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*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])
________________________________________________________________________________________
Rubi [A] time = 0.453834, antiderivative size = 246, normalized size of antiderivative = 1.,
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 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 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 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]
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*}
Mathematica [B] time = 10.2652, 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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Maple [F] time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sinh \left ( bx+a \right ) \right ) ^{3} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
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 = 1.47258, size = 266, normalized size = 1.08 \begin{align*} \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} \end{align*}
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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Fricas [B] time = 3.01363, size = 3312, normalized size = 13.46 \begin{align*} \text{result too large to display} \end{align*}
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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac{3}{2}}}\, dx \end{align*}
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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
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)