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

Optimal. Leaf size=166 \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{d} e^{\frac{2 b c}{d}-2 a} \text{Erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} e^{2 a-\frac{2 b c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{3/2}}+\frac{\sqrt{c+d x} \sinh (2 a+2 b x)}{4 b}+\frac{(c+d x)^{3/2}}{3 d} \]

[Out]

(c + d*x)^(3/2)/(3*d) + (Sqrt[d]*E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])
/(16*b^(3/2)) - (Sqrt[d]*E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(
3/2)) + (Sqrt[c + d*x]*Sinh[2*a + 2*b*x])/(4*b)

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Rubi [A]  time = 0.269155, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3312, 3296, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{d} e^{\frac{2 b c}{d}-2 a} \text{Erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} e^{2 a-\frac{2 b c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{3/2}}+\frac{\sqrt{c+d x} \sinh (2 a+2 b x)}{4 b}+\frac{(c+d x)^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]*Cosh[a + b*x]^2,x]

[Out]

(c + d*x)^(3/2)/(3*d) + (Sqrt[d]*E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])
/(16*b^(3/2)) - (Sqrt[d]*E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(
3/2)) + (Sqrt[c + d*x]*Sinh[2*a + 2*b*x])/(4*b)

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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 \sqrt{c+d x} \cosh ^2(a+b x) \, dx &=\int \left (\frac{1}{2} \sqrt{c+d x}+\frac{1}{2} \sqrt{c+d x} \cosh (2 a+2 b x)\right ) \, dx\\ &=\frac{(c+d x)^{3/2}}{3 d}+\frac{1}{2} \int \sqrt{c+d x} \cosh (2 a+2 b x) \, dx\\ &=\frac{(c+d x)^{3/2}}{3 d}+\frac{\sqrt{c+d x} \sinh (2 a+2 b x)}{4 b}-\frac{d \int \frac{\sinh (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{8 b}\\ &=\frac{(c+d x)^{3/2}}{3 d}+\frac{\sqrt{c+d x} \sinh (2 a+2 b x)}{4 b}-\frac{d \int \frac{e^{-i (2 i a+2 i b x)}}{\sqrt{c+d x}} \, dx}{16 b}+\frac{d \int \frac{e^{i (2 i a+2 i b x)}}{\sqrt{c+d x}} \, dx}{16 b}\\ &=\frac{(c+d x)^{3/2}}{3 d}+\frac{\sqrt{c+d x} \sinh (2 a+2 b x)}{4 b}+\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 )}{8 b}-\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 )}{8 b}\\ &=\frac{(c+d x)^{3/2}}{3 d}+\frac{\sqrt{d} 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 )}{16 b^{3/2}}-\frac{\sqrt{d} 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 )}{16 b^{3/2}}+\frac{\sqrt{c+d x} \sinh (2 a+2 b x)}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.439951, size = 129, normalized size = 0.78 \[ \frac{1}{48} \sqrt{c+d x} \left (\frac{3 \sqrt{2} e^{2 a-\frac{2 b c}{d}} \text{Gamma}\left (\frac{3}{2},-\frac{2 b (c+d x)}{d}\right )}{b \sqrt{-\frac{b (c+d x)}{d}}}-\frac{3 \sqrt{2} e^{\frac{2 b c}{d}-2 a} \text{Gamma}\left (\frac{3}{2},\frac{2 b (c+d x)}{d}\right )}{b \sqrt{\frac{b (c+d x)}{d}}}+\frac{16 (c+d x)}{d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x]*((16*(c + d*x))/d + (3*Sqrt[2]*E^(2*a - (2*b*c)/d)*Gamma[3/2, (-2*b*(c + d*x))/d])/(b*Sqrt[-((b
*(c + d*x))/d)]) - (3*Sqrt[2]*E^(-2*a + (2*b*c)/d)*Gamma[3/2, (2*b*(c + d*x))/d])/(b*Sqrt[(b*(c + d*x))/d])))/
48

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \begin{align*} \int \left ( \cosh \left ( bx+a \right ) \right ) ^{2}\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 = 1.54858, size = 255, normalized size = 1.54 \begin{align*} -\frac{\frac{3 \, \sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (2 \, a - \frac{2 \, b c}{d}\right )}}{b \sqrt{-\frac{b}{d}}} - \frac{3 \, \sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-2 \, a + \frac{2 \, b c}{d}\right )}}{b \sqrt{\frac{b}{d}}} - 32 \,{\left (d x + c\right )}^{\frac{3}{2}} - \frac{12 \, \sqrt{d x + c} d e^{\left (2 \, a + \frac{2 \,{\left (d x + c\right )} b}{d} - \frac{2 \, b c}{d}\right )}}{b} + \frac{12 \, \sqrt{d x + c} d e^{\left (-2 \, a - \frac{2 \,{\left (d x + c\right )} b}{d} + \frac{2 \, b c}{d}\right )}}{b}}{96 \, 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]

-1/96*(3*sqrt(2)*sqrt(pi)*d*erf(sqrt(2)*sqrt(d*x + c)*sqrt(-b/d))*e^(2*a - 2*b*c/d)/(b*sqrt(-b/d)) - 3*sqrt(2)
*sqrt(pi)*d*erf(sqrt(2)*sqrt(d*x + c)*sqrt(b/d))*e^(-2*a + 2*b*c/d)/(b*sqrt(b/d)) - 32*(d*x + c)^(3/2) - 12*sq
rt(d*x + c)*d*e^(2*a + 2*(d*x + c)*b/d - 2*b*c/d)/b + 12*sqrt(d*x + c)*d*e^(-2*a - 2*(d*x + c)*b/d + 2*b*c/d)/
b)/d

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Fricas [B]  time = 2.21522, size = 1438, normalized size = 8.66 \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)^2*(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

1/96*(3*sqrt(2)*sqrt(pi)*(d^2*cosh(b*x + a)^2*cosh(-2*(b*c - a*d)/d) - d^2*cosh(b*x + a)^2*sinh(-2*(b*c - a*d)
/d) + (d^2*cosh(-2*(b*c - a*d)/d) - d^2*sinh(-2*(b*c - a*d)/d))*sinh(b*x + a)^2 + 2*(d^2*cosh(b*x + a)*cosh(-2
*(b*c - a*d)/d) - d^2*cosh(b*x + a)*sinh(-2*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(2)*sqrt(d*x + c)
*sqrt(b/d)) + 3*sqrt(2)*sqrt(pi)*(d^2*cosh(b*x + a)^2*cosh(-2*(b*c - a*d)/d) + d^2*cosh(b*x + a)^2*sinh(-2*(b*
c - a*d)/d) + (d^2*cosh(-2*(b*c - a*d)/d) + d^2*sinh(-2*(b*c - a*d)/d))*sinh(b*x + a)^2 + 2*(d^2*cosh(b*x + a)
*cosh(-2*(b*c - a*d)/d) + d^2*cosh(b*x + a)*sinh(-2*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(-b/d)*erf(sqrt(2)*sqrt
(d*x + c)*sqrt(-b/d)) + 4*(3*b*d*cosh(b*x + a)^4 + 12*b*d*cosh(b*x + a)*sinh(b*x + a)^3 + 3*b*d*sinh(b*x + a)^
4 + 8*(b^2*d*x + b^2*c)*cosh(b*x + a)^2 + 2*(4*b^2*d*x + 9*b*d*cosh(b*x + a)^2 + 4*b^2*c)*sinh(b*x + a)^2 - 3*
b*d + 4*(3*b*d*cosh(b*x + a)^3 + 4*(b^2*d*x + b^2*c)*cosh(b*x + a))*sinh(b*x + a))*sqrt(d*x + c))/(b^2*d*cosh(
b*x + a)^2 + 2*b^2*d*cosh(b*x + a)*sinh(b*x + a) + b^2*d*sinh(b*x + a)^2)

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

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