ei \left (1, 3 b x +3 a -\frac {3 \left (d a -c b \right )}{d}\right )}{8 d^{2}}-\frac {3 b \,{\mathrm e}^{-b x -a}}{8 d \left (b d x +c b \right )}+\frac {3 b \,{\mathrm e}^{-\frac {d a -c b}{d}} \Ei \left (1, b x +a -\frac {d a -c b}{d}\right )}{8 d^{2}}-\frac {3 b \,{\mathrm e}^{b x +a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {3 b \,{\mathrm e}^{\frac {d a -c b}{d}} \Ei \left (1, -b x -a -\frac {-d a +c b}{d}\right )}{8 d^{2}}-\frac {b \,{\mathrm e}^{3 b x +3 a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {3 b \,{\mathrm e}^{\frac {3 d a -3 c b}{d}} \Ei \left (1, -3 b x -3 a -\frac {3 \left (-d a +c b \right )}{d}\right )}{8 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b*exp(-3*b*x-3*a)/(b*d*x+b*c)/d+3/8*b/d^2*exp(-3*(a*d-b*c)/d)*Ei(1,3*b*x+3*a-3*(a*d-b*c)/d)-3/8*b*exp(-b*
x-a)/d/(b*d*x+b*c)+3/8*b/d^2*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)-3/8*b/d^2*exp(b*x+a)/(b*c/d+b*x)-3/8*b/
d^2*exp((a*d-b*c)/d)*Ei(1,-b*x-a-(-a*d+b*c)/d)-1/8*b/d^2*exp(3*b*x+3*a)/(b*c/d+b*x)-3/8*b/d^2*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 = 0.46, size = 145, normalized size = 1.00 \[ -\frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{2}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{2}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^3/(c + d*x)^2,x)

[Out]

int(cosh(a + b*x)^3/(c + d*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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