3.55 \(\int \sqrt{c+d x} \sinh ^3(a+b x) \, dx\)

Optimal. Leaf size=275 \[ \frac{3 \sqrt{\pi } \sqrt{d} e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{3/2}}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} e^{\frac{3 b c}{d}-3 a} \text{Erf}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{48 b^{3/2}}+\frac{3 \sqrt{\pi } \sqrt{d} e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{3/2}}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} e^{3 a-\frac{3 b c}{d}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{48 b^{3/2}}-\frac{3 \sqrt{c+d x} \cosh (a+b x)}{4 b}+\frac{\sqrt{c+d x} \cosh (3 a+3 b x)}{12 b} \]

[Out]

(-3*Sqrt[c + d*x]*Cosh[a + b*x])/(4*b) + (Sqrt[c + d*x]*Cosh[3*a + 3*b*x])/(12*b) + (3*Sqrt[d]*E^(-a + (b*c)/d
)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(3/2)) - (Sqrt[d]*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(
Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2)) + (3*Sqrt[d]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt
[c + d*x])/Sqrt[d]])/(16*b^(3/2)) - (Sqrt[d]*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x
])/Sqrt[d]])/(48*b^(3/2))

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Rubi [A]  time = 0.521909, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3312, 3296, 3307, 2180, 2204, 2205} \[ \frac{3 \sqrt{\pi } \sqrt{d} e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{3/2}}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} e^{\frac{3 b c}{d}-3 a} \text{Erf}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{48 b^{3/2}}+\frac{3 \sqrt{\pi } \sqrt{d} e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{3/2}}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} e^{3 a-\frac{3 b c}{d}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{48 b^{3/2}}-\frac{3 \sqrt{c+d x} \cosh (a+b x)}{4 b}+\frac{\sqrt{c+d x} \cosh (3 a+3 b x)}{12 b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]*Sinh[a + b*x]^3,x]

[Out]

(-3*Sqrt[c + d*x]*Cosh[a + b*x])/(4*b) + (Sqrt[c + d*x]*Cosh[3*a + 3*b*x])/(12*b) + (3*Sqrt[d]*E^(-a + (b*c)/d
)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(3/2)) - (Sqrt[d]*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(
Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2)) + (3*Sqrt[d]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt
[c + d*x])/Sqrt[d]])/(16*b^(3/2)) - (Sqrt[d]*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x
])/Sqrt[d]])/(48*b^(3/2))

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 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 \sqrt{c+d x} \sinh ^3(a+b x) \, dx &=i \int \left (\frac{3}{4} i \sqrt{c+d x} \sinh (a+b x)-\frac{1}{4} i \sqrt{c+d x} \sinh (3 a+3 b x)\right ) \, dx\\ &=\frac{1}{4} \int \sqrt{c+d x} \sinh (3 a+3 b x) \, dx-\frac{3}{4} \int \sqrt{c+d x} \sinh (a+b x) \, dx\\ &=-\frac{3 \sqrt{c+d x} \cosh (a+b x)}{4 b}+\frac{\sqrt{c+d x} \cosh (3 a+3 b x)}{12 b}-\frac{d \int \frac{\cosh (3 a+3 b x)}{\sqrt{c+d x}} \, dx}{24 b}+\frac{(3 d) \int \frac{\cosh (a+b x)}{\sqrt{c+d x}} \, dx}{8 b}\\ &=-\frac{3 \sqrt{c+d x} \cosh (a+b x)}{4 b}+\frac{\sqrt{c+d x} \cosh (3 a+3 b x)}{12 b}-\frac{d \int \frac{e^{-i (3 i a+3 i b x)}}{\sqrt{c+d x}} \, dx}{48 b}-\frac{d \int \frac{e^{i (3 i a+3 i b x)}}{\sqrt{c+d x}} \, dx}{48 b}+\frac{(3 d) \int \frac{e^{-i (i a+i b x)}}{\sqrt{c+d x}} \, dx}{16 b}+\frac{(3 d) \int \frac{e^{i (i a+i b x)}}{\sqrt{c+d x}} \, dx}{16 b}\\ &=-\frac{3 \sqrt{c+d x} \cosh (a+b x)}{4 b}+\frac{\sqrt{c+d x} \cosh (3 a+3 b x)}{12 b}-\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 )}{24 b}-\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 )}{24 b}+\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 )}{8 b}+\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 )}{8 b}\\ &=-\frac{3 \sqrt{c+d x} \cosh (a+b x)}{4 b}+\frac{\sqrt{c+d x} \cosh (3 a+3 b x)}{12 b}+\frac{3 \sqrt{d} e^{-a+\frac{b c}{d}} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{3/2}}-\frac{\sqrt{d} 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 )}{48 b^{3/2}}+\frac{3 \sqrt{d} e^{a-\frac{b c}{d}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{3/2}}-\frac{\sqrt{d} 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 )}{48 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.291349, size = 209, normalized size = 0.76 \[ \frac{\sqrt{c+d x} e^{-3 \left (a+\frac{b c}{d}\right )} \left (\sqrt{3} e^{6 a} \sqrt{\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{3}{2},-\frac{3 b (c+d x)}{d}\right )-27 e^{4 a+\frac{2 b c}{d}} \sqrt{\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{3}{2},-\frac{b (c+d x)}{d}\right )+e^{\frac{4 b c}{d}} \sqrt{-\frac{b (c+d x)}{d}} \left (\sqrt{3} e^{\frac{2 b c}{d}} \text{Gamma}\left (\frac{3}{2},\frac{3 b (c+d x)}{d}\right )-27 e^{2 a} \text{Gamma}\left (\frac{3}{2},\frac{b (c+d x)}{d}\right )\right )\right )}{72 b \sqrt{-\frac{b^2 (c+d x)^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x]*(Sqrt[3]*E^(6*a)*Sqrt[(b*(c + d*x))/d]*Gamma[3/2, (-3*b*(c + d*x))/d] - 27*E^(4*a + (2*b*c)/d)*
Sqrt[(b*(c + d*x))/d]*Gamma[3/2, -((b*(c + d*x))/d)] + E^((4*b*c)/d)*Sqrt[-((b*(c + d*x))/d)]*(-27*E^(2*a)*Gam
ma[3/2, (b*(c + d*x))/d] + Sqrt[3]*E^((2*b*c)/d)*Gamma[3/2, (3*b*(c + d*x))/d])))/(72*b*E^(3*(a + (b*c)/d))*Sq
rt[-((b^2*(c + d*x)^2)/d^2)])

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \begin{align*} \int \left ( \sinh \left ( bx+a \right ) \right ) ^{3}\sqrt{dx+c}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.88603, size = 450, normalized size = 1.64 \begin{align*} -\frac{\frac{\sqrt{3} \sqrt{\pi } d \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (3 \, a - \frac{3 \, b c}{d}\right )}}{b \sqrt{-\frac{b}{d}}} + \frac{\sqrt{3} \sqrt{\pi } d \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )}}{b \sqrt{\frac{b}{d}}} - \frac{27 \, \sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (a - \frac{b c}{d}\right )}}{b \sqrt{-\frac{b}{d}}} - \frac{27 \, \sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-a + \frac{b c}{d}\right )}}{b \sqrt{\frac{b}{d}}} - \frac{6 \, \sqrt{d x + c} d e^{\left (3 \, a + \frac{3 \,{\left (d x + c\right )} b}{d} - \frac{3 \, b c}{d}\right )}}{b} + \frac{54 \, \sqrt{d x + c} d e^{\left (a + \frac{{\left (d x + c\right )} b}{d} - \frac{b c}{d}\right )}}{b} + \frac{54 \, \sqrt{d x + c} d e^{\left (-a - \frac{{\left (d x + c\right )} b}{d} + \frac{b c}{d}\right )}}{b} - \frac{6 \, \sqrt{d x + c} d e^{\left (-3 \, a - \frac{3 \,{\left (d x + c\right )} b}{d} + \frac{3 \, b c}{d}\right )}}{b}}{144 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(sqrt(3)*sqrt(pi)*d*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-b/d))*e^(3*a - 3*b*c/d)/(b*sqrt(-b/d)) + sqrt(3)*sq
rt(pi)*d*erf(sqrt(3)*sqrt(d*x + c)*sqrt(b/d))*e^(-3*a + 3*b*c/d)/(b*sqrt(b/d)) - 27*sqrt(pi)*d*erf(sqrt(d*x +
c)*sqrt(-b/d))*e^(a - b*c/d)/(b*sqrt(-b/d)) - 27*sqrt(pi)*d*erf(sqrt(d*x + c)*sqrt(b/d))*e^(-a + b*c/d)/(b*sqr
t(b/d)) - 6*sqrt(d*x + c)*d*e^(3*a + 3*(d*x + c)*b/d - 3*b*c/d)/b + 54*sqrt(d*x + c)*d*e^(a + (d*x + c)*b/d -
b*c/d)/b + 54*sqrt(d*x + c)*d*e^(-a - (d*x + c)*b/d + b*c/d)/b - 6*sqrt(d*x + c)*d*e^(-3*a - 3*(d*x + c)*b/d +
3*b*c/d)/b)/d

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Fricas [B]  time = 2.99826, size = 2966, normalized size = 10.79 \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)^(1/2),x, algorithm="fricas")

[Out]

-1/144*(sqrt(3)*sqrt(pi)*(d*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) - d*cosh(b*x + a)^3*sinh(-3*(b*c - a*d)/d)
+ (d*cosh(-3*(b*c - a*d)/d) - d*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d*cosh(b*x + a)*cosh(-3*(b*c - a*
d)/d) - d*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d*cosh(b*x + a)^2*cosh(-3*(b*c - a*d)/d)
- d*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)) - sq
rt(3)*sqrt(pi)*(d*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) + d*cosh(b*x + a)^3*sinh(-3*(b*c - a*d)/d) + (d*cosh(
-3*(b*c - a*d)/d) + d*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d*cosh(b*x + a)*cosh(-3*(b*c - a*d)/d) + d*
cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d*cosh(b*x + a)^2*cosh(-3*(b*c - a*d)/d) + d*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)) - 27*sqrt(pi
)*(d*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) - d*cosh(b*x + a)^3*sinh(-(b*c - a*d)/d) + (d*cosh(-(b*c - a*d)/d) -
d*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d*cosh(b*x + a)*cosh(-(b*c - a*d)/d) - d*cosh(b*x + a)*sinh(-(b*
c - a*d)/d))*sinh(b*x + a)^2 + 3*(d*cosh(b*x + a)^2*cosh(-(b*c - a*d)/d) - d*cosh(b*x + a)^2*sinh(-(b*c - a*d)
/d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(d*x + c)*sqrt(b/d)) + 27*sqrt(pi)*(d*cosh(b*x + a)^3*cosh(-(b*c - a*d)/
d) + d*cosh(b*x + a)^3*sinh(-(b*c - a*d)/d) + (d*cosh(-(b*c - a*d)/d) + d*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^
3 + 3*(d*cosh(b*x + a)*cosh(-(b*c - a*d)/d) + d*cosh(b*x + a)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d*cos
h(b*x + a)^2*cosh(-(b*c - a*d)/d) + d*cosh(b*x + a)^2*sinh(-(b*c - a*d)/d))*sinh(b*x + a))*sqrt(-b/d)*erf(sqrt
(d*x + c)*sqrt(-b/d)) - 6*(b*cosh(b*x + a)^6 + 6*b*cosh(b*x + a)*sinh(b*x + a)^5 + b*sinh(b*x + a)^6 - 9*b*cos
h(b*x + a)^4 + 3*(5*b*cosh(b*x + a)^2 - 3*b)*sinh(b*x + a)^4 + 4*(5*b*cosh(b*x + a)^3 - 9*b*cosh(b*x + a))*sin
h(b*x + a)^3 - 9*b*cosh(b*x + a)^2 + 3*(5*b*cosh(b*x + a)^4 - 18*b*cosh(b*x + a)^2 - 3*b)*sinh(b*x + a)^2 + 6*
(b*cosh(b*x + a)^5 - 6*b*cosh(b*x + a)^3 - 3*b*cosh(b*x + a))*sinh(b*x + a) + b)*sqrt(d*x + c))/(b^2*cosh(b*x
+ a)^3 + 3*b^2*cosh(b*x + a)^2*sinh(b*x + a) + 3*b^2*cosh(b*x + a)*sinh(b*x + a)^2 + b^2*sinh(b*x + a)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c + d*x)*sinh(a + b*x)**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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