\(\int x (a+i a \sinh (e+f x))^{3/2} \, dx\) [127]

Optimal result

Integrand size = 19, antiderivative size = 185 \[ \int x (a+i a \sinh (e+f x))^{3/2} \, dx=-\frac {16 a \sqrt {a+i a \sinh (e+f x)}}{3 f^2}-\frac {8 a \cosh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{9 f^2}+\frac {4 a x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{3 f}+\frac {8 a x \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 f} \] Output:

-16/3*a*(a+I*a*sinh(f*x+e))^(1/2)/f^2-8/9*a*cosh(1/2*e+1/4*I*Pi+1/2*f*x)^2 
*(a+I*a*sinh(f*x+e))^(1/2)/f^2+4/3*a*x*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*sinh(1 
/2*e+1/4*I*Pi+1/2*f*x)*(a+I*a*sinh(f*x+e))^(1/2)/f+8/3*a*x*(a+I*a*sinh(f*x 
+e))^(1/2)*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/f
 

Mathematica [A] (verified)

Time = 3.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.75 \[ \int x (a+i a \sinh (e+f x))^{3/2} \, dx=-\frac {a \left (27 (2 i+f x) \cosh \left (\frac {1}{2} (e+f x)\right )+(-2 i+3 f x) \cosh \left (\frac {3}{2} (e+f x)\right )+2 i (28 i-12 f x+(2 i+3 f x) \cosh (e+f x)) \sinh \left (\frac {1}{2} (e+f x)\right )\right ) (-i+\sinh (e+f x)) \sqrt {a+i a \sinh (e+f x)}}{9 f^2 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^3} \] Input:

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

Output:

-1/9*(a*(27*(2*I + f*x)*Cosh[(e + f*x)/2] + (-2*I + 3*f*x)*Cosh[(3*(e + f* 
x))/2] + (2*I)*(28*I - 12*f*x + (2*I + 3*f*x)*Cosh[e + f*x])*Sinh[(e + f*x 
)/2])*(-I + Sinh[e + f*x])*Sqrt[a + I*a*Sinh[e + f*x]])/(f^2*(Cosh[(e + f* 
x)/2] + I*Sinh[(e + f*x)/2])^3)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {3042, 3800, 3042, 3791, 3042, 3777, 26, 3042, 26, 3118}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int x (a+a \sin (i e+i f x))^{3/2}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \int x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx\)

\(\Big \downarrow \) 3791

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \int x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 i \int -i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int -i \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \left (\frac {2 i \int \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

Input:

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

Output:

2*a*Sech[e/2 + (I/4)*Pi + (f*x)/2]*((-4*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^3)/ 
(9*f^2) + (2*x*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^2*Sinh[e/2 + (I/4)*Pi + (f*x 
)/2])/(3*f) + (2*((-4*Cosh[e/2 + (I/4)*Pi + (f*x)/2])/f^2 + (2*x*Sinh[e/2 
+ (I/4)*Pi + (f*x)/2])/f))/3)*Sqrt[a + I*a*Sinh[e + f*x]]
 

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 3042

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

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, 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 3791

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e 
/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n]))   Int[(c + d*x)^m*Sin[e/2 + a 
*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
 

Maple [F]

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

Input:

int(x*(a+I*a*sinh(f*x+e))^(3/2),x)
 

Output:

int(x*(a+I*a*sinh(f*x+e))^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int x (a+i a \sinh (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

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

integrate(x*(a+I*a*sinh(f*x+e))**(3/2),x)
 

Output:

Integral(x*(I*a*(sinh(e + f*x) - I))**(3/2), x)
 

Maxima [F]

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

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

Output:

integrate((I*a*sinh(f*x + e) + a)^(3/2)*x, x)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.60 \[ \int x (a+i a \sinh (e+f x))^{3/2} \, dx=-\frac {{\left (-\left (3 i - 3\right ) \, a^{\frac {3}{2}} f x e^{\left (3 \, f x + 3 \, e\right )} - \left (27 i + 27\right ) \, a^{\frac {3}{2}} f x e^{\left (2 \, f x + 2 \, e\right )} - \left (27 i - 27\right ) \, a^{\frac {3}{2}} f x e^{\left (f x + e\right )} - \left (3 i + 3\right ) \, a^{\frac {3}{2}} f x + \left (2 i - 2\right ) \, a^{\frac {3}{2}} e^{\left (3 \, f x + 3 \, e\right )} + \left (54 i + 54\right ) \, a^{\frac {3}{2}} e^{\left (2 \, f x + 2 \, e\right )} - \left (54 i - 54\right ) \, a^{\frac {3}{2}} e^{\left (f x + e\right )} - \left (2 i + 2\right ) \, a^{\frac {3}{2}}\right )} e^{\left (-\frac {3}{2} \, f x - \frac {3}{2} \, e\right )}}{18 \, f^{2}} \] Input:

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

Output:

-1/18*(-(3*I - 3)*a^(3/2)*f*x*e^(3*f*x + 3*e) - (27*I + 27)*a^(3/2)*f*x*e^ 
(2*f*x + 2*e) - (27*I - 27)*a^(3/2)*f*x*e^(f*x + e) - (3*I + 3)*a^(3/2)*f* 
x + (2*I - 2)*a^(3/2)*e^(3*f*x + 3*e) + (54*I + 54)*a^(3/2)*e^(2*f*x + 2*e 
) - (54*I - 54)*a^(3/2)*e^(f*x + e) - (2*I + 2)*a^(3/2))*e^(-3/2*f*x - 3/2 
*e)/f^2
 

Mupad [F(-1)]

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

int(x*(a + a*sinh(e + f*x)*1i)^(3/2),x)
 

Output:

int(x*(a + a*sinh(e + f*x)*1i)^(3/2), x)
 

Reduce [F]

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

int(x*(a+I*a*sinh(f*x+e))^(3/2),x)
 

Output:

sqrt(a)*a*(int(sqrt(sinh(e + f*x)*i + 1)*sinh(e + f*x)*x,x)*i + int(sqrt(s 
inh(e + f*x)*i + 1)*x,x))