\(\int \sinh ^3(\frac {b x}{c+d x}) \, dx\) [294]

Optimal result

Integrand size = 13, antiderivative size = 143 \[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx=-\frac {3 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}+\frac {3 b c \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sinh ^3\left (\frac {b x}{c+d x}\right )}{d}+\frac {3 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}-\frac {3 b c \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5726, 3394, 3384, 3379, 3382} \[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx=-\frac {3 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}+\frac {3 b c \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2}+\frac {3 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}-\frac {3 b c \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sinh ^3\left (\frac {b x}{c+d x}\right )}{d} \]

[In]

Int[Sinh[(b*x)/(c + d*x)]^3,x]

[Out]

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

Rule 3379

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 3382

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 3384

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 3394

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 5726

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]

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.62 \[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx=\frac {-c d e^{-\frac {3 b x}{c+d x}}+3 c d e^{-\frac {b x}{c+d x}}-3 c d e^{\frac {b x}{c+d x}}+c d e^{\frac {3 b x}{c+d x}}-d^2 e^{-\frac {3 b x}{c+d x}} x+d^2 e^{\frac {3 b x}{c+d x}} x-6 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{c d+d^2 x}\right )+6 b c \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 b c}{c d+d^2 x}\right )-6 d^2 x \sinh \left (\frac {b x}{c+d x}\right )+6 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{c d+d^2 x}\right )-6 b c \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c}{c d+d^2 x}\right )}{8 d^2} \]

[In]

Integrate[Sinh[(b*x)/(c + d*x)]^3,x]

[Out]

(-((c*d)/E^((3*b*x)/(c + d*x))) + (3*c*d)/E^((b*x)/(c + d*x)) - 3*c*d*E^((b*x)/(c + d*x)) + c*d*E^((3*b*x)/(c
+ d*x)) - (d^2*x)/E^((3*b*x)/(c + d*x)) + d^2*E^((3*b*x)/(c + d*x))*x - 6*b*c*Cosh[b/d]*CoshIntegral[(b*c)/(c*
d + d^2*x)] + 6*b*c*Cosh[(3*b)/d]*CoshIntegral[(3*b*c)/(c*d + d^2*x)] - 6*d^2*x*Sinh[(b*x)/(c + d*x)] + 6*b*c*
Sinh[b/d]*SinhIntegral[(b*c)/(c*d + d^2*x)] - 6*b*c*Sinh[(3*b)/d]*SinhIntegral[(3*b*c)/(c*d + d^2*x)])/(8*d^2)

Maple [A] (verified)

Time = 3.35 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.75

method	result	size
risch	\(\frac {3 b c \,{\mathrm e}^{\frac {b}{d}} \operatorname {Ei}_{1}\left (\frac {b c}{d \left (d x +c \right )}\right )}{8 d^{2}}+\frac {3 \,{\mathrm e}^{-\frac {b x}{d x +c}} x}{8}+\frac {3 b c \,{\mathrm e}^{-\frac {b}{d}} \operatorname {Ei}_{1}\left (-\frac {b c}{d \left (d x +c \right )}\right )}{8 d^{2}}-\frac {3 \,{\mathrm e}^{\frac {b x}{d x +c}} x}{8}+\frac {{\mathrm e}^{\frac {3 b x}{d x +c}} x}{8}-\frac {3 \,{\mathrm e}^{\frac {3 b}{d}} \operatorname {Ei}_{1}\left (\frac {3 b c}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}-\frac {{\mathrm e}^{-\frac {3 b x}{d x +c}} x}{8}-\frac {3 \,{\mathrm e}^{-\frac {3 b}{d}} \operatorname {Ei}_{1}\left (-\frac {3 b c}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}+\frac {3 \,{\mathrm e}^{-\frac {b x}{d x +c}} c}{8 d}-\frac {3 c \,{\mathrm e}^{\frac {b x}{d x +c}}}{8 d}+\frac {{\mathrm e}^{\frac {3 b x}{d x +c}} c}{8 d}-\frac {{\mathrm e}^{-\frac {3 b x}{d x +c}} c}{8 d}\)	\(250\)

[In]

int(sinh(b*x/(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 701 vs. \(2 (135) = 270\).

Time = 0.28 (sec) , antiderivative size = 701, normalized size of antiderivative = 4.90 \[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx=\frac {3 \, {\left (b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {3 \, b}{d}\right ) - b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b}{d}\right )\right )} \sinh \left (\frac {b x}{d x + c}\right )^{4} + 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b x}{d x + c}\right )^{3} - 6 \, {\left (b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} \cosh \left (\frac {3 \, b}{d}\right ) - b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} \cosh \left (\frac {b}{d}\right )\right )} \sinh \left (\frac {b x}{d x + c}\right )^{2} + 3 \, {\left (b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{4} + b c {\rm Ei}\left (\frac {3 \, b c}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {3 \, b}{d}\right ) - 3 \, {\left (b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{4} + b c {\rm Ei}\left (\frac {b c}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {b}{d}\right ) - 6 \, {\left (d^{2} x - {\left (d^{2} x + c d\right )} \cosh \left (\frac {b x}{d x + c}\right )^{2} + c d\right )} \sinh \left (\frac {b x}{d x + c}\right ) + 3 \, {\left (b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{4} - 2 \, b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} \sinh \left (\frac {b x}{d x + c}\right )^{2} + b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \sinh \left (\frac {b x}{d x + c}\right )^{4} - b c {\rm Ei}\left (\frac {3 \, b c}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {3 \, b}{d}\right ) - 3 \, {\left (b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{4} - 2 \, b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} \sinh \left (\frac {b x}{d x + c}\right )^{2} + b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \sinh \left (\frac {b x}{d x + c}\right )^{4} - b c {\rm Ei}\left (\frac {b c}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b}{d}\right )}{8 \, {\left (d^{2} \cosh \left (\frac {b x}{d x + c}\right )^{4} - 2 \, d^{2} \cosh \left (\frac {b x}{d x + c}\right )^{2} \sinh \left (\frac {b x}{d x + c}\right )^{2} + d^{2} \sinh \left (\frac {b x}{d x + c}\right )^{4}\right )}} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x}{d x + c}\right )^{3} \,d x } \]

[In]

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

[Out]

-3/8*b*c*integrate(x*e^(3*b*c/(d^2*x + c*d))/(d^2*x^2*e^(3*b/d) + 2*c*d*x*e^(3*b/d) + c^2*e^(3*b/d)), x) + 3/8
*b*c*integrate(x*e^(b*c/(d^2*x + c*d))/(d^2*x^2*e^(b/d) + 2*c*d*x*e^(b/d) + c^2*e^(b/d)), x) + 3/8*b*c*integra
te(x*e^(-b*c/(d^2*x + c*d) + b/d)/(d^2*x^2 + 2*c*d*x + c^2), x) - 3/8*b*c*integrate(x*e^(-3*b*c/(d^2*x + c*d)
+ 3*b/d)/(d^2*x^2 + 2*c*d*x + c^2), x) - 1/8*(x*e^(3*b*c/(d^2*x + c*d)) - 3*x*e^(b*c/(d^2*x + c*d) + 2*b/d) +
3*x*e^(-b*c/(d^2*x + c*d) + 4*b/d) - x*e^(-3*b*c/(d^2*x + c*d) + 6*b/d))*e^(-3*b/d)

Giac [F]

\[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x}{d x + c}\right )^{3} \,d x } \]

[In]

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

[Out]

integrate(sinh(b*x/(d*x + c))^3, x)

Mupad [F(-1)]

Timed out. \[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx=\int {\mathrm {sinh}\left (\frac {b\,x}{c+d\,x}\right )}^3 \,d x \]

[In]

int(sinh((b*x)/(c + d*x))^3,x)

[Out]

int(sinh((b*x)/(c + d*x))^3, x)