\(\int \sqrt {c+d x} \cosh ^2(a+b x) \, dx\) [50]

Optimal result

Integrand size = 18, antiderivative size = 166 \[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\frac {(c+d x)^{3/2}}{3 d}+\frac {\sqrt {d} e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {d} e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}+\frac {\sqrt {c+d x} \sinh (2 a+2 b x)}{4 b} \]

[Out]

1/3*(d*x+c)^(3/2)/d+1/32*exp(-2*a+2*b*c/d)*erf(2^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*d^(1/2)*2^(1/2)*Pi^(1/2)
/b^(3/2)-1/32*exp(2*a-2*b*c/d)*erfi(2^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))*d^(1/2)*2^(1/2)*Pi^(1/2)/b^(3/2)+1/
4*sinh(2*b*x+2*a)*(d*x+c)^(1/2)/b

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3393, 3377, 3389, 2211, 2235, 2236} \[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} e^{\frac {2 b c}{d}-2 a} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} e^{2 a-\frac {2 b c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}+\frac {\sqrt {c+d x} \sinh (2 a+2 b x)}{4 b}+\frac {(c+d x)^{3/2}}{3 d} \]

[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 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 3377

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 3389

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 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 {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 {\text {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 {\text {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] (verified)

Time = 0.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.78 \[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\frac {1}{48} \sqrt {c+d x} \left (\frac {16 (c+d x)}{d}+\frac {3 \sqrt {2} e^{2 a-\frac {2 b c}{d}} \Gamma \left (\frac {3}{2},-\frac {2 b (c+d x)}{d}\right )}{b \sqrt {-\frac {b (c+d x)}{d}}}-\frac {3 \sqrt {2} e^{-2 a+\frac {2 b c}{d}} \Gamma \left (\frac {3}{2},\frac {2 b (c+d x)}{d}\right )}{b \sqrt {\frac {b (c+d x)}{d}}}\right ) \]

[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

Maple [F]

\[\int \cosh \left (b x +a \right )^{2} \sqrt {d x +c}d x\]

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (122) = 244\).

Time = 0.26 (sec) , antiderivative size = 590, normalized size of antiderivative = 3.55 \[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\frac {3 \, \sqrt {2} \sqrt {\pi } {\left (d^{2} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{2} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (d^{2} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{2} \cosh \left (b x + a\right ) \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) + 3 \, \sqrt {2} \sqrt {\pi } {\left (d^{2} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{2} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (d^{2} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{2} \cosh \left (b x + a\right ) \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) + 4 \, {\left (3 \, b d \cosh \left (b x + a\right )^{4} + 12 \, b d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \, b d \sinh \left (b x + a\right )^{4} + 8 \, {\left (b^{2} d x + b^{2} c\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (4 \, b^{2} d x + 9 \, b d \cosh \left (b x + a\right )^{2} + 4 \, b^{2} c\right )} \sinh \left (b x + a\right )^{2} - 3 \, b d + 4 \, {\left (3 \, b d \cosh \left (b x + a\right )^{3} + 4 \, {\left (b^{2} d x + b^{2} c\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {d x + c}}{96 \, {\left (b^{2} d \cosh \left (b x + a\right )^{2} + 2 \, b^{2} d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} d \sinh \left (b x + a\right )^{2}\right )}} \]

[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)

Sympy [F]

\[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\int \sqrt {c + d x} \cosh ^{2}{\left (a + b x \right )}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.14 \[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=-\frac {\frac {3 \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )}}{b \sqrt {-\frac {b}{d}}} - \frac {3 \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )}}{b \sqrt {\frac {b}{d}}} - 32 \, {\left (d x + c\right )}^{\frac {3}{2}} - \frac {12 \, \sqrt {d x + c} d e^{\left (2 \, a + \frac {2 \, {\left (d x + c\right )} b}{d} - \frac {2 \, b c}{d}\right )}}{b} + \frac {12 \, \sqrt {d x + c} d e^{\left (-2 \, a - \frac {2 \, {\left (d x + c\right )} b}{d} + \frac {2 \, b c}{d}\right )}}{b}}{96 \, d} \]

[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

Giac [F]

\[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\int { \sqrt {d x + c} \cosh \left (b x + a\right )^{2} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\int {\mathrm {cosh}\left (a+b\,x\right )}^2\,\sqrt {c+d\,x} \,d x \]

[In]

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

[Out]

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