Integrand size = 18, antiderivative size = 166 \[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\frac {(c+d x)^{3/2}}{3 d}+\frac {\sqrt {d} e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {d} e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}+\frac {\sqrt {c+d x} \sinh (2 a+2 b x)}{4 b} \] Output:
1/3*(d*x+c)^(3/2)/d+1/32*d^(1/2)*exp(-2*a+2*b*c/d)*2^(1/2)*Pi^(1/2)*erf(2^ (1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(3/2)-1/32*d^(1/2)*exp(2*a-2*b*c/d) *2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(3/2)+1/4* (d*x+c)^(1/2)*sinh(2*b*x+2*a)/b
Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.78 \[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\frac {1}{48} \sqrt {c+d x} \left (\frac {16 (c+d x)}{d}+\frac {3 \sqrt {2} e^{2 a-\frac {2 b c}{d}} \Gamma \left (\frac {3}{2},-\frac {2 b (c+d x)}{d}\right )}{b \sqrt {-\frac {b (c+d x)}{d}}}-\frac {3 \sqrt {2} e^{-2 a+\frac {2 b c}{d}} \Gamma \left (\frac {3}{2},\frac {2 b (c+d x)}{d}\right )}{b \sqrt {\frac {b (c+d x)}{d}}}\right ) \] Input:
Integrate[Sqrt[c + d*x]*Cosh[a + b*x]^2,x]
Output:
(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
Time = 0.51 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3793, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {c+d x} \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \int \left (\frac {1}{2} \sqrt {c+d x} \cosh (2 a+2 b x)+\frac {1}{2} \sqrt {c+d x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\frac {\pi }{2}} \sqrt {d} e^{\frac {2 b c}{d}-2 a} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} e^{2 a-\frac {2 b c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}+\frac {\sqrt {c+d x} \sinh (2 a+2 b x)}{4 b}+\frac {(c+d x)^{3/2}}{3 d}\) |
Input:
Int[Sqrt[c + d*x]*Cosh[a + b*x]^2,x]
Output:
(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)
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[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]))
\[\int \sqrt {d x +c}\, \cosh \left (b x +a \right )^{2}d x\]
Input:
int((d*x+c)^(1/2)*cosh(b*x+a)^2,x)
Output:
int((d*x+c)^(1/2)*cosh(b*x+a)^2,x)
Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (122) = 244\).
Time = 0.10 (sec) , antiderivative size = 590, normalized size of antiderivative = 3.55 \[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(1/2)*cosh(b*x+a)^2,x, algorithm="fricas")
Output:
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))*s qrt(b/d)*erf(sqrt(2)*sqrt(d*x + c)*sqrt(b/d)) + 3*sqrt(2)*sqrt(pi)*(d^2*co sh(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)
\[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\int \sqrt {c + d x} \cosh ^{2}{\left (a + b x \right )}\, dx \] Input:
integrate((d*x+c)**(1/2)*cosh(b*x+a)**2,x)
Output:
Integral(sqrt(c + d*x)*cosh(a + b*x)**2, x)
Time = 0.14 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.14 \[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=-\frac {\frac {3 \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )}}{b \sqrt {-\frac {b}{d}}} - \frac {3 \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )}}{b \sqrt {\frac {b}{d}}} - 32 \, {\left (d x + c\right )}^{\frac {3}{2}} - \frac {12 \, \sqrt {d x + c} d e^{\left (2 \, a + \frac {2 \, {\left (d x + c\right )} b}{d} - \frac {2 \, b c}{d}\right )}}{b} + \frac {12 \, \sqrt {d x + c} d e^{\left (-2 \, a - \frac {2 \, {\left (d x + c\right )} b}{d} + \frac {2 \, b c}{d}\right )}}{b}}{96 \, d} \] Input:
integrate((d*x+c)^(1/2)*cosh(b*x+a)^2,x, algorithm="maxima")
Output:
-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*sqrt (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
\[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\int { \sqrt {d x + c} \cosh \left (b x + a\right )^{2} \,d x } \] Input:
integrate((d*x+c)^(1/2)*cosh(b*x+a)^2,x, algorithm="giac")
Output:
integrate(sqrt(d*x + c)*cosh(b*x + a)^2, x)
Timed out. \[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\int {\mathrm {cosh}\left (a+b\,x\right )}^2\,\sqrt {c+d\,x} \,d x \] Input:
int(cosh(a + b*x)^2*(c + d*x)^(1/2),x)
Output:
int(cosh(a + b*x)^2*(c + d*x)^(1/2), x)
\[ \int \sqrt {c+d x} \cosh ^2(a+b x) \, dx=\int \sqrt {d x +c}\, \cosh \left (b x +a \right )^{2}d x \] Input:
int((d*x+c)^(1/2)*cosh(b*x+a)^2,x)
Output:
int(sqrt(c + d*x)*cosh(a + b*x)**2,x)