Integrand size = 18, antiderivative size = 326 \[ \int (c+d x)^{3/2} \cosh ^3(a+b x) \, dx=-\frac {d \sqrt {c+d x} \cosh (a+b x)}{b^2}-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {9 d^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {d^{3/2} e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {9 d^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {d^{3/2} e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {2 (c+d x)^{3/2} \sinh (a+b x)}{3 b}+\frac {(c+d x)^{3/2} \cosh ^2(a+b x) \sinh (a+b x)}{3 b} \]
2/3*(d*x+c)^(3/2)*sinh(b*x+a)/b+1/3*(d*x+c)^(3/2)*cosh(b*x+a)^2*sinh(b*x+a )/b+1/288*d^(3/2)*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^(5/2)+1/288*d^(3/2)*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^(5/2)+9/32*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)+9/32*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)-d*cosh(b *x+a)*(d*x+c)^(1/2)/b^2-1/6*d*cosh(b*x+a)^3*(d*x+c)^(1/2)/b^2
Time = 0.33 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.64 \[ \int (c+d x)^{3/2} \cosh ^3(a+b x) \, dx=\frac {e^{-3 \left (a+\frac {b c}{d}\right )} (c+d x)^{5/2} \left (\sqrt {3} e^{6 a} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {3 b (c+d x)}{d}\right )+81 e^{4 a+\frac {2 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {b (c+d x)}{d}\right )+e^{\frac {4 b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \left (81 e^{2 a} \Gamma \left (\frac {5}{2},\frac {b (c+d x)}{d}\right )+\sqrt {3} e^{\frac {2 b c}{d}} \Gamma \left (\frac {5}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{216 d \left (-\frac {b^2 (c+d x)^2}{d^2}\right )^{3/2}} \]
((c + d*x)^(5/2)*(Sqrt[3]*E^(6*a)*Sqrt[(b*(c + d*x))/d]*Gamma[5/2, (-3*b*( c + d*x))/d] + 81*E^(4*a + (2*b*c)/d)*Sqrt[(b*(c + d*x))/d]*Gamma[5/2, -(( b*(c + d*x))/d)] + E^((4*b*c)/d)*Sqrt[-((b*(c + d*x))/d)]*(81*E^(2*a)*Gamm a[5/2, (b*(c + d*x))/d] + Sqrt[3]*E^((2*b*c)/d)*Gamma[5/2, (3*b*(c + d*x)) /d])))/(216*d*E^(3*(a + (b*c)/d))*(-((b^2*(c + d*x)^2)/d^2))^(3/2))
Result contains complex when optimal does not.
Time = 1.64 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.43, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {3042, 3792, 3042, 3777, 26, 3042, 26, 3777, 3042, 3788, 26, 2611, 2633, 2634, 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.
| \(\displaystyle \int (c+d x)^{3/2} \cosh ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^{3/2} \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {d^2 \int \frac {\cosh ^3(a+b x)}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \int (c+d x)^{3/2} \cosh (a+b x)dx-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \int (c+d x)^{3/2} \sin \left (i a+i b x+\frac {\pi }{2}\right )dx-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}-\frac {3 i d \int -i \sqrt {c+d x} \sinh (a+b x)dx}{2 b}\right )-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}-\frac {3 d \int \sqrt {c+d x} \sinh (a+b x)dx}{2 b}\right )-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}-\frac {3 d \int -i \sqrt {c+d x} \sin (i a+i b x)dx}{2 b}\right )-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \int \sqrt {c+d x} \sin (i a+i b x)dx}{2 b}\right )-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \int \frac {\cosh (a+b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\right )-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\right )-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {1}{2} i \int -\frac {i e^{a+b x}}{\sqrt {c+d x}}dx-\frac {1}{2} i \int \frac {i e^{-a-b x}}{\sqrt {c+d x}}dx\right )}{2 b}\right )}{2 b}\right )-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {1}{2} \int \frac {e^{-a-b x}}{\sqrt {c+d x}}dx+\frac {1}{2} \int \frac {e^{a+b x}}{\sqrt {c+d x}}dx\right )}{2 b}\right )}{2 b}\right )-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {\int e^{-a-\frac {b (c+d x)}{d}+\frac {b c}{d}}d\sqrt {c+d x}}{d}+\frac {\int e^{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{d}\right )}{2 b}\right )}{2 b}\right )-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {\int e^{-a-\frac {b (c+d x)}{d}+\frac {b c}{d}}d\sqrt {c+d x}}{d}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}\right )-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {d^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{12 b^2}-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {\sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}\right )+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {d^2 \int \left (\frac {3 \cosh (a+b x)}{4 \sqrt {c+d x}}+\frac {\cosh (3 a+3 b x)}{4 \sqrt {c+d x}}\right )dx}{12 b^2}-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {\sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}\right )+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \left (\frac {3 \sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{3}} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {3 \sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{3}} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}\right )}{12 b^2}-\frac {d \sqrt {c+d x} \cosh ^3(a+b x)}{6 b^2}+\frac {2}{3} \left (\frac {(c+d x)^{3/2} \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i \sqrt {c+d x} \cosh (a+b x)}{b}-\frac {i d \left (\frac {\sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}\right )+\frac {(c+d x)^{3/2} \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
-1/6*(d*Sqrt[c + d*x]*Cosh[a + b*x]^3)/b^2 + (d^2*((3*E^(-a + (b*c)/d)*Sqr t[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]*Erfi[(Sqrt[b]*Sqrt[c + d*x] )/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sq rt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d])))/(12*b^2) + (( c + d*x)^(3/2)*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b) + (2*((((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
3.1.57.3.1 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[((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]))
\[\int \left (d x +c \right )^{\frac {3}{2}} \cosh \left (b x +a \right )^{3}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1545 vs. \(2 (246) = 492\).
Time = 0.29 (sec) , antiderivative size = 1545, normalized size of antiderivative = 4.74 \[ \int (c+d x)^{3/2} \cosh ^3(a+b x) \, dx=\text {Too large to display} \]
1/288*(sqrt(3)*sqrt(pi)*(d^2*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) - d^2* cosh(b*x + a)^3*sinh(-3*(b*c - a*d)/d) + (d^2*cosh(-3*(b*c - a*d)/d) - d^2 *sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d^2*cosh(b*x + a)*cosh(-3*(b *c - a*d)/d) - d^2*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d^2*cosh(b*x + a)^2*cosh(-3*(b*c - a*d)/d) - d^2*cosh(b*x + a)^2*sinh( -3*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(3)*sqrt(d*x + c)*sqrt (b/d)) - sqrt(3)*sqrt(pi)*(d^2*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) + d^ 2*cosh(b*x + a)^3*sinh(-3*(b*c - a*d)/d) + (d^2*cosh(-3*(b*c - a*d)/d) + d ^2*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d^2*cosh(b*x + a)*cosh(-3* (b*c - a*d)/d) + d^2*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d^2*cosh(b*x + a)^2*cosh(-3*(b*c - a*d)/d) + d^2*cosh(b*x + a)^2*sin h(-3*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(-b/d)*erf(sqrt(3)*sqrt(d*x + c)*s qrt(-b/d)) + 81*sqrt(pi)*(d^2*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) - d^2*c osh(b*x + a)^3*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)^3 + 3*(d^2*cosh(b*x + a)*cosh(-(b*c - a*d) /d) - d^2*cosh(b*x + a)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d^2*cos h(b*x + a)^2*cosh(-(b*c - a*d)/d) - d^2*cosh(b*x + a)^2*sinh(-(b*c - a*d)/ d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(d*x + c)*sqrt(b/d)) - 81*sqrt(pi)*(d ^2*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) + d^2*cosh(b*x + a)^3*sinh(-(b*c - a*d)/d) + (d^2*cosh(-(b*c - a*d)/d) + d^2*sinh(-(b*c - a*d)/d))*sinh(b...
\[ \int (c+d x)^{3/2} \cosh ^3(a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \cosh ^{3}{\left (a + b x \right )}\, dx \]
Time = 0.28 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.32 \[ \int (c+d x)^{3/2} \cosh ^3(a+b x) \, dx=\frac {\frac {\sqrt {3} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )}}{b^{2} \sqrt {-\frac {b}{d}}} + \frac {\sqrt {3} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{b^{2} \sqrt {\frac {b}{d}}} + \frac {81 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b^{2} \sqrt {-\frac {b}{d}}} + \frac {81 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b^{2} \sqrt {\frac {b}{d}}} - \frac {54 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (\frac {b c}{d}\right )} + 3 \, \sqrt {d x + c} d^{2} e^{\left (\frac {b c}{d}\right )}\right )} e^{\left (-a - \frac {{\left (d x + c\right )} b}{d}\right )}}{b^{2}} - \frac {6 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (\frac {3 \, b c}{d}\right )} + \sqrt {d x + c} d^{2} e^{\left (\frac {3 \, b c}{d}\right )}\right )} e^{\left (-3 \, a - \frac {3 \, {\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac {6 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (3 \, a\right )} - \sqrt {d x + c} d^{2} e^{\left (3 \, a\right )}\right )} e^{\left (\frac {3 \, {\left (d x + c\right )} b}{d} - \frac {3 \, b c}{d}\right )}}{b^{2}} + \frac {54 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{a} - 3 \, \sqrt {d x + c} d^{2} e^{a}\right )} e^{\left (\frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b^{2}}}{288 \, d} \]
1/288*(sqrt(3)*sqrt(pi)*d^2*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-b/d))*e^(3*a - 3*b*c/d)/(b^2*sqrt(-b/d)) + sqrt(3)*sqrt(pi)*d^2*erf(sqrt(3)*sqrt(d*x + c )*sqrt(b/d))*e^(-3*a + 3*b*c/d)/(b^2*sqrt(b/d)) + 81*sqrt(pi)*d^2*erf(sqrt (d*x + c)*sqrt(-b/d))*e^(a - b*c/d)/(b^2*sqrt(-b/d)) + 81*sqrt(pi)*d^2*erf (sqrt(d*x + c)*sqrt(b/d))*e^(-a + b*c/d)/(b^2*sqrt(b/d)) - 54*(2*(d*x + c) ^(3/2)*b*d*e^(b*c/d) + 3*sqrt(d*x + c)*d^2*e^(b*c/d))*e^(-a - (d*x + c)*b/ d)/b^2 - 6*(2*(d*x + c)^(3/2)*b*d*e^(3*b*c/d) + sqrt(d*x + c)*d^2*e^(3*b*c /d))*e^(-3*a - 3*(d*x + c)*b/d)/b^2 + 6*(2*(d*x + c)^(3/2)*b*d*e^(3*a) - s qrt(d*x + c)*d^2*e^(3*a))*e^(3*(d*x + c)*b/d - 3*b*c/d)/b^2 + 54*(2*(d*x + c)^(3/2)*b*d*e^a - 3*sqrt(d*x + c)*d^2*e^a)*e^((d*x + c)*b/d - b*c/d)/b^2 )/d
\[ \int (c+d x)^{3/2} \cosh ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{\frac {3}{2}} \cosh \left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int (c+d x)^{3/2} \cosh ^3(a+b x) \, dx=\int {\mathrm {cosh}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{3/2} \,d x \]