3.3.97 \(\int \sinh ^3(\frac {a+b x}{c+d x}) \, dx\) [297]

Optimal. Leaf size=194 \[ -\frac {3 (b c-a d) \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}+\frac {3 (b c-a d) \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sinh ^3\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {3 (b c-a d) \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 (b c-a d) \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2} \]

[Out]

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

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Rubi [A]
time = 0.25, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5726, 3394, 3384, 3379, 3382} \begin {gather*} -\frac {3 \cosh \left (\frac {b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}+\frac {3 \cosh \left (\frac {3 b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac {3 \sinh \left (\frac {b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 \sinh \left (\frac {3 b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sinh ^3\left (\frac {a+b x}{c+d x}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[(a + b*x)/(c + d*x)]^3,x]

[Out]

(-3*(b*c - a*d)*Cosh[b/d]*CoshIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) + (3*(b*c - a*d)*Cosh[(3*b)/d]*Cosh
Integral[(3*(b*c - a*d))/(d*(c + d*x))])/(4*d^2) + ((c + d*x)*Sinh[(a + b*x)/(c + d*x)]^3)/d + (3*(b*c - a*d)*
Sinh[b/d]*SinhIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) - (3*(b*c - a*d)*Sinh[(3*b)/d]*SinhIntegral[(3*(b*c
- a*d))/(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*} \int \sinh ^3\left (\frac {a+b x}{c+d x}\right ) \, dx &=-\frac {\text {Subst}\left (\int \frac {\sinh ^3\left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d}\\ &=\frac {(c+d x) \sinh ^3\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(3 (b c-a d)) \text {Subst}\left (\int \left (-\frac {\cosh \left (\frac {3 b}{d}-\frac {3 (b c-a d) x}{d}\right )}{4 x}+\frac {\cosh \left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{4 x}\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(c+d x) \sinh ^3\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {(3 (b c-a d)) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 b}{d}-\frac {3 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}-\frac {(3 (b c-a d)) \text {Subst}\left (\int \frac {\cosh \left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}\\ &=\frac {(c+d x) \sinh ^3\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {\left (3 (b c-a d) \cosh \left (\frac {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}+\frac {\left (3 (b c-a d) \cosh \left (\frac {3 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}+\frac {\left (3 (b c-a d) \sinh \left (\frac {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}-\frac {\left (3 (b c-a d) \sinh \left (\frac {3 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}\\ &=-\frac {3 (b c-a d) \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}+\frac {3 (b c-a d) \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sinh ^3\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {3 (b c-a d) \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 (b c-a d) \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(599\) vs. \(2(194)=388\).
time = 0.93, size = 599, normalized size = 3.09 \begin {gather*} \frac {6 (b c-a d) \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right )-3 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{c d+d^2 x}\right )+3 a d \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{c d+d^2 x}\right )+3 b c \text {Chi}\left (\frac {b c-a d}{c d+d^2 x}\right ) \sinh \left (\frac {b}{d}\right )-3 a d \text {Chi}\left (\frac {b c-a d}{c d+d^2 x}\right ) \sinh \left (\frac {b}{d}\right )-3 (b c-a d) \text {Chi}\left (\frac {-b c+a d}{d (c+d x)}\right ) \left (\cosh \left (\frac {b}{d}\right )+\sinh \left (\frac {b}{d}\right )\right )-6 c d \sinh \left (\frac {a+b x}{c+d x}\right )-6 d^2 x \sinh \left (\frac {a+b x}{c+d x}\right )+2 c d \sinh \left (\frac {3 (a+b x)}{c+d x}\right )+2 d^2 x \sinh \left (\frac {3 (a+b x)}{c+d x}\right )-3 b c \cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )+3 a d \cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )-3 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )+3 a d \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )+6 b c \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right )-6 a d \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right )-3 b c \cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )+3 a d \cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )+3 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )-3 a d \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )}{8 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(b*c - a*d)*Cosh[(3*b)/d]*CoshIntegral[(3*(-(b*c) + a*d))/(d*(c + d*x))] - 3*b*c*Cosh[b/d]*CoshIntegral[(b*
c - a*d)/(c*d + d^2*x)] + 3*a*d*Cosh[b/d]*CoshIntegral[(b*c - a*d)/(c*d + d^2*x)] + 3*b*c*CoshIntegral[(b*c -
a*d)/(c*d + d^2*x)]*Sinh[b/d] - 3*a*d*CoshIntegral[(b*c - a*d)/(c*d + d^2*x)]*Sinh[b/d] - 3*(b*c - a*d)*CoshIn
tegral[(-(b*c) + a*d)/(d*(c + d*x))]*(Cosh[b/d] + Sinh[b/d]) - 6*c*d*Sinh[(a + b*x)/(c + d*x)] - 6*d^2*x*Sinh[
(a + b*x)/(c + d*x)] + 2*c*d*Sinh[(3*(a + b*x))/(c + d*x)] + 2*d^2*x*Sinh[(3*(a + b*x))/(c + d*x)] - 3*b*c*Cos
h[b/d]*SinhIntegral[(-(b*c) + a*d)/(d*(c + d*x))] + 3*a*d*Cosh[b/d]*SinhIntegral[(-(b*c) + a*d)/(d*(c + d*x))]
- 3*b*c*Sinh[b/d]*SinhIntegral[(-(b*c) + a*d)/(d*(c + d*x))] + 3*a*d*Sinh[b/d]*SinhIntegral[(-(b*c) + a*d)/(d
*(c + d*x))] + 6*b*c*Sinh[(3*b)/d]*SinhIntegral[(3*(-(b*c) + a*d))/(d*(c + d*x))] - 6*a*d*Sinh[(3*b)/d]*SinhIn
tegral[(3*(-(b*c) + a*d))/(d*(c + d*x))] - 3*b*c*Cosh[b/d]*SinhIntegral[(b*c - a*d)/(c*d + d^2*x)] + 3*a*d*Cos
h[b/d]*SinhIntegral[(b*c - a*d)/(c*d + d^2*x)] + 3*b*c*Sinh[b/d]*SinhIntegral[(b*c - a*d)/(c*d + d^2*x)] - 3*a
*d*Sinh[b/d]*SinhIntegral[(b*c - a*d)/(c*d + d^2*x)])/(8*d^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(699\) vs. \(2(186)=372\).
time = 8.94, size = 700, normalized size = 3.61

method	result	size
risch	\(-\frac {{\mathrm e}^{-\frac {3 \left (b x +a \right )}{d x +c}} a}{8 \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {3 \left (b x +a \right )}{d x +c}} b c}{8 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {3 \,{\mathrm e}^{-\frac {3 b}{d}} \expIntegral \left (1, \frac {3 a d -3 b c}{d \left (d x +c \right )}\right ) a}{8 d}-\frac {3 \,{\mathrm e}^{-\frac {3 b}{d}} \expIntegral \left (1, \frac {3 a d -3 b c}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}+\frac {3 \,{\mathrm e}^{-\frac {b x +a}{d x +c}} a}{8 \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}-\frac {3 \,{\mathrm e}^{-\frac {b x +a}{d x +c}} b c}{8 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}-\frac {3 \,{\mathrm e}^{-\frac {b}{d}} \expIntegral \left (1, \frac {a d -b c}{d \left (d x +c \right )}\right ) a}{8 d}+\frac {3 \,{\mathrm e}^{-\frac {b}{d}} \expIntegral \left (1, \frac {a d -b c}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}+\frac {d \,{\mathrm e}^{\frac {3 b x +3 a}{d x +c}} x a}{8 a d -8 b c}-\frac {{\mathrm e}^{\frac {3 b x +3 a}{d x +c}} x b c}{8 \left (a d -b c \right )}+\frac {{\mathrm e}^{\frac {3 b x +3 a}{d x +c}} c a}{8 a d -8 b c}-\frac {{\mathrm e}^{\frac {3 b x +3 a}{d x +c}} c^{2} b}{8 d \left (a d -b c \right )}+\frac {3 \,{\mathrm e}^{\frac {3 b}{d}} \expIntegral \left (1, -\frac {3 \left (a d -b c \right )}{d \left (d x +c \right )}\right ) a}{8 d}-\frac {3 \,{\mathrm e}^{\frac {3 b}{d}} \expIntegral \left (1, -\frac {3 \left (a d -b c \right )}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}-\frac {3 d \,{\mathrm e}^{\frac {b x +a}{d x +c}} x a}{8 \left (a d -b c \right )}+\frac {3 \,{\mathrm e}^{\frac {b x +a}{d x +c}} x b c}{8 \left (a d -b c \right )}-\frac {3 \,{\mathrm e}^{\frac {b x +a}{d x +c}} c a}{8 \left (a d -b c \right )}+\frac {3 \,{\mathrm e}^{\frac {b x +a}{d x +c}} c^{2} b}{8 d \left (a d -b c \right )}-\frac {3 \,{\mathrm e}^{\frac {b}{d}} \expIntegral \left (1, -\frac {a d -b c}{d \left (d x +c \right )}\right ) a}{8 d}+\frac {3 \,{\mathrm e}^{\frac {b}{d}} \expIntegral \left (1, -\frac {a d -b c}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}\)	\(700\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 717 vs. \(2 (186) = 372\).
time = 0.49, size = 717, normalized size = 3.70 \begin {gather*} -\frac {6 \, {\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} \cosh \left (\frac {3 \, b}{d}\right ) \sinh \left (\frac {b x + a}{d x + c}\right )^{2} - 3 \, {\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {3 \, b}{d}\right ) \sinh \left (\frac {b x + a}{d x + c}\right )^{4} - 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right )^{3} - 3 \, {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{4} + {\left (b c - a d\right )} {\rm Ei}\left (\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {3 \, b}{d}\right ) + 3 \, {\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) + {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{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 + a}{d x + c}\right )^{2} + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right ) - 3 \, {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{4} - 2 \, {\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} \sinh \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \sinh \left (\frac {b x + a}{d x + c}\right )^{4} - {\left (b c - a d\right )} {\rm Ei}\left (\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {3 \, b}{d}\right ) - 3 \, {\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b}{d}\right )}{8 \, {\left (d^{2} \cosh \left (\frac {b x + a}{d x + c}\right )^{4} - 2 \, d^{2} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} \sinh \left (\frac {b x + a}{d x + c}\right )^{2} + d^{2} \sinh \left (\frac {b x + a}{d x + c}\right )^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1383 vs. \(2 (186) = 372\).
time = 8.62, size = 1383, normalized size = 7.13 \begin {gather*} \frac {{\left (3 \, b^{3} c^{2} {\rm Ei}\left (-\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {3 \, b}{d}\right )} - 6 \, a b^{2} c d {\rm Ei}\left (-\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {3 \, b}{d}\right )} - \frac {3 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (-\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {3 \, b}{d}\right )}}{d x + c} + 3 \, a^{2} b d^{2} {\rm Ei}\left (-\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {3 \, b}{d}\right )} + \frac {6 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (-\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {3 \, b}{d}\right )}}{d x + c} - \frac {3 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (-\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {3 \, b}{d}\right )}}{d x + c} - 3 \, b^{3} c^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} + 6 \, a b^{2} c d {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} + \frac {3 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} - 3 \, a^{2} b d^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} - \frac {6 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} + \frac {3 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} - 3 \, b^{3} c^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} + 6 \, a b^{2} c d {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} + \frac {3 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} - 3 \, a^{2} b d^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} - \frac {6 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} + \frac {3 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} + 3 \, b^{3} c^{2} {\rm Ei}\left (\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {3 \, b}{d}\right )} - 6 \, a b^{2} c d {\rm Ei}\left (\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {3 \, b}{d}\right )} - \frac {3 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {3 \, b}{d}\right )}}{d x + c} + 3 \, a^{2} b d^{2} {\rm Ei}\left (\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {3 \, b}{d}\right )} + \frac {6 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {3 \, b}{d}\right )}}{d x + c} - \frac {3 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (\frac {3 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {3 \, b}{d}\right )}}{d x + c} + b^{2} c^{2} d e^{\left (\frac {3 \, {\left (b x + a\right )}}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (\frac {3 \, {\left (b x + a\right )}}{d x + c}\right )} + a^{2} d^{3} e^{\left (\frac {3 \, {\left (b x + a\right )}}{d x + c}\right )} - 3 \, b^{2} c^{2} d e^{\left (\frac {b x + a}{d x + c}\right )} + 6 \, a b c d^{2} e^{\left (\frac {b x + a}{d x + c}\right )} - 3 \, a^{2} d^{3} e^{\left (\frac {b x + a}{d x + c}\right )} + 3 \, b^{2} c^{2} d e^{\left (-\frac {b x + a}{d x + c}\right )} - 6 \, a b c d^{2} e^{\left (-\frac {b x + a}{d x + c}\right )} + 3 \, a^{2} d^{3} e^{\left (-\frac {b x + a}{d x + c}\right )} - b^{2} c^{2} d e^{\left (-\frac {3 \, {\left (b x + a\right )}}{d x + c}\right )} + 2 \, a b c d^{2} e^{\left (-\frac {3 \, {\left (b x + a\right )}}{d x + c}\right )} - a^{2} d^{3} e^{\left (-\frac {3 \, {\left (b x + a\right )}}{d x + c}\right )}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{8 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh((a + b*x)/(c + d*x))^3,x)

[Out]

int(sinh((a + b*x)/(c + d*x))^3, x)

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