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

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

[Out]

-3/4*cosh(a-b*c/d)*Shi(b*c/d+b*x)/d+1/4*cosh(3*a-3*b*c/d)*Shi(3*b*c/d+3*b*x)/d+1/4*Chi(3*b*c/d+3*b*x)*sinh(3*a
-3*b*c/d)/d-3/4*Chi(b*c/d+b*x)*sinh(a-b*c/d)/d

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Rubi [A]  time = 0.28, antiderivative size = 121, 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 = {3312, 3303, 3298, 3301} \[ \frac {\sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}-\frac {3 \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d}-\frac {3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]^3/(c + d*x),x]

[Out]

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

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])
)

Rubi steps

\begin {align*} \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx &=i \int \left (\frac {3 i \sinh (a+b x)}{4 (c+d x)}-\frac {i \sinh (3 a+3 b x)}{4 (c+d x)}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\sinh (3 a+3 b x)}{c+d x} \, dx-\frac {3}{4} \int \frac {\sinh (a+b x)}{c+d x} \, dx\\ &=\frac {1}{4} \cosh \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 \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \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 {\cosh \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 {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx\\ &=\frac {\text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{4 d}-\frac {3 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{4 d}-\frac {3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\cosh \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*}

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Mathematica [A]  time = 0.19, size = 102, normalized size = 0.84 \[ \frac {\sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b (c+d x)}{d}\right )-3 \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right )-3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )+\cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b (c+d x)}{d}\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.77, size = 188, normalized size = 1.55 \[ -\frac {3 \, {\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left ({\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, {\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - {\left ({\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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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giac [A]  time = 0.19, size = 113, normalized size = 0.93 \[ \frac {{\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} - 3 \, {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 3 \, {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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

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maple [A]  time = 0.15, size = 166, normalized size = 1.37 \[ \frac {{\mathrm e}^{-\frac {3 \left (d a -c b \right )}{d}} \Ei \left (1, 3 b x +3 a -\frac {3 \left (d a -c b \right )}{d}\right )}{8 d}-\frac {3 \,{\mathrm e}^{-\frac {d a -c b}{d}} \Ei \left (1, b x +a -\frac {d a -c b}{d}\right )}{8 d}+\frac {3 \,{\mathrm e}^{\frac {d a -c b}{d}} \Ei \left (1, -b x -a -\frac {-d a +c b}{d}\right )}{8 d}-\frac {{\mathrm e}^{\frac {3 d a -3 c b}{d}} \Ei \left (1, -3 b x -3 a -\frac {3 \left (-d a +c b \right )}{d}\right )}{8 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(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 = 0.46, size = 117, normalized size = 0.97 \[ \frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{1}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} + \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{1}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)^3/(c + d*x),x)

[Out]

int(sinh(a + b*x)^3/(c + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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