3.57 \(\int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=246 \[ -\frac {3 \sqrt {\pi } \sqrt {b} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {3 \pi } \sqrt {b} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {3 \sqrt {\pi } \sqrt {b} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {3 \pi } \sqrt {b} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}} \]

[Out]

-3/4*exp(-a+b*c/d)*erf(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*b^(1/2)*Pi^(1/2)/d^(3/2)-3/4*exp(a-b*c/d)*erfi(b^(1/2)*(
d*x+c)^(1/2)/d^(1/2))*b^(1/2)*Pi^(1/2)/d^(3/2)+1/4*exp(-3*a+3*b*c/d)*erf(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2)
)*b^(1/2)*3^(1/2)*Pi^(1/2)/d^(3/2)+1/4*exp(3*a-3*b*c/d)*erfi(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*b^(1/2)*3^
(1/2)*Pi^(1/2)/d^(3/2)-2*sinh(b*x+a)^3/d/(d*x+c)^(1/2)

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Rubi [A]  time = 0.45, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3313, 3307, 2180, 2204, 2205} \[ -\frac {3 \sqrt {\pi } \sqrt {b} e^{\frac {b c}{d}-a} \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {3 \pi } \sqrt {b} e^{\frac {3 b c}{d}-3 a} \text {Erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {3 \sqrt {\pi } \sqrt {b} e^{a-\frac {b c}{d}} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {3 \pi } \sqrt {b} e^{3 a-\frac {3 b c}{d}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[b]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) + (Sqrt[b]*E^(-3*a + (
3*b*c)/d)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) - (3*Sqrt[b]*E^(a - (b*c)/d)*Sq
rt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) + (Sqrt[b]*E^(3*a - (3*b*c)/d)*Sqrt[3*Pi]*Erfi[(Sqrt
[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) - (2*Sinh[a + b*x]^3)/(d*Sqrt[c + d*x])

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

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 {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx &=-\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}}-\frac {(6 b) \int \left (\frac {\cosh (a+b x)}{4 \sqrt {c+d x}}-\frac {\cosh (3 a+3 b x)}{4 \sqrt {c+d x}}\right ) \, dx}{d}\\ &=-\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}}-\frac {(3 b) \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx}{2 d}+\frac {(3 b) \int \frac {\cosh (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{2 d}\\ &=-\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}}-\frac {(3 b) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{4 d}-\frac {(3 b) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{4 d}+\frac {(3 b) \int \frac {e^{-i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{4 d}+\frac {(3 b) \int \frac {e^{i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{4 d}\\ &=-\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}}+\frac {(3 b) \operatorname {Subst}\left (\int e^{i \left (3 i a-\frac {3 i b c}{d}\right )-\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d^2}-\frac {(3 b) \operatorname {Subst}\left (\int e^{i \left (i a-\frac {i b c}{d}\right )-\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d^2}-\frac {(3 b) \operatorname {Subst}\left (\int e^{-i \left (i a-\frac {i b c}{d}\right )+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d^2}+\frac {(3 b) \operatorname {Subst}\left (\int e^{-i \left (3 i a-\frac {3 i b c}{d}\right )+\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d^2}\\ &=-\frac {3 \sqrt {b} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {b} e^{-3 a+\frac {3 b c}{d}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {3 \sqrt {b} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {b} e^{3 a-\frac {3 b c}{d}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}}\\ \end {align*}

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Mathematica [B]  time = 10.11, size = 2058, normalized size = 8.37 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(Cosh[a]*(-(((-((1 + E^((2*b*(c + d*x))/d))/E^((b*(c + d*x))/d)) + Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, -((
b*(c + d*x))/d)] + Sqrt[(b*(c + d*x))/d]*Gamma[1/2, (b*(c + d*x))/d])*Sinh[(b*c)/d])/(d*Sqrt[c + d*x])) + (Cos
h[(b*c)/d]*(Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, -((b*(c + d*x))/d)] - Sqrt[(b*(c + d*x))/d]*Gamma[1/2, (b*(c +
d*x))/d] - 2*Sinh[(b*(c + d*x))/d]))/(d*Sqrt[c + d*x])) + Sinh[a]*((Cosh[(b*c)/d]*(-((1 + E^((2*b*(c + d*x))/
d))/E^((b*(c + d*x))/d)) + Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, -((b*(c + d*x))/d)] + Sqrt[(b*(c + d*x))/d]*Gam
ma[1/2, (b*(c + d*x))/d]))/(d*Sqrt[c + d*x]) + (Sinh[(b*c)/d]*(-(Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, -((b*(c +
d*x))/d)]) + Sqrt[(b*(c + d*x))/d]*Gamma[1/2, (b*(c + d*x))/d] + 2*Sinh[(b*(c + d*x))/d]))/(d*Sqrt[c + d*x]))
))/4 + (-(Sinh[3*a]*(-(((1 + 2*Cosh[(2*b*c)/d])*(-((1 + E^((6*b*(c + d*x))/d))/E^((3*b*(c + d*x))/d)) + Sqrt[3
]*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, (-3*b*(c + d*x))/d] + Sqrt[3]*Sqrt[(b*(c + d*x))/d]*Gamma[1/2, (3*b*(c +
d*x))/d])*Sinh[(b*c)/d])/(d*Sqrt[c + d*x])) + (Cosh[(b*c)/d]*(-1 + 2*Cosh[(2*b*c)/d])*((Sqrt[b]*Sqrt[6*Pi]*(E
rf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]))/Sqrt[d] - (2*Sqr
t[2]*Sinh[(3*b*(c + d*x))/d])/Sqrt[c + d*x]))/(Sqrt[2]*d))) - Cosh[3*a]*((Cosh[(b*c)/d]*(-1 + 2*Cosh[(2*b*c)/d
])*(-((1 + E^((6*b*(c + d*x))/d))/E^((3*b*(c + d*x))/d)) + Sqrt[3]*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, (-3*b*(
c + d*x))/d] + Sqrt[3]*Sqrt[(b*(c + d*x))/d]*Gamma[1/2, (3*b*(c + d*x))/d]))/(d*Sqrt[c + d*x]) - ((1 + 2*Cosh[
(2*b*c)/d])*Sinh[(b*c)/d]*((Sqrt[b]*Sqrt[6*Pi]*(Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + Erfi[(Sqrt[3]*S
qrt[b]*Sqrt[c + d*x])/Sqrt[d]]))/Sqrt[d] - (2*Sqrt[2]*Sinh[(3*b*(c + d*x))/d])/Sqrt[c + d*x]))/(Sqrt[2]*d)))/8
+ (Sinh[3*a]*(-(((1 + 2*Cosh[(2*b*c)/d])*(-((1 + E^((6*b*(c + d*x))/d))/E^((3*b*(c + d*x))/d)) + Sqrt[3]*Sqrt
[-((b*(c + d*x))/d)]*Gamma[1/2, (-3*b*(c + d*x))/d] + Sqrt[3]*Sqrt[(b*(c + d*x))/d]*Gamma[1/2, (3*b*(c + d*x))
/d])*Sinh[(b*c)/d])/(d*Sqrt[c + d*x])) + (Cosh[(b*c)/d]*(-1 + 2*Cosh[(2*b*c)/d])*((Sqrt[b]*Sqrt[6*Pi]*(Erf[(Sq
rt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]))/Sqrt[d] - (2*Sqrt[2]*S
inh[(3*b*(c + d*x))/d])/Sqrt[c + d*x]))/(Sqrt[2]*d)) + Cosh[3*a]*((Cosh[(b*c)/d]*(-1 + 2*Cosh[(2*b*c)/d])*(-((
1 + E^((6*b*(c + d*x))/d))/E^((3*b*(c + d*x))/d)) + Sqrt[3]*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, (-3*b*(c + d*x
))/d] + Sqrt[3]*Sqrt[(b*(c + d*x))/d]*Gamma[1/2, (3*b*(c + d*x))/d]))/(d*Sqrt[c + d*x]) - ((1 + 2*Cosh[(2*b*c)
/d])*Sinh[(b*c)/d]*((Sqrt[b]*Sqrt[6*Pi]*(Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + Erfi[(Sqrt[3]*Sqrt[b]*
Sqrt[c + d*x])/Sqrt[d]]))/Sqrt[d] - (2*Sqrt[2]*Sinh[(3*b*(c + d*x))/d])/Sqrt[c + d*x]))/(Sqrt[2]*d)))/8 + (Cos
h[3*a]*(-(((1 + 2*Cosh[(2*b*c)/d])*(-((1 + E^((6*b*(c + d*x))/d))/E^((3*b*(c + d*x))/d)) + Sqrt[3]*Sqrt[-((b*(
c + d*x))/d)]*Gamma[1/2, (-3*b*(c + d*x))/d] + Sqrt[3]*Sqrt[(b*(c + d*x))/d]*Gamma[1/2, (3*b*(c + d*x))/d])*Si
nh[(b*c)/d])/(d*Sqrt[c + d*x])) + (Cosh[(b*c)/d]*(-1 + 2*Cosh[(2*b*c)/d])*((Sqrt[b]*Sqrt[6*Pi]*(Erf[(Sqrt[3]*S
qrt[b]*Sqrt[c + d*x])/Sqrt[d]] + Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]))/Sqrt[d] - (2*Sqrt[2]*Sinh[(3*
b*(c + d*x))/d])/Sqrt[c + d*x]))/(Sqrt[2]*d)) + Sinh[3*a]*((Cosh[(b*c)/d]*(-1 + 2*Cosh[(2*b*c)/d])*(-((1 + E^(
(6*b*(c + d*x))/d))/E^((3*b*(c + d*x))/d)) + Sqrt[3]*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, (-3*b*(c + d*x))/d] +
Sqrt[3]*Sqrt[(b*(c + d*x))/d]*Gamma[1/2, (3*b*(c + d*x))/d]))/(d*Sqrt[c + d*x]) - ((1 + 2*Cosh[(2*b*c)/d])*Si
nh[(b*c)/d]*((Sqrt[b]*Sqrt[6*Pi]*(Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c
+ d*x])/Sqrt[d]]))/Sqrt[d] - (2*Sqrt[2]*Sinh[(3*b*(c + d*x))/d])/Sqrt[c + d*x]))/(Sqrt[2]*d)))/4

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fricas [B]  time = 0.49, size = 1346, normalized size = 5.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(3)*sqrt(pi)*((d*x + c)*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)^3*sinh(-3*(b
*c - a*d)/d) + ((d*x + c)*cosh(-3*(b*c - a*d)/d) - (d*x + c)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((d*x
+ c)*cosh(b*x + a)*cosh(-3*(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 +
3*((d*x + c)*cosh(b*x + a)^2*cosh(-3*(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)^2*sinh(-3*(b*c - a*d)/d))*sinh(
b*x + a))*sqrt(b/d)*erf(sqrt(3)*sqrt(d*x + c)*sqrt(b/d)) - sqrt(3)*sqrt(pi)*((d*x + c)*cosh(b*x + a)^3*cosh(-3
*(b*c - a*d)/d) + (d*x + c)*cosh(b*x + a)^3*sinh(-3*(b*c - a*d)/d) + ((d*x + c)*cosh(-3*(b*c - a*d)/d) + (d*x
+ c)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((d*x + c)*cosh(b*x + a)*cosh(-3*(b*c - a*d)/d) + (d*x + c)*c
osh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*((d*x + c)*cosh(b*x + a)^2*cosh(-3*(b*c - a*d)/d) + (
d*x + c)*cosh(b*x + a)^2*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(-b/d)*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-b/d
)) - 3*sqrt(pi)*((d*x + c)*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)^3*sinh(-(b*c - a*d)/
d) + ((d*x + c)*cosh(-(b*c - a*d)/d) - (d*x + c)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((d*x + c)*cosh(b*x
+ a)*cosh(-(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*((d*x + c)*cosh
(b*x + a)^2*cosh(-(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)^2*sinh(-(b*c - a*d)/d))*sinh(b*x + a))*sqrt(b/d)*er
f(sqrt(d*x + c)*sqrt(b/d)) + 3*sqrt(pi)*((d*x + c)*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) + (d*x + c)*cosh(b*x +
a)^3*sinh(-(b*c - a*d)/d) + ((d*x + c)*cosh(-(b*c - a*d)/d) + (d*x + c)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^3
+ 3*((d*x + c)*cosh(b*x + a)*cosh(-(b*c - a*d)/d) + (d*x + c)*cosh(b*x + a)*sinh(-(b*c - a*d)/d))*sinh(b*x +
a)^2 + 3*((d*x + c)*cosh(b*x + a)^2*cosh(-(b*c - a*d)/d) + (d*x + c)*cosh(b*x + a)^2*sinh(-(b*c - a*d)/d))*sin
h(b*x + a))*sqrt(-b/d)*erf(sqrt(d*x + c)*sqrt(-b/d)) - (cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + si
nh(b*x + a)^6 + 3*(5*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^4 - 3*cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 - 3*cosh(
b*x + a))*sinh(b*x + a)^3 + 3*(5*cosh(b*x + a)^4 - 6*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 3*cosh(b*x + a)^2
+ 6*(cosh(b*x + a)^5 - 2*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) - 1)*sqrt(d*x + c))/((d^2*x + c*d)*cos
h(b*x + a)^3 + 3*(d^2*x + c*d)*cosh(b*x + a)^2*sinh(b*x + a) + 3*(d^2*x + c*d)*cosh(b*x + a)*sinh(b*x + a)^2 +
(d^2*x + c*d)*sinh(b*x + a)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{3}\left (b x +a \right )}{\left (d x +c \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 197, normalized size = 0.80 \[ \frac {\frac {\sqrt {3} \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {3 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} - \frac {\sqrt {3} \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} - \frac {3 \, \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (-a + \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {3 \, \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (a - \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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