3.59 \(\int \frac{\cosh ^3(a+b x)}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=228 \[ \frac{3 \sqrt{\pi } e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\frac{\pi }{3}} e^{\frac{3 b c}{d}-3 a} \text{Erf}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}}+\frac{3 \sqrt{\pi } e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\frac{\pi }{3}} e^{3 a-\frac{3 b c}{d}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}} \]

[Out]

(3*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (E^(-3*a + (3*b*c)/d)
*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (3*E^(a - (b*c)/d)*Sqrt[Pi]*Er
fi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[
b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d])

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Rubi [A]  time = 0.373702, antiderivative size = 228, 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 = {3312, 3307, 2180, 2204, 2205} \[ \frac{3 \sqrt{\pi } e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\frac{\pi }{3}} e^{\frac{3 b c}{d}-3 a} \text{Erf}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}}+\frac{3 \sqrt{\pi } e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\frac{\pi }{3}} e^{3 a-\frac{3 b c}{d}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^3/Sqrt[c + d*x],x]

[Out]

(3*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (E^(-3*a + (3*b*c)/d)
*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (3*E^(a - (b*c)/d)*Sqrt[Pi]*Er
fi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[
b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && 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{\cosh ^3(a+b x)}{\sqrt{c+d x}} \, dx &=\int \left (\frac{3 \cosh (a+b x)}{4 \sqrt{c+d x}}+\frac{\cosh (3 a+3 b x)}{4 \sqrt{c+d x}}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\cosh (3 a+3 b x)}{\sqrt{c+d x}} \, dx+\frac{3}{4} \int \frac{\cosh (a+b x)}{\sqrt{c+d x}} \, dx\\ &=\frac{1}{8} \int \frac{e^{-i (3 i a+3 i b x)}}{\sqrt{c+d x}} \, dx+\frac{1}{8} \int \frac{e^{i (3 i a+3 i b x)}}{\sqrt{c+d x}} \, dx+\frac{3}{8} \int \frac{e^{-i (i a+i b x)}}{\sqrt{c+d x}} \, dx+\frac{3}{8} \int \frac{e^{i (i a+i b x)}}{\sqrt{c+d x}} \, dx\\ &=\frac{\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 )}{4 d}+\frac{\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 )}{4 d}+\frac{3 \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 )}{4 d}+\frac{3 \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 )}{4 d}\\ &=\frac{3 e^{-a+\frac{b c}{d}} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}}+\frac{e^{-3 a+\frac{3 b c}{d}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}}+\frac{3 e^{a-\frac{b c}{d}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}}+\frac{e^{3 a-\frac{3 b c}{d}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{8 \sqrt{b} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.206166, size = 192, normalized size = 0.84 \[ \frac{e^{-3 \left (a+\frac{b c}{d}\right )} \left (\sqrt{3} e^{6 a} \sqrt{-\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{1}{2},-\frac{3 b (c+d x)}{d}\right )+9 e^{4 a+\frac{2 b c}{d}} \sqrt{-\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{1}{2},-\frac{b (c+d x)}{d}\right )-e^{\frac{4 b c}{d}} \sqrt{\frac{b (c+d x)}{d}} \left (9 e^{2 a} \text{Gamma}\left (\frac{1}{2},\frac{b (c+d x)}{d}\right )+\sqrt{3} e^{\frac{2 b c}{d}} \text{Gamma}\left (\frac{1}{2},\frac{3 b (c+d x)}{d}\right )\right )\right )}{24 b \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^3/Sqrt[c + d*x],x]

[Out]

(Sqrt[3]*E^(6*a)*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, (-3*b*(c + d*x))/d] + 9*E^(4*a + (2*b*c)/d)*Sqrt[-((b*(c
+ d*x))/d)]*Gamma[1/2, -((b*(c + d*x))/d)] - E^((4*b*c)/d)*Sqrt[(b*(c + d*x))/d]*(9*E^(2*a)*Gamma[1/2, (b*(c +
d*x))/d] + Sqrt[3]*E^((2*b*c)/d)*Gamma[1/2, (3*b*(c + d*x))/d]))/(24*b*E^(3*(a + (b*c)/d))*Sqrt[c + d*x])

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Maple [F]  time = 0.123, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}{\frac{1}{\sqrt{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^3/(d*x+c)^(1/2),x)

[Out]

int(cosh(b*x+a)^3/(d*x+c)^(1/2),x)

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Maxima [A]  time = 1.62049, size = 239, normalized size = 1.05 \begin{align*} \frac{\frac{\sqrt{3} \sqrt{\pi } \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (3 \, a - \frac{3 \, b c}{d}\right )}}{\sqrt{-\frac{b}{d}}} + \frac{\sqrt{3} \sqrt{\pi } \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )}}{\sqrt{\frac{b}{d}}} + \frac{9 \, \sqrt{\pi } \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (a - \frac{b c}{d}\right )}}{\sqrt{-\frac{b}{d}}} + \frac{9 \, \sqrt{\pi } \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-a + \frac{b c}{d}\right )}}{\sqrt{\frac{b}{d}}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

1/24*(sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-b/d))*e^(3*a - 3*b*c/d)/sqrt(-b/d) + sqrt(3)*sqrt(pi)*e
rf(sqrt(3)*sqrt(d*x + c)*sqrt(b/d))*e^(-3*a + 3*b*c/d)/sqrt(b/d) + 9*sqrt(pi)*erf(sqrt(d*x + c)*sqrt(-b/d))*e^
(a - b*c/d)/sqrt(-b/d) + 9*sqrt(pi)*erf(sqrt(d*x + c)*sqrt(b/d))*e^(-a + b*c/d)/sqrt(b/d))/d

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Fricas [A]  time = 2.08736, size = 594, normalized size = 2.61 \begin{align*} \frac{\sqrt{3} \sqrt{\pi } \sqrt{\frac{b}{d}}{\left (\cosh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) - \sinh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right )\right )} \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) - \sqrt{3} \sqrt{\pi } \sqrt{-\frac{b}{d}}{\left (\cosh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) + \sinh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right )\right )} \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) + 9 \, \sqrt{\pi } \sqrt{\frac{b}{d}}{\left (\cosh \left (-\frac{b c - a d}{d}\right ) - \sinh \left (-\frac{b c - a d}{d}\right )\right )} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) - 9 \, \sqrt{\pi } \sqrt{-\frac{b}{d}}{\left (\cosh \left (-\frac{b c - a d}{d}\right ) + \sinh \left (-\frac{b c - a d}{d}\right )\right )} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right )}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

1/24*(sqrt(3)*sqrt(pi)*sqrt(b/d)*(cosh(-3*(b*c - a*d)/d) - sinh(-3*(b*c - a*d)/d))*erf(sqrt(3)*sqrt(d*x + c)*s
qrt(b/d)) - sqrt(3)*sqrt(pi)*sqrt(-b/d)*(cosh(-3*(b*c - a*d)/d) + sinh(-3*(b*c - a*d)/d))*erf(sqrt(3)*sqrt(d*x
+ c)*sqrt(-b/d)) + 9*sqrt(pi)*sqrt(b/d)*(cosh(-(b*c - a*d)/d) - sinh(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(
b/d)) - 9*sqrt(pi)*sqrt(-b/d)*(cosh(-(b*c - a*d)/d) + sinh(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-b/d)))/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh ^{3}{\left (a + b x \right )}}{\sqrt{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**3/(d*x+c)**(1/2),x)

[Out]

Integral(cosh(a + b*x)**3/sqrt(c + d*x), 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}}{\sqrt{d x + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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