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

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

[Out]

(-2*Cosh[a + b*x]^3)/(d*Sqrt[c + d*x]) - (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)*Sqrt[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))

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Rubi [A]  time = 0.421196, 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, 3308, 2180, 2204, 2205} \[ -\frac{3 \sqrt{\pi } \sqrt{b} e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 d^{3/2}}-\frac{\sqrt{3 \pi } \sqrt{b} e^{\frac{3 b c}{d}-3 a} \text{Erf}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 d^{3/2}}+\frac{3 \sqrt{\pi } \sqrt{b} e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 d^{3/2}}+\frac{\sqrt{3 \pi } \sqrt{b} e^{3 a-\frac{3 b c}{d}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 d^{3/2}}-\frac{2 \cosh ^3(a+b x)}{d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cosh[a + b*x]^3)/(d*Sqrt[c + d*x]) - (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)*Sqrt[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))

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 3308

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

Rubi steps

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

Mathematica [B]  time = 2.73577, size = 717, normalized size = 2.91 \[ \frac{e^{-\frac{3 b (c+d x)}{d}} \left (\sqrt{3} \sqrt{d} e^{\frac{3 b (c+d x)}{d}} \sqrt{-\frac{b (c+d x)}{d}} \cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Gamma}\left (\frac{1}{2},-\frac{3 b (c+d x)}{d}\right )+\sqrt{3} \sqrt{d} e^{\frac{3 b (c+d x)}{d}} \sqrt{\frac{b (c+d x)}{d}} \cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Gamma}\left (\frac{1}{2},\frac{3 b (c+d x)}{d}\right )+3 \sqrt{d} e^{\frac{3 b (c+d x)}{d}} \sqrt{\frac{b (c+d x)}{d}} \cosh \left (a-\frac{b c}{d}\right ) \text{Gamma}\left (\frac{1}{2},\frac{b (c+d x)}{d}\right )-3 \sqrt{d} e^{\frac{3 b (c+d x)}{d}} \sqrt{\frac{b (c+d x)}{d}} \sinh \left (a-\frac{b c}{d}\right ) \text{Gamma}\left (\frac{1}{2},\frac{b (c+d x)}{d}\right )+3 \sqrt{d} e^{\frac{3 b (c+d x)}{d}} \sqrt{-\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{1}{2},-\frac{b (c+d x)}{d}\right ) \left (\sinh \left (a-\frac{b c}{d}\right )+\cosh \left (a-\frac{b c}{d}\right )\right )+\sqrt{3 \pi } \sqrt{b} \sqrt{c+d x} e^{\frac{3 b (c+d x)}{d}} \sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Erf}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )+\sqrt{3 \pi } \sqrt{b} \sqrt{c+d x} e^{\frac{3 b (c+d x)}{d}} \sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Erfi}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )-\sqrt{d} e^{\frac{6 b (c+d x)}{d}} \sinh \left (3 a-\frac{3 b c}{d}\right )+3 \sqrt{d} e^{\frac{2 b (c+d x)}{d}} \sinh \left (a-\frac{b c}{d}\right )-3 \sqrt{d} e^{\frac{4 b (c+d x)}{d}} \sinh \left (a-\frac{b c}{d}\right )-\sqrt{d} e^{\frac{6 b (c+d x)}{d}} \cosh \left (3 a-\frac{3 b c}{d}\right )-3 \sqrt{d} e^{\frac{2 b (c+d x)}{d}} \cosh \left (a-\frac{b c}{d}\right )-3 \sqrt{d} e^{\frac{4 b (c+d x)}{d}} \cosh \left (a-\frac{b c}{d}\right )+\sqrt{d} \sinh \left (3 a-\frac{3 b c}{d}\right )-\sqrt{d} \cosh \left (3 a-\frac{3 b c}{d}\right )\right )}{4 d^{3/2} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[d]*Cosh[3*a - (3*b*c)/d]) - Sqrt[d]*E^((6*b*(c + d*x))/d)*Cosh[3*a - (3*b*c)/d] - 3*Sqrt[d]*E^((2*b*(c
+ d*x))/d)*Cosh[a - (b*c)/d] - 3*Sqrt[d]*E^((4*b*(c + d*x))/d)*Cosh[a - (b*c)/d] + Sqrt[3]*Sqrt[d]*E^((3*b*(c
+ d*x))/d)*Sqrt[-((b*(c + d*x))/d)]*Cosh[3*a - (3*b*c)/d]*Gamma[1/2, (-3*b*(c + d*x))/d] + 3*Sqrt[d]*E^((3*b*
(c + d*x))/d)*Sqrt[(b*(c + d*x))/d]*Cosh[a - (b*c)/d]*Gamma[1/2, (b*(c + d*x))/d] + Sqrt[3]*Sqrt[d]*E^((3*b*(c
+ d*x))/d)*Sqrt[(b*(c + d*x))/d]*Cosh[3*a - (3*b*c)/d]*Gamma[1/2, (3*b*(c + d*x))/d] + Sqrt[d]*Sinh[3*a - (3*
b*c)/d] - Sqrt[d]*E^((6*b*(c + d*x))/d)*Sinh[3*a - (3*b*c)/d] + Sqrt[b]*E^((3*b*(c + d*x))/d)*Sqrt[3*Pi]*Sqrt[
c + d*x]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]*Sinh[3*a - (3*b*c)/d] + Sqrt[b]*E^((3*b*(c + d*x))/d)*Sq
rt[3*Pi]*Sqrt[c + d*x]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]*Sinh[3*a - (3*b*c)/d] + 3*Sqrt[d]*E^((2*b
*(c + d*x))/d)*Sinh[a - (b*c)/d] - 3*Sqrt[d]*E^((4*b*(c + d*x))/d)*Sinh[a - (b*c)/d] - 3*Sqrt[d]*E^((3*b*(c +
d*x))/d)*Sqrt[(b*(c + d*x))/d]*Gamma[1/2, (b*(c + d*x))/d]*Sinh[a - (b*c)/d] + 3*Sqrt[d]*E^((3*b*(c + d*x))/d)
*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, -((b*(c + d*x))/d)]*(Cosh[a - (b*c)/d] + Sinh[a - (b*c)/d]))/(4*d^(3/2)*E
^((3*b*(c + d*x))/d)*Sqrt[c + d*x])

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Maple [F]  time = 0.155, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cosh \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(cosh(b*x+a)^3/(d*x+c)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.36259, size = 265, 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(cosh(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)*sqr
t(-(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 = 2.37642, size = 3313, normalized size = 13.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(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)*
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)) + 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)*cos
h(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)*e
rf(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))*si
nh(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 + s
inh(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)*co
sh(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{\cosh ^{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(cosh(b*x+a)**3/(d*x+c)**(3/2),x)

[Out]

Integral(cosh(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{\cosh \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(cosh(b*x+a)^3/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

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