\(\int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx\) [113]

Optimal result

Integrand size = 20, antiderivative size = 81 \[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\frac {8 a^2 (5 i A+3 B) \cosh (x)}{15 \sqrt {a+i a \sinh (x)}}+\frac {2}{15} a (5 i A+3 B) \cosh (x) \sqrt {a+i a \sinh (x)}+\frac {2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2} \] Output:

8/15*a^2*(5*I*A+3*B)*cosh(x)/(a+I*a*sinh(x))^(1/2)+2/15*a*(5*I*A+3*B)*cosh 
(x)*(a+I*a*sinh(x))^(1/2)+2/5*B*cosh(x)*(a+I*a*sinh(x))^(3/2)
 

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02 \[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=-\frac {a \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right ) \sqrt {a+i a \sinh (x)} (-50 i A-39 B+3 B \cosh (2 x)+2 (5 A-9 i B) \sinh (x))}{15 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )} \] Input:

Integrate[(a + I*a*Sinh[x])^(3/2)*(A + B*Sinh[x]),x]
 

Output:

-1/15*(a*(Cosh[x/2] - I*Sinh[x/2])*Sqrt[a + I*a*Sinh[x]]*((-50*I)*A - 39*B 
 + 3*B*Cosh[2*x] + 2*(5*A - (9*I)*B)*Sinh[x]))/(Cosh[x/2] + I*Sinh[x/2])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3042, 3230, 3042, 3126, 3042, 3125}

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.

\(\displaystyle \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+a \sin (i x))^{3/2} (A-i B \sin (i x))dx\)

\(\Big \downarrow \) 3230

\(\displaystyle \frac {1}{5} (5 A-3 i B) \int (i \sinh (x) a+a)^{3/2}dx+\frac {2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} (5 A-3 i B) \int (\sin (i x) a+a)^{3/2}dx+\frac {2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2}\)

\(\Big \downarrow \) 3126

\(\displaystyle \frac {1}{5} (5 A-3 i B) \left (\frac {4}{3} a \int \sqrt {i \sinh (x) a+a}dx+\frac {2}{3} i a \cosh (x) \sqrt {a+i a \sinh (x)}\right )+\frac {2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} (5 A-3 i B) \left (\frac {4}{3} a \int \sqrt {\sin (i x) a+a}dx+\frac {2}{3} i a \cosh (x) \sqrt {a+i a \sinh (x)}\right )+\frac {2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2}\)

\(\Big \downarrow \) 3125

\(\displaystyle \frac {1}{5} (5 A-3 i B) \left (\frac {8 i a^2 \cosh (x)}{3 \sqrt {a+i a \sinh (x)}}+\frac {2}{3} i a \cosh (x) \sqrt {a+i a \sinh (x)}\right )+\frac {2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2}\)

Input:

Int[(a + I*a*Sinh[x])^(3/2)*(A + B*Sinh[x]),x]
 

Output:

(2*B*Cosh[x]*(a + I*a*Sinh[x])^(3/2))/5 + ((5*A - (3*I)*B)*((((8*I)/3)*a^2 
*Cosh[x])/Sqrt[a + I*a*Sinh[x]] + ((2*I)/3)*a*Cosh[x]*Sqrt[a + I*a*Sinh[x] 
]))/5
 

Defintions of rubi rules used

rule 3042

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

rule 3125

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos 
[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq 
Q[a^2 - b^2, 0]
 

rule 3126

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos 
[c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) 
 Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ 
a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
 

rule 3230

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( 
f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1))   Int[(a + b*Sin[e 
 + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] 
&& EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)]
 

Maple [F]

\[\int \left (a +i a \sinh \left (x \right )\right )^{\frac {3}{2}} \left (A +B \sinh \left (x \right )\right )d x\]

Input:

int((a+I*a*sinh(x))^(3/2)*(A+B*sinh(x)),x)
 

Output:

int((a+I*a*sinh(x))^(3/2)*(A+B*sinh(x)),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.01 \[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\frac {1}{30} \, {\left (3 i \, B a e^{\left (5 \, x\right )} - 5 \, {\left (-2 i \, A - 3 \, B\right )} a e^{\left (4 \, x\right )} + 30 \, {\left (3 \, A - 2 i \, B\right )} a e^{\left (3 \, x\right )} - 30 \, {\left (-3 i \, A - 2 \, B\right )} a e^{\left (2 \, x\right )} + 5 \, {\left (2 \, A - 3 i \, B\right )} a e^{x} - 3 \, B a\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} e^{\left (-2 \, x\right )} \] Input:

integrate((a+I*a*sinh(x))^(3/2)*(A+B*sinh(x)),x, algorithm="fricas")
 

Output:

1/30*(3*I*B*a*e^(5*x) - 5*(-2*I*A - 3*B)*a*e^(4*x) + 30*(3*A - 2*I*B)*a*e^ 
(3*x) - 30*(-3*I*A - 2*B)*a*e^(2*x) + 5*(2*A - 3*I*B)*a*e^x - 3*B*a)*sqrt( 
1/2*I*a*e^(-x))*e^(-2*x)
 

Sympy [F]

\[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int \left (i a \left (\sinh {\left (x \right )} - i\right )\right )^{\frac {3}{2}} \left (A + B \sinh {\left (x \right )}\right )\, dx \] Input:

integrate((a+I*a*sinh(x))**(3/2)*(A+B*sinh(x)),x)
 

Output:

Integral((I*a*(sinh(x) - I))**(3/2)*(A + B*sinh(x)), x)
 

Maxima [F]

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

integrate((a+I*a*sinh(x))^(3/2)*(A+B*sinh(x)),x, algorithm="maxima")
 

Output:

integrate((B*sinh(x) + A)*(I*a*sinh(x) + a)^(3/2), x)
 

Giac [F]

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

integrate((a+I*a*sinh(x))^(3/2)*(A+B*sinh(x)),x, algorithm="giac")
 

Output:

integrate((B*sinh(x) + A)*(I*a*sinh(x) + a)^(3/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int \left (A+B\,\mathrm {sinh}\left (x\right )\right )\,{\left (a+a\,\mathrm {sinh}\left (x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \] Input:

int((A + B*sinh(x))*(a + a*sinh(x)*1i)^(3/2),x)
 

Output:

int((A + B*sinh(x))*(a + a*sinh(x)*1i)^(3/2), x)
 

Reduce [F]

\[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\sqrt {a}\, a \left (\left (\int \sqrt {\sinh \left (x \right ) i +1}d x \right ) a +\left (\int \sqrt {\sinh \left (x \right ) i +1}\, \sinh \left (x \right )^{2}d x \right ) b i +\left (\int \sqrt {\sinh \left (x \right ) i +1}\, \sinh \left (x \right )d x \right ) a i +\left (\int \sqrt {\sinh \left (x \right ) i +1}\, \sinh \left (x \right )d x \right ) b \right ) \] Input:

int((a+I*a*sinh(x))^(3/2)*(A+B*sinh(x)),x)
 

Output:

sqrt(a)*a*(int(sqrt(sinh(x)*i + 1),x)*a + int(sqrt(sinh(x)*i + 1)*sinh(x)* 
*2,x)*b*i + int(sqrt(sinh(x)*i + 1)*sinh(x),x)*a*i + int(sqrt(sinh(x)*i + 
1)*sinh(x),x)*b)