Integrand size = 16, antiderivative size = 146 \[ \int (c+d x)^{3/2} \cosh (a+b x) \, dx=-\frac {3 d \sqrt {c+d x} \cosh (a+b x)}{2 b^2}+\frac {3 d^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {3 d^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {(c+d x)^{3/2} \sinh (a+b x)}{b} \]
(d*x+c)^(3/2)*sinh(b*x+a)/b+3/8*d^(3/2)*exp(-a+b*c/d)*erf(b^(1/2)*(d*x+c)^ (1/2)/d^(1/2))*Pi^(1/2)/b^(5/2)+3/8*d^(3/2)*exp(a-b*c/d)*erfi(b^(1/2)*(d*x +c)^(1/2)/d^(1/2))*Pi^(1/2)/b^(5/2)-3/2*d*cosh(b*x+a)*(d*x+c)^(1/2)/b^2
Time = 0.07 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.73 \[ \int (c+d x)^{3/2} \cosh (a+b x) \, dx=\frac {d e^{-a-\frac {b c}{d}} \sqrt {c+d x} \left (-\frac {e^{2 a} \Gamma \left (\frac {5}{2},-\frac {b (c+d x)}{d}\right )}{\sqrt {-\frac {b (c+d x)}{d}}}-\frac {e^{\frac {2 b c}{d}} \Gamma \left (\frac {5}{2},\frac {b (c+d x)}{d}\right )}{\sqrt {\frac {b (c+d x)}{d}}}\right )}{2 b^2} \]
(d*E^(-a - (b*c)/d)*Sqrt[c + d*x]*(-((E^(2*a)*Gamma[5/2, -((b*(c + d*x))/d )])/Sqrt[-((b*(c + d*x))/d)]) - (E^((2*b*c)/d)*Gamma[5/2, (b*(c + d*x))/d] )/Sqrt[(b*(c + d*x))/d]))/(2*b^2)
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3777, 26, 3042, 26, 3777, 3042, 3788, 26, 2611, 2633, 2634}
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.
| \(\displaystyle \int (c+d x)^{3/2} \cosh (a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^{3/2} \sin \left (i a+i b x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sinh (a+b x)}{b}-\frac {3 i d \int -i \sqrt {c+d x} \sinh (a+b x)dx}{2 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sinh (a+b x)}{b}-\frac {3 d \int \sqrt {c+d x} \sinh (a+b x)dx}{2 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sinh (a+b x)}{b}-\frac {3 d \int -i \sqrt {c+d x} \sin (i a+i b x)dx}{2 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \int \sqrt {c+d x} \sin (i a+i b x)dx}{2 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \int \frac {\cosh (a+b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {1}{2} i \int -\frac {i e^{a+b x}}{\sqrt {c+d x}}dx-\frac {1}{2} i \int \frac {i e^{-a-b x}}{\sqrt {c+d x}}dx\right )}{2 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {1}{2} \int \frac {e^{-a-b x}}{\sqrt {c+d x}}dx+\frac {1}{2} \int \frac {e^{a+b x}}{\sqrt {c+d x}}dx\right )}{2 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {\int e^{-a-\frac {b (c+d x)}{d}+\frac {b c}{d}}d\sqrt {c+d x}}{d}+\frac {\int e^{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{d}\right )}{2 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {\int e^{-a-\frac {b (c+d x)}{d}+\frac {b c}{d}}d\sqrt {c+d x}}{d}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {\sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}\) |
(((3*I)/2)*d*((I*Sqrt[c + d*x]*Cosh[a + b*x])/b - ((I/2)*d*((E^(-a + (b*c) /d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) + ( E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b] *Sqrt[d])))/b))/b + ((c + d*x)^(3/2)*Sinh[a + b*x])/b
3.1.42.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[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]
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]
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] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [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]
\[\int \left (d x +c \right )^{\frac {3}{2}} \cosh \left (b x +a \right )d x\]
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (110) = 220\).
Time = 0.27 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.65 \[ \int (c+d x)^{3/2} \cosh (a+b x) \, dx=\frac {3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) - d^{2} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left (d^{2} \cosh \left (-\frac {b c - a d}{d}\right ) - d^{2} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - 3 \, \sqrt {\pi } {\left (d^{2} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) + d^{2} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left (d^{2} \cosh \left (-\frac {b c - a d}{d}\right ) + d^{2} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) - 2 \, {\left (2 \, b^{2} d x + 2 \, b^{2} c - {\left (2 \, b^{2} d x + 2 \, b^{2} c - 3 \, b d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (2 \, b^{2} d x + 2 \, b^{2} c - 3 \, b d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (2 \, b^{2} d x + 2 \, b^{2} c - 3 \, b d\right )} \sinh \left (b x + a\right )^{2} + 3 \, b d\right )} \sqrt {d x + c}}{8 \, {\left (b^{3} \cosh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )\right )}} \]
1/8*(3*sqrt(pi)*(d^2*cosh(b*x + a)*cosh(-(b*c - a*d)/d) - d^2*cosh(b*x + a )*sinh(-(b*c - a*d)/d) + (d^2*cosh(-(b*c - a*d)/d) - d^2*sinh(-(b*c - a*d) /d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(d*x + c)*sqrt(b/d)) - 3*sqrt(pi)*(d ^2*cosh(b*x + a)*cosh(-(b*c - a*d)/d) + d^2*cosh(b*x + a)*sinh(-(b*c - a*d )/d) + (d^2*cosh(-(b*c - a*d)/d) + d^2*sinh(-(b*c - a*d)/d))*sinh(b*x + a) )*sqrt(-b/d)*erf(sqrt(d*x + c)*sqrt(-b/d)) - 2*(2*b^2*d*x + 2*b^2*c - (2*b ^2*d*x + 2*b^2*c - 3*b*d)*cosh(b*x + a)^2 - 2*(2*b^2*d*x + 2*b^2*c - 3*b*d )*cosh(b*x + a)*sinh(b*x + a) - (2*b^2*d*x + 2*b^2*c - 3*b*d)*sinh(b*x + a )^2 + 3*b*d)*sqrt(d*x + c))/(b^3*cosh(b*x + a) + b^3*sinh(b*x + a))
\[ \int (c+d x)^{3/2} \cosh (a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \cosh {\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (110) = 220\).
Time = 0.19 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.84 \[ \int (c+d x)^{3/2} \cosh (a+b x) \, dx=\frac {16 \, {\left (d x + c\right )}^{\frac {5}{2}} \cosh \left (b x + a\right ) + \frac {{\left (\frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b^{3} \sqrt {-\frac {b}{d}}} + \frac {15 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b^{3} \sqrt {\frac {b}{d}}} - \frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{\left (\frac {b c}{d}\right )} + 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{\left (\frac {b c}{d}\right )} + 15 \, \sqrt {d x + c} d^{3} e^{\left (\frac {b c}{d}\right )}\right )} e^{\left (-a - \frac {{\left (d x + c\right )} b}{d}\right )}}{b^{3}} - \frac {2 \, {\left (4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d e^{a} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} e^{a} + 15 \, \sqrt {d x + c} d^{3} e^{a}\right )} e^{\left (\frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b^{3}}\right )} b}{d}}{40 \, d} \]
1/40*(16*(d*x + c)^(5/2)*cosh(b*x + a) + (15*sqrt(pi)*d^3*erf(sqrt(d*x + c )*sqrt(-b/d))*e^(a - b*c/d)/(b^3*sqrt(-b/d)) + 15*sqrt(pi)*d^3*erf(sqrt(d* x + c)*sqrt(b/d))*e^(-a + b*c/d)/(b^3*sqrt(b/d)) - 2*(4*(d*x + c)^(5/2)*b^ 2*d*e^(b*c/d) + 10*(d*x + c)^(3/2)*b*d^2*e^(b*c/d) + 15*sqrt(d*x + c)*d^3* e^(b*c/d))*e^(-a - (d*x + c)*b/d)/b^3 - 2*(4*(d*x + c)^(5/2)*b^2*d*e^a - 1 0*(d*x + c)^(3/2)*b*d^2*e^a + 15*sqrt(d*x + c)*d^3*e^a)*e^((d*x + c)*b/d - b*c/d)/b^3)*b/d)/d
Time = 0.30 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.38 \[ \int (c+d x)^{3/2} \cosh (a+b x) \, dx=-\frac {\frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {b d} \sqrt {d x + c}}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )}}{\sqrt {b d} b^{2}} + \frac {3 \, \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {\sqrt {-b d} \sqrt {d x + c}}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )}}{\sqrt {-b d} b^{2}} - \frac {2 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b^{2}} + \frac {2 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d + 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (-\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}}{b^{2}}}{8 \, d} \]
-1/8*(3*sqrt(pi)*d^3*erf(-sqrt(b*d)*sqrt(d*x + c)/d)*e^((b*c - a*d)/d)/(sq rt(b*d)*b^2) + 3*sqrt(pi)*d^3*erf(-sqrt(-b*d)*sqrt(d*x + c)/d)*e^(-(b*c - a*d)/d)/(sqrt(-b*d)*b^2) - 2*(2*(d*x + c)^(3/2)*b*d - 3*sqrt(d*x + c)*d^2) *e^(((d*x + c)*b - b*c + a*d)/d)/b^2 + 2*(2*(d*x + c)^(3/2)*b*d + 3*sqrt(d *x + c)*d^2)*e^(-((d*x + c)*b - b*c + a*d)/d)/b^2)/d
Timed out. \[ \int (c+d x)^{3/2} \cosh (a+b x) \, dx=\int \mathrm {cosh}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{3/2} \,d x \]