3.51 \(\int \frac{\cosh ^2(a+b x)}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=138 \[ \frac{\sqrt{\frac{\pi }{2}} e^{\frac{2 b c}{d}-2 a} \text{Erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{2 b c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 \sqrt{b} \sqrt{d}}+\frac{\sqrt{c+d x}}{d} \]

[Out]

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

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Rubi [A]  time = 0.214058, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{\sqrt{\frac{\pi }{2}} e^{\frac{2 b c}{d}-2 a} \text{Erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{2 b c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 \sqrt{b} \sqrt{d}}+\frac{\sqrt{c+d x}}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^2/Sqrt[c + d*x],x]

[Out]

Sqrt[c + d*x]/d + (E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*Sqrt[b]*Sq
rt[d]) + (E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*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 ^2(a+b x)}{\sqrt{c+d x}} \, dx &=\int \left (\frac{1}{2 \sqrt{c+d x}}+\frac{\cosh (2 a+2 b x)}{2 \sqrt{c+d x}}\right ) \, dx\\ &=\frac{\sqrt{c+d x}}{d}+\frac{1}{2} \int \frac{\cosh (2 a+2 b x)}{\sqrt{c+d x}} \, dx\\ &=\frac{\sqrt{c+d x}}{d}+\frac{1}{4} \int \frac{e^{-i (2 i a+2 i b x)}}{\sqrt{c+d x}} \, dx+\frac{1}{4} \int \frac{e^{i (2 i a+2 i b x)}}{\sqrt{c+d x}} \, dx\\ &=\frac{\sqrt{c+d x}}{d}+\frac{\operatorname{Subst}\left (\int e^{i \left (2 i a-\frac{2 i b c}{d}\right )-\frac{2 b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{2 d}+\frac{\operatorname{Subst}\left (\int e^{-i \left (2 i a-\frac{2 i b c}{d}\right )+\frac{2 b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{2 d}\\ &=\frac{\sqrt{c+d x}}{d}+\frac{e^{-2 a+\frac{2 b c}{d}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 \sqrt{b} \sqrt{d}}+\frac{e^{2 a-\frac{2 b c}{d}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4 \sqrt{b} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.118473, size = 141, normalized size = 1.02 \[ \frac{e^{2 a-\frac{2 b c}{d}} \sqrt{-\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{1}{2},-\frac{2 b (c+d x)}{d}\right )}{4 \sqrt{2} b \sqrt{c+d x}}-\frac{e^{\frac{2 b c}{d}-2 a} \sqrt{\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{1}{2},\frac{2 b (c+d x)}{d}\right )}{4 \sqrt{2} b \sqrt{c+d x}}+\frac{\sqrt{c+d x}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^2/Sqrt[c + d*x],x]

[Out]

Sqrt[c + d*x]/d + (E^(2*a - (2*b*c)/d)*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, (-2*b*(c + d*x))/d])/(4*Sqrt[2]*b*S
qrt[c + d*x]) - (E^(-2*a + (2*b*c)/d)*Sqrt[(b*(c + d*x))/d]*Gamma[1/2, (2*b*(c + d*x))/d])/(4*Sqrt[2]*b*Sqrt[c
+ d*x])

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Maple [F]  time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}{\frac{1}{\sqrt{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.56816, size = 144, normalized size = 1.04 \begin{align*} \frac{\frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (2 \, a - \frac{2 \, b c}{d}\right )}}{\sqrt{-\frac{b}{d}}} + \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-2 \, a + \frac{2 \, b c}{d}\right )}}{\sqrt{\frac{b}{d}}} + 8 \, \sqrt{d x + c}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(sqrt(2)*sqrt(pi)*erf(sqrt(2)*sqrt(d*x + c)*sqrt(-b/d))*e^(2*a - 2*b*c/d)/sqrt(-b/d) + sqrt(2)*sqrt(pi)*er
f(sqrt(2)*sqrt(d*x + c)*sqrt(b/d))*e^(-2*a + 2*b*c/d)/sqrt(b/d) + 8*sqrt(d*x + c))/d

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Fricas [A]  time = 2.21131, size = 369, normalized size = 2.67 \begin{align*} \frac{\sqrt{2} \sqrt{\pi }{\left (d \cosh \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) - d \sinh \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right )\right )} \sqrt{\frac{b}{d}} \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) - \sqrt{2} \sqrt{\pi }{\left (d \cosh \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) + d \sinh \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right )\right )} \sqrt{-\frac{b}{d}} \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) + 8 \, \sqrt{d x + c} b}{8 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(sqrt(2)*sqrt(pi)*(d*cosh(-2*(b*c - a*d)/d) - d*sinh(-2*(b*c - a*d)/d))*sqrt(b/d)*erf(sqrt(2)*sqrt(d*x + c
)*sqrt(b/d)) - sqrt(2)*sqrt(pi)*(d*cosh(-2*(b*c - a*d)/d) + d*sinh(-2*(b*c - a*d)/d))*sqrt(-b/d)*erf(sqrt(2)*s
qrt(d*x + c)*sqrt(-b/d)) + 8*sqrt(d*x + c)*b)/(b*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh ^{2}{\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)**2/(d*x+c)**(1/2),x)

[Out]

Integral(cosh(a + b*x)**2/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 )^{2}}{\sqrt{d x + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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