Integrand size = 18, antiderivative size = 211 \[ \int (c+d x)^{3/2} \cosh ^2(a+b x) \, dx=\frac {3 d \sqrt {c+d x}}{16 b^2}+\frac {(c+d x)^{5/2}}{5 d}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {3 d^{3/2} e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}+\frac {3 d^{3/2} e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{5/2}}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b} \] Output:
3/16*d*(d*x+c)^(1/2)/b^2+1/5*(d*x+c)^(5/2)/d-3/8*d*(d*x+c)^(1/2)*cosh(b*x+ a)^2/b^2+3/128*d^(3/2)*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^(5/2)+3/128*d^(3/2)*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^(5/2)+1/2*(d*x+c)^( 3/2)*cosh(b*x+a)*sinh(b*x+a)/b
Time = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.65 \[ \int (c+d x)^{3/2} \cosh ^2(a+b x) \, dx=\frac {\sqrt {c+d x} \left (32 (c+d x)^2-\frac {5 \sqrt {2} d^2 e^{2 a-\frac {2 b c}{d}} \Gamma \left (\frac {5}{2},-\frac {2 b (c+d x)}{d}\right )}{b^2 \sqrt {-\frac {b (c+d x)}{d}}}-\frac {5 \sqrt {2} d^2 e^{-2 a+\frac {2 b c}{d}} \Gamma \left (\frac {5}{2},\frac {2 b (c+d x)}{d}\right )}{b^2 \sqrt {\frac {b (c+d x)}{d}}}\right )}{160 d} \] Input:
Integrate[(c + d*x)^(3/2)*Cosh[a + b*x]^2,x]
Output:
(Sqrt[c + d*x]*(32*(c + d*x)^2 - (5*Sqrt[2]*d^2*E^(2*a - (2*b*c)/d)*Gamma[ 5/2, (-2*b*(c + d*x))/d])/(b^2*Sqrt[-((b*(c + d*x))/d)]) - (5*Sqrt[2]*d^2* E^(-2*a + (2*b*c)/d)*Gamma[5/2, (2*b*(c + d*x))/d])/(b^2*Sqrt[(b*(c + d*x) )/d])))/(160*d)
Time = 0.64 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3792, 17, 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.
| \(\displaystyle \int (c+d x)^{3/2} \cosh ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^{3/2} \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {3 d^2 \int \frac {\cosh ^2(a+b x)}{\sqrt {c+d x}}dx}{16 b^2}+\frac {1}{2} \int (c+d x)^{3/2}dx-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {3 d^2 \int \frac {\cosh ^2(a+b x)}{\sqrt {c+d x}}dx}{16 b^2}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^2}{\sqrt {c+d x}}dx}{16 b^2}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {3 d^2 \int \left (\frac {\cosh (2 a+2 b x)}{2 \sqrt {c+d x}}+\frac {1}{2 \sqrt {c+d x}}\right )dx}{16 b^2}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 d^2 \left (\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 b c}{d}-2 a} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {2 b c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 \sqrt {b} \sqrt {d}}+\frac {\sqrt {c+d x}}{d}\right )}{16 b^2}-\frac {3 d \sqrt {c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^{5/2}}{5 d}\) |
Input:
Int[(c + d*x)^(3/2)*Cosh[a + b*x]^2,x]
Output:
(c + d*x)^(5/2)/(5*d) - (3*d*Sqrt[c + d*x]*Cosh[a + b*x]^2)/(8*b^2) + (3*d ^2*(Sqrt[c + d*x]/d + (E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b ]*Sqrt[c + d*x])/Sqrt[d]])/(4*Sqrt[b]*Sqrt[d]) + (E^(2*a - (2*b*c)/d)*Sqrt [Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*Sqrt[b]*Sqrt[d])) )/(16*b^2) + ((c + d*x)^(3/2)*Cosh[a + b*x]*Sinh[a + b*x])/(2*b)
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
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]))
\[\int \left (d x +c \right )^{\frac {3}{2}} \cosh \left (b x +a \right )^{2}d x\]
Input:
int((d*x+c)^(3/2)*cosh(b*x+a)^2,x)
Output:
int((d*x+c)^(3/2)*cosh(b*x+a)^2,x)
Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (159) = 318\).
Time = 0.12 (sec) , antiderivative size = 755, normalized size of antiderivative = 3.58 \[ \int (c+d x)^{3/2} \cosh ^2(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(3/2)*cosh(b*x+a)^2,x, algorithm="fricas")
Output:
1/640*(15*sqrt(2)*sqrt(pi)*(d^3*cosh(b*x + a)^2*cosh(-2*(b*c - a*d)/d) - d ^3*cosh(b*x + a)^2*sinh(-2*(b*c - a*d)/d) + (d^3*cosh(-2*(b*c - a*d)/d) - d^3*sinh(-2*(b*c - a*d)/d))*sinh(b*x + a)^2 + 2*(d^3*cosh(b*x + a)*cosh(-2 *(b*c - a*d)/d) - d^3*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)) - 15*sqrt(2)*sqrt(pi)*(d^3 *cosh(b*x + a)^2*cosh(-2*(b*c - a*d)/d) + d^3*cosh(b*x + a)^2*sinh(-2*(b*c - a*d)/d) + (d^3*cosh(-2*(b*c - a*d)/d) + d^3*sinh(-2*(b*c - a*d)/d))*sin h(b*x + a)^2 + 2*(d^3*cosh(b*x + a)*cosh(-2*(b*c - a*d)/d) + d^3*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*(20*b^2*d^2*x - 5*(4*b^2*d^2*x + 4*b^2*c*d - 3*b*d^ 2)*cosh(b*x + a)^4 - 20*(4*b^2*d^2*x + 4*b^2*c*d - 3*b*d^2)*cosh(b*x + a)* sinh(b*x + a)^3 - 5*(4*b^2*d^2*x + 4*b^2*c*d - 3*b*d^2)*sinh(b*x + a)^4 + 20*b^2*c*d + 15*b*d^2 - 32*(b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*cosh(b*x + a)^2 - 2*(16*b^3*d^2*x^2 + 32*b^3*c*d*x + 16*b^3*c^2 + 15*(4*b^2*d^2*x + 4*b^2*c*d - 3*b*d^2)*cosh(b*x + a)^2)*sinh(b*x + a)^2 - 4*(5*(4*b^2*d^2*x + 4*b^2*c*d - 3*b*d^2)*cosh(b*x + a)^3 + 16*(b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*cosh(b*x + a))*sinh(b*x + a))*sqrt(d*x + c))/(b^3*d*cosh(b*x + a) ^2 + 2*b^3*d*cosh(b*x + a)*sinh(b*x + a) + b^3*d*sinh(b*x + a)^2)
\[ \int (c+d x)^{3/2} \cosh ^2(a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \cosh ^{2}{\left (a + b x \right )}\, dx \] Input:
integrate((d*x+c)**(3/2)*cosh(b*x+a)**2,x)
Output:
Integral((c + d*x)**(3/2)*cosh(a + b*x)**2, x)
Time = 0.15 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.13 \[ \int (c+d x)^{3/2} \cosh ^2(a+b x) \, dx=\frac {128 \, {\left (d x + c\right )}^{\frac {5}{2}} + \frac {15 \, \sqrt {2} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )}}{b^{2} \sqrt {-\frac {b}{d}}} + \frac {15 \, \sqrt {2} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )}}{b^{2} \sqrt {\frac {b}{d}}} - \frac {20 \, {\left (4 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (\frac {2 \, b c}{d}\right )} + 3 \, \sqrt {d x + c} d^{2} e^{\left (\frac {2 \, b c}{d}\right )}\right )} e^{\left (-2 \, a - \frac {2 \, {\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac {20 \, {\left (4 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (2 \, a\right )} - 3 \, \sqrt {d x + c} d^{2} e^{\left (2 \, a\right )}\right )} e^{\left (\frac {2 \, {\left (d x + c\right )} b}{d} - \frac {2 \, b c}{d}\right )}}{b^{2}}}{640 \, d} \] Input:
integrate((d*x+c)^(3/2)*cosh(b*x+a)^2,x, algorithm="maxima")
Output:
1/640*(128*(d*x + c)^(5/2) + 15*sqrt(2)*sqrt(pi)*d^2*erf(sqrt(2)*sqrt(d*x + c)*sqrt(-b/d))*e^(2*a - 2*b*c/d)/(b^2*sqrt(-b/d)) + 15*sqrt(2)*sqrt(pi)* d^2*erf(sqrt(2)*sqrt(d*x + c)*sqrt(b/d))*e^(-2*a + 2*b*c/d)/(b^2*sqrt(b/d) ) - 20*(4*(d*x + c)^(3/2)*b*d*e^(2*b*c/d) + 3*sqrt(d*x + c)*d^2*e^(2*b*c/d ))*e^(-2*a - 2*(d*x + c)*b/d)/b^2 + 20*(4*(d*x + c)^(3/2)*b*d*e^(2*a) - 3* sqrt(d*x + c)*d^2*e^(2*a))*e^(2*(d*x + c)*b/d - 2*b*c/d)/b^2)/d
\[ \int (c+d x)^{3/2} \cosh ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{\frac {3}{2}} \cosh \left (b x + a\right )^{2} \,d x } \] Input:
integrate((d*x+c)^(3/2)*cosh(b*x+a)^2,x, algorithm="giac")
Output:
integrate((d*x + c)^(3/2)*cosh(b*x + a)^2, x)
Timed out. \[ \int (c+d x)^{3/2} \cosh ^2(a+b x) \, dx=\int {\mathrm {cosh}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{3/2} \,d x \] Input:
int(cosh(a + b*x)^2*(c + d*x)^(3/2),x)
Output:
int(cosh(a + b*x)^2*(c + d*x)^(3/2), x)
\[ \int (c+d x)^{3/2} \cosh ^2(a+b x) \, dx=\left (\int \sqrt {d x +c}\, \cosh \left (b x +a \right )^{2} x d x \right ) d +\left (\int \sqrt {d x +c}\, \cosh \left (b x +a \right )^{2}d x \right ) c \] Input:
int((d*x+c)^(3/2)*cosh(b*x+a)^2,x)
Output:
int(sqrt(c + d*x)*cosh(a + b*x)**2*x,x)*d + int(sqrt(c + d*x)*cosh(a + b*x )**2,x)*c