\(\int \frac {\cosh ^3(a+b x)}{\sqrt {c+d x}} \, dx\) [59]

Optimal result

Integrand size = 18, antiderivative size = 228 \[ \int \frac {\cosh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\frac {3 e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {3 e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}} \]

[Out]

1/24*exp(-3*a+3*b*c/d)*erf(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*3^(1/2)*Pi^(1/2)/b^(1/2)/d^(1/2)+1/24*exp(3*
a-3*b*c/d)*erfi(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*3^(1/2)*Pi^(1/2)/b^(1/2)/d^(1/2)+3/8*exp(-a+b*c/d)*erf(
b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*Pi^(1/2)/b^(1/2)/d^(1/2)+3/8*exp(a-b*c/d)*erfi(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*P
i^(1/2)/b^(1/2)/d^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3393, 3388, 2211, 2235, 2236} \[ \int \frac {\cosh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\frac {3 \sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{3}} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {3 \sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{3}} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}} \]

[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 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2236

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]

Rule 3388

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 3393

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])
)

Rubi steps \begin{align*} \text {integral}& = \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 {\text {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 {\text {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 \text {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 \text {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] (verified)

Time = 0.15 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.84 \[ \int \frac {\cosh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\frac {e^{-3 \left (a+\frac {b c}{d}\right )} \left (\sqrt {3} e^{6 a} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {3 b (c+d x)}{d}\right )+9 e^{4 a+\frac {2 b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )-e^{\frac {4 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \left (9 e^{2 a} \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )+\sqrt {3} e^{\frac {2 b c}{d}} \Gamma \left (\frac {1}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{24 b \sqrt {c+d x}} \]

[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])

Maple [F]

\[\int \frac {\cosh \left (b x +a \right )^{3}}{\sqrt {d x +c}}d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\frac {\sqrt {3} \sqrt {\pi } \sqrt {\frac {b}{d}} {\left (\cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - \sqrt {3} \sqrt {\pi } \sqrt {-\frac {b}{d}} {\left (\cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) + 9 \, \sqrt {\pi } \sqrt {\frac {b}{d}} {\left (\cosh \left (-\frac {b c - a d}{d}\right ) - \sinh \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - 9 \, \sqrt {\pi } \sqrt {-\frac {b}{d}} {\left (\cosh \left (-\frac {b c - a d}{d}\right ) + \sinh \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right )}{24 \, b} \]

[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

Sympy [F]

\[ \int \frac {\cosh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {\cosh ^{3}{\left (a + b x \right )}}{\sqrt {c + d x}}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.78 \[ \int \frac {\cosh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\frac {\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )}}{\sqrt {-\frac {b}{d}}} + \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{\sqrt {\frac {b}{d}}} + \frac {9 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{\sqrt {-\frac {b}{d}}} + \frac {9 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{\sqrt {\frac {b}{d}}}}{24 \, d} \]

[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

Giac [F]

\[ \int \frac {\cosh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3}}{\sqrt {d x + c}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{\sqrt {c+d\,x}} \,d x \]

[In]

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

[Out]

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