\(\int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx\) [54]

Optimal result

Integrand size = 18, antiderivative size = 325 \[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=-\frac {2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac {9 d^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}-\frac {d^{3/2} e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}-\frac {9 d^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{5/2}}+\frac {d^{3/2} e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {d \sqrt {c+d x} \sinh (a+b x)}{b^2}+\frac {(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2} \] Output:

-2/3*(d*x+c)^(3/2)*cosh(b*x+a)/b+9/32*d^(3/2)*exp(-a+b*c/d)*Pi^(1/2)*erf(b 
^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(5/2)-1/288*d^(3/2)*exp(-3*a+3*b*c/d)*3^(1 
/2)*Pi^(1/2)*erf(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(5/2)-9/32*d^(3/ 
2)*exp(a-b*c/d)*Pi^(1/2)*erfi(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(5/2)+1/288 
*d^(3/2)*exp(3*a-3*b*c/d)*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*b^(1/2)*(d*x+c)^(1 
/2)/d^(1/2))/b^(5/2)+d*(d*x+c)^(1/2)*sinh(b*x+a)/b^2+1/3*(d*x+c)^(3/2)*cos 
h(b*x+a)*sinh(b*x+a)^2/b-1/6*d*(d*x+c)^(1/2)*sinh(b*x+a)^3/b^2
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.65 \[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=\frac {d e^{-3 \left (a+\frac {b c}{d}\right )} \sqrt {c+d x} \left (-\sqrt {3} e^{6 a} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {3 b (c+d x)}{d}\right )+81 e^{4 a+\frac {2 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {b (c+d x)}{d}\right )+e^{\frac {4 b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \left (-81 e^{2 a} \Gamma \left (\frac {5}{2},\frac {b (c+d x)}{d}\right )+\sqrt {3} e^{\frac {2 b c}{d}} \Gamma \left (\frac {5}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{216 b^2 \sqrt {-\frac {b^2 (c+d x)^2}{d^2}}} \] Input:

Integrate[(c + d*x)^(3/2)*Sinh[a + b*x]^3,x]
 

Output:

(d*Sqrt[c + d*x]*(-(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)* 
Gamma[5/2, (b*(c + d*x))/d] + Sqrt[3]*E^((2*b*c)/d)*Gamma[5/2, (3*b*(c + d 
*x))/d])))/(216*b^2*E^(3*(a + (b*c)/d))*Sqrt[-((b^2*(c + d*x)^2)/d^2)])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.70 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.49, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 26, 3792, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3789, 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.

\(\displaystyle \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int i (c+d x)^{3/2} \sin (i a+i b x)^3dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int (c+d x)^{3/2} \sin (i a+i b x)^3dx\)

\(\Big \downarrow \) 3792

\(\displaystyle i \left (\frac {d^2 \int -\frac {i \sinh ^3(a+b x)}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \int i (c+d x)^{3/2} \sinh (a+b x)dx+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (-\frac {i d^2 \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} i \int (c+d x)^{3/2} \sinh (a+b x)dx+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (-\frac {i d^2 \int \frac {i \sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} i \int -i (c+d x)^{3/2} \sin (i a+i b x)dx+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (\frac {d^2 \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \int (c+d x)^{3/2} \sin (i a+i b x)dx+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle i \left (\frac {d^2 \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \int \sqrt {c+d x} \cosh (a+b x)dx}{2 b}\right )+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (\frac {d^2 \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \int \sqrt {c+d x} \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{2 b}\right )+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle i \left (\frac {d^2 \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}-\frac {i d \int -\frac {i \sinh (a+b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\right )+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (\frac {d^2 \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}-\frac {d \int \frac {\sinh (a+b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\right )+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (\frac {d^2 \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}-\frac {d \int -\frac {i \sin (i a+i b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\right )+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (\frac {d^2 \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \int \frac {\sin (i a+i b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}\right )+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 3789

\(\displaystyle i \left (\frac {d^2 \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \left (\frac {1}{2} i \int \frac {e^{a+b x}}{\sqrt {c+d x}}dx-\frac {1}{2} i \int \frac {e^{-a-b x}}{\sqrt {c+d x}}dx\right )}{2 b}\right )}{2 b}\right )+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle i \left (\frac {d^2 \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \left (\frac {i \int e^{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{d}-\frac {i \int e^{-a-\frac {b (c+d x)}{d}+\frac {b c}{d}}d\sqrt {c+d x}}{d}\right )}{2 b}\right )}{2 b}\right )+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle i \left (\frac {d^2 \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \left (\frac {i \sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {i \int e^{-a-\frac {b (c+d x)}{d}+\frac {b c}{d}}d\sqrt {c+d x}}{d}\right )}{2 b}\right )}{2 b}\right )+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle i \left (\frac {d^2 \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \left (\frac {i \sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {i \sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}\right )-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 3793

\(\displaystyle i \left (\frac {d^2 \int \left (\frac {3 i \sinh (a+b x)}{4 \sqrt {c+d x}}-\frac {i \sinh (3 a+3 b x)}{4 \sqrt {c+d x}}\right )dx}{12 b^2}+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \left (\frac {i \sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {i \sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}\right )-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle i \left (\frac {d^2 \left (-\frac {3 i \sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {i \sqrt {\frac {\pi }{3}} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {3 i \sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {i \sqrt {\frac {\pi }{3}} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}\right )}{12 b^2}+\frac {i d \sqrt {c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^{3/2} \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {\sqrt {c+d x} \sinh (a+b x)}{b}+\frac {i d \left (\frac {i \sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {i \sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}\right )-\frac {i (c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\)

Input:

Int[(c + d*x)^(3/2)*Sinh[a + b*x]^3,x]
 

Output:

I*((d^2*((((-3*I)/8)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x]) 
/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) + ((I/8)*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[ 
(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) + (((3*I)/8)*E 
^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sq 
rt[d]) - ((I/8)*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[ 
c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d])))/(12*b^2) - ((I/3)*(c + d*x)^(3/2)* 
Cosh[a + b*x]*Sinh[a + b*x]^2)/b + ((I/6)*d*Sqrt[c + d*x]*Sinh[a + b*x]^3) 
/b^2 + (2*((I*(c + d*x)^(3/2)*Cosh[a + b*x])/b - (((3*I)/2)*d*(((I/2)*d*(( 
(-1/2*I)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/( 
Sqrt[b]*Sqrt[d]) + ((I/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + 
d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d])))/b + (Sqrt[c + d*x]*Sinh[a + b*x])/b))/ 
b))/3)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2611

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]
 

rule 2633

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 2634

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3777

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]
 

rule 3789

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
/2   Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2   Int[(c + d*x)^m*E 
^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
 

rule 3792

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]
 

rule 3793

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]))
 

Maple [F]

\[\int \left (d x +c \right )^{\frac {3}{2}} \sinh \left (b x +a \right )^{3}d x\]

Input:

int((d*x+c)^(3/2)*sinh(b*x+a)^3,x)
 

Output:

int((d*x+c)^(3/2)*sinh(b*x+a)^3,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1543 vs. \(2 (245) = 490\).

Time = 0.11 (sec) , antiderivative size = 1543, normalized size of antiderivative = 4.75 \[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(3/2)*sinh(b*x+a)^3,x, algorithm="fricas")
 

Output:

-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)*sqr 
t(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*si 
nh(-3*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(-b/d)*erf(sqrt(3)*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*sin 
h(-(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*co 
sh(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(...
 

Sympy [F]

\[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \sinh ^{3}{\left (a + b x \right )}\, dx \] Input:

integrate((d*x+c)**(3/2)*sinh(b*x+a)**3,x)
 

Output:

Integral((c + d*x)**(3/2)*sinh(a + b*x)**3, x)
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.32 \[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=\frac {\frac {\sqrt {3} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )}}{b^{2} \sqrt {-\frac {b}{d}}} - \frac {\sqrt {3} \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{b^{2} \sqrt {\frac {b}{d}}} - \frac {81 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b^{2} \sqrt {-\frac {b}{d}}} + \frac {81 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b^{2} \sqrt {\frac {b}{d}}} - \frac {54 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (\frac {b c}{d}\right )} + 3 \, \sqrt {d x + c} d^{2} e^{\left (\frac {b c}{d}\right )}\right )} e^{\left (-a - \frac {{\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac {6 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (\frac {3 \, b c}{d}\right )} + \sqrt {d x + c} d^{2} e^{\left (\frac {3 \, b c}{d}\right )}\right )} e^{\left (-3 \, a - \frac {3 \, {\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac {6 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{\left (3 \, a\right )} - \sqrt {d x + c} d^{2} e^{\left (3 \, a\right )}\right )} e^{\left (\frac {3 \, {\left (d x + c\right )} b}{d} - \frac {3 \, b c}{d}\right )}}{b^{2}} - \frac {54 \, {\left (2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d e^{a} - 3 \, \sqrt {d x + c} d^{2} e^{a}\right )} e^{\left (\frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b^{2}}}{288 \, d} \] Input:

integrate((d*x+c)^(3/2)*sinh(b*x+a)^3,x, algorithm="maxima")
 

Output:

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
 

Giac [F]

\[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{\frac {3}{2}} \sinh \left (b x + a\right )^{3} \,d x } \] Input:

integrate((d*x+c)^(3/2)*sinh(b*x+a)^3,x, algorithm="giac")
 

Output:

integrate((d*x + c)^(3/2)*sinh(b*x + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=\int {\mathrm {sinh}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{3/2} \,d x \] Input:

int(sinh(a + b*x)^3*(c + d*x)^(3/2),x)
 

Output:

int(sinh(a + b*x)^3*(c + d*x)^(3/2), x)
 

Reduce [F]

\[ \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx=\left (\int \sqrt {d x +c}\, \sinh \left (b x +a \right )^{3} x d x \right ) d +\left (\int \sqrt {d x +c}\, \sinh \left (b x +a \right )^{3}d x \right ) c \] Input:

int((d*x+c)^(3/2)*sinh(b*x+a)^3,x)
 

Output:

int(sqrt(c + d*x)*sinh(a + b*x)**3*x,x)*d + int(sqrt(c + d*x)*sinh(a + b*x 
)**3,x)*c