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

Optimal. Leaf size=226 \[ -\frac{3 f (b c-a d) \cosh \left (\frac{b f}{d}+e\right ) \text{Chi}\left (\frac{(b c-a d) f}{d (c+d x)}\right )}{4 d^2}+\frac{3 f (b c-a d) \cosh \left (3 \left (\frac{b f}{d}+e\right )\right ) \text{Chi}\left (\frac{3 (b c-a d) f}{d (c+d x)}\right )}{4 d^2}+\frac{3 f (b c-a d) \sinh \left (\frac{b f}{d}+e\right ) \text{Shi}\left (\frac{(b c-a d) f}{d (c+d x)}\right )}{4 d^2}-\frac{3 f (b c-a d) \sinh \left (3 \left (\frac{b f}{d}+e\right )\right ) \text{Shi}\left (\frac{3 (b c-a d) f}{d (c+d x)}\right )}{4 d^2}+\frac{(c+d x) \sinh ^3\left (\frac{a f+b f x+c e+d e x}{c+d x}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.476477, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5609, 5607, 3313, 3303, 3298, 3301} \[ -\frac{3 f (b c-a d) \cosh \left (\frac{b f}{d}+e\right ) \text{Chi}\left (\frac{(b c-a d) f}{d (c+d x)}\right )}{4 d^2}+\frac{3 f (b c-a d) \cosh \left (3 \left (\frac{b f}{d}+e\right )\right ) \text{Chi}\left (\frac{3 (b c-a d) f}{d (c+d x)}\right )}{4 d^2}+\frac{3 f (b c-a d) \sinh \left (\frac{b f}{d}+e\right ) \text{Shi}\left (\frac{(b c-a d) f}{d (c+d x)}\right )}{4 d^2}-\frac{3 f (b c-a d) \sinh \left (3 \left (\frac{b f}{d}+e\right )\right ) \text{Shi}\left (\frac{3 (b c-a d) f}{d (c+d x)}\right )}{4 d^2}+\frac{(c+d x) \sinh ^3\left (\frac{a f+b f x+c e+d e x}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5609

Int[Sinh[u_]^(n_.), x_Symbol] :> With[{lst = QuotientOfLinearsParts[u, x]}, Int[Sinh[(lst[[1]] + lst[[2]]*x)/(
lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n, 0] && QuotientOfLinearsQ[u, x]

Rule 5607

Int[Sinh[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol] :> -Dist[d^(-1), Subst[Int[Sinh[(
b*e)/d - (e*(b*c - a*d)*x)/d]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*
c - a*d, 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]

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 \sinh ^3\left (e+\frac{f (a+b x)}{c+d x}\right ) \, dx &=\int \sinh ^3\left (\frac{c e+a f+(d e+b f) x}{c+d x}\right ) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^3\left (\frac{d e+b f}{d}-\frac{(-d (c e+a f)+c (d e+b f)) x}{d}\right )}{x^2} \, dx,x,\frac{1}{c+d x}\right )}{d}\\ &=\frac{(c+d x) \sinh ^3\left (\frac{c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac{(3 (b c-a d) f) \operatorname{Subst}\left (\int \left (-\frac{\cosh \left (3 \left (e+\frac{b f}{d}\right )-\frac{3 (b c-a d) f x}{d}\right )}{4 x}+\frac{\cosh \left (e+\frac{b f}{d}-\frac{(b c-a d) f x}{d}\right )}{4 x}\right ) \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \sinh ^3\left (\frac{c e+a f+d e x+b f x}{c+d x}\right )}{d}+\frac{(3 (b c-a d) f) \operatorname{Subst}\left (\int \frac{\cosh \left (3 \left (e+\frac{b f}{d}\right )-\frac{3 (b c-a d) f x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}-\frac{(3 (b c-a d) f) \operatorname{Subst}\left (\int \frac{\cosh \left (e+\frac{b f}{d}-\frac{(b c-a d) f x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}\\ &=\frac{(c+d x) \sinh ^3\left (\frac{c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac{\left (3 (b c-a d) f \cosh \left (e+\frac{b f}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{(b c-a d) f x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}+\frac{\left (3 (b c-a d) f \cosh \left (3 \left (e+\frac{b f}{d}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 (b c-a d) f x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}+\frac{\left (3 (b c-a d) f \sinh \left (e+\frac{b f}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{(b c-a d) f x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}-\frac{\left (3 (b c-a d) f \sinh \left (3 \left (e+\frac{b f}{d}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 (b c-a d) f x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{4 d^2}\\ &=-\frac{3 (b c-a d) f \cosh \left (e+\frac{b f}{d}\right ) \text{Chi}\left (\frac{(b c-a d) f}{d (c+d x)}\right )}{4 d^2}+\frac{3 (b c-a d) f \cosh \left (3 \left (e+\frac{b f}{d}\right )\right ) \text{Chi}\left (\frac{3 (b c-a d) f}{d (c+d x)}\right )}{4 d^2}+\frac{(c+d x) \sinh ^3\left (\frac{c e+a f+d e x+b f x}{c+d x}\right )}{d}+\frac{3 (b c-a d) f \sinh \left (e+\frac{b f}{d}\right ) \text{Shi}\left (\frac{(b c-a d) f}{d (c+d x)}\right )}{4 d^2}-\frac{3 (b c-a d) f \sinh \left (3 \left (e+\frac{b f}{d}\right )\right ) \text{Shi}\left (\frac{3 (b c-a d) f}{d (c+d x)}\right )}{4 d^2}\\ \end{align*}

Mathematica [B]  time = 6.18071, size = 671, normalized size = 2.97 \[ \frac{-6 a d f \cosh \left (3 \left (\frac{b f}{d}+e\right )\right ) \text{Chi}\left (\frac{3 (a d f-b c f)}{d (c+d x)}\right )+6 b c f \cosh \left (3 \left (\frac{b f}{d}+e\right )\right ) \text{Chi}\left (\frac{3 (a d f-b c f)}{d (c+d x)}\right )+3 f (b c-a d) \left (\sinh \left (\frac{b f}{d}+e\right )-\cosh \left (\frac{b f}{d}+e\right )\right ) \text{Chi}\left (\frac{(b c-a d) f}{d (c+d x)}\right )-3 f (b c-a d) \left (\sinh \left (\frac{b f}{d}+e\right )+\cosh \left (\frac{b f}{d}+e\right )\right ) \text{Chi}\left (\frac{a d f-b c f}{d (c+d x)}\right )-6 d^2 x \sinh \left (\frac{a f+b f x+c e+d e x}{c+d x}\right )+2 d^2 x \sinh \left (\frac{3 (a f+b f x+c e+d e x)}{c+d x}\right )-3 a d f \sinh \left (\frac{b f}{d}+e\right ) \text{Shi}\left (\frac{(b c-a d) f}{d (c+d x)}\right )+3 a d f \sinh \left (\frac{b f}{d}+e\right ) \text{Shi}\left (\frac{a d f-b c f}{d (c+d x)}\right )-6 a d f \sinh \left (3 \left (\frac{b f}{d}+e\right )\right ) \text{Shi}\left (\frac{3 (a d f-b c f)}{d (c+d x)}\right )+3 b c f \sinh \left (\frac{b f}{d}+e\right ) \text{Shi}\left (\frac{(b c-a d) f}{d (c+d x)}\right )-3 b c f \sinh \left (\frac{b f}{d}+e\right ) \text{Shi}\left (\frac{a d f-b c f}{d (c+d x)}\right )+6 b c f \sinh \left (3 \left (\frac{b f}{d}+e\right )\right ) \text{Shi}\left (\frac{3 (a d f-b c f)}{d (c+d x)}\right )+3 a d f \cosh \left (\frac{b f}{d}+e\right ) \text{Shi}\left (\frac{(b c-a d) f}{d (c+d x)}\right )+3 a d f \cosh \left (\frac{b f}{d}+e\right ) \text{Shi}\left (\frac{a d f-b c f}{d (c+d x)}\right )-3 b c f \cosh \left (\frac{b f}{d}+e\right ) \text{Shi}\left (\frac{(b c-a d) f}{d (c+d x)}\right )-3 b c f \cosh \left (\frac{b f}{d}+e\right ) \text{Shi}\left (\frac{a d f-b c f}{d (c+d x)}\right )-6 c d \sinh \left (\frac{a f+b f x+c e+d e x}{c+d x}\right )+2 c d \sinh \left (\frac{3 (a f+b f x+c e+d e x)}{c+d x}\right )}{8 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*b*c*f*Cosh[3*(e + (b*f)/d)]*CoshIntegral[(3*(-(b*c*f) + a*d*f))/(d*(c + d*x))] - 6*a*d*f*Cosh[3*(e + (b*f)/
d)]*CoshIntegral[(3*(-(b*c*f) + a*d*f))/(d*(c + d*x))] + 3*(b*c - a*d)*f*CoshIntegral[((b*c - a*d)*f)/(d*(c +
d*x))]*(-Cosh[e + (b*f)/d] + Sinh[e + (b*f)/d]) - 3*(b*c - a*d)*f*CoshIntegral[(-(b*c*f) + a*d*f)/(d*(c + d*x)
)]*(Cosh[e + (b*f)/d] + Sinh[e + (b*f)/d]) - 6*c*d*Sinh[(c*e + a*f + d*e*x + b*f*x)/(c + d*x)] - 6*d^2*x*Sinh[
(c*e + a*f + d*e*x + b*f*x)/(c + d*x)] + 2*c*d*Sinh[(3*(c*e + a*f + d*e*x + b*f*x))/(c + d*x)] + 2*d^2*x*Sinh[
(3*(c*e + a*f + d*e*x + b*f*x))/(c + d*x)] - 3*b*c*f*Cosh[e + (b*f)/d]*SinhIntegral[((b*c - a*d)*f)/(d*(c + d*
x))] + 3*a*d*f*Cosh[e + (b*f)/d]*SinhIntegral[((b*c - a*d)*f)/(d*(c + d*x))] + 3*b*c*f*Sinh[e + (b*f)/d]*SinhI
ntegral[((b*c - a*d)*f)/(d*(c + d*x))] - 3*a*d*f*Sinh[e + (b*f)/d]*SinhIntegral[((b*c - a*d)*f)/(d*(c + d*x))]
- 3*b*c*f*Cosh[e + (b*f)/d]*SinhIntegral[(-(b*c*f) + a*d*f)/(d*(c + d*x))] + 3*a*d*f*Cosh[e + (b*f)/d]*SinhIn
tegral[(-(b*c*f) + a*d*f)/(d*(c + d*x))] - 3*b*c*f*Sinh[e + (b*f)/d]*SinhIntegral[(-(b*c*f) + a*d*f)/(d*(c + d
*x))] + 3*a*d*f*Sinh[e + (b*f)/d]*SinhIntegral[(-(b*c*f) + a*d*f)/(d*(c + d*x))] + 6*b*c*f*Sinh[3*(e + (b*f)/d
)]*SinhIntegral[(3*(-(b*c*f) + a*d*f))/(d*(c + d*x))] - 6*a*d*f*Sinh[3*(e + (b*f)/d)]*SinhIntegral[(3*(-(b*c*f
) + a*d*f))/(d*(c + d*x))])/(8*d^2)

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Maple [B]  time = 0.105, size = 922, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(e+f*(b*x+a)/(d*x+c))^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.55016, size = 2047, normalized size = 9.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(6*(b*c - a*d)*f*Ei(-3*(b*c - a*d)*f/(d^2*x + c*d))*cosh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2*cosh(3*
(d*e + b*f)/d)*sinh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2 - 3*(b*c - a*d)*f*Ei(-3*(b*c - a*d)*f/(d^2*x + c*
d))*cosh(3*(d*e + b*f)/d)*sinh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^4 - 2*(d^2*x + c*d)*sinh((c*e + a*f + (d
*e + b*f)*x)/(d*x + c))^3 - 3*((b*c - a*d)*f*Ei(-3*(b*c - a*d)*f/(d^2*x + c*d))*cosh((c*e + a*f + (d*e + b*f)*
x)/(d*x + c))^4 + (b*c - a*d)*f*Ei(3*(b*c - a*d)*f/(d^2*x + c*d)))*cosh(3*(d*e + b*f)/d) + 3*((b*c - a*d)*f*Ei
((b*c - a*d)*f/(d^2*x + c*d)) + (b*c - a*d)*f*Ei(-(b*c - a*d)*f/(d^2*x + c*d)))*cosh((d*e + b*f)/d) + 6*(d^2*x
- (d^2*x + c*d)*cosh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2 + c*d)*sinh((c*e + a*f + (d*e + b*f)*x)/(d*x +
c)) - 3*((b*c - a*d)*f*Ei(-3*(b*c - a*d)*f/(d^2*x + c*d))*cosh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^4 - 2*(b
*c - a*d)*f*Ei(-3*(b*c - a*d)*f/(d^2*x + c*d))*cosh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2*sinh((c*e + a*f +
(d*e + b*f)*x)/(d*x + c))^2 + (b*c - a*d)*f*Ei(-3*(b*c - a*d)*f/(d^2*x + c*d))*sinh((c*e + a*f + (d*e + b*f)*
x)/(d*x + c))^4 - (b*c - a*d)*f*Ei(3*(b*c - a*d)*f/(d^2*x + c*d)))*sinh(3*(d*e + b*f)/d) - 3*((b*c - a*d)*f*Ei
((b*c - a*d)*f/(d^2*x + c*d)) - (b*c - a*d)*f*Ei(-(b*c - a*d)*f/(d^2*x + c*d)))*sinh((d*e + b*f)/d))/(d^2*cosh
((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^4 - 2*d^2*cosh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2*sinh((c*e + a*
f + (d*e + b*f)*x)/(d*x + c))^2 + d^2*sinh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^4)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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