\(\int x (a+i a \sinh (c+d x))^{5/2} \, dx\) [132]

Optimal result

Integrand size = 19, antiderivative size = 312 \[ \int x (a+i a \sinh (c+d x))^{5/2} \, dx=-\frac {128 a^2 \sqrt {a+i a \sinh (c+d x)}}{15 d^2}-\frac {64 a^2 \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {16 a^2 \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}+\frac {32 a^2 x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d}+\frac {8 a^2 x \cosh ^3\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}+\frac {64 a^2 x \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{15 d} \] Output:

-128/15*a^2*(a+I*a*sinh(d*x+c))^(1/2)/d^2-64/45*a^2*cosh(1/2*c+1/4*I*Pi+1/ 
2*d*x)^2*(a+I*a*sinh(d*x+c))^(1/2)/d^2-16/25*a^2*cosh(1/2*c+1/4*I*Pi+1/2*d 
*x)^4*(a+I*a*sinh(d*x+c))^(1/2)/d^2+32/15*a^2*x*cosh(1/2*c+1/4*I*Pi+1/2*d* 
x)*sinh(1/2*c+1/4*I*Pi+1/2*d*x)*(a+I*a*sinh(d*x+c))^(1/2)/d+8/5*a^2*x*cosh 
(1/2*c+1/4*I*Pi+1/2*d*x)^3*sinh(1/2*c+1/4*I*Pi+1/2*d*x)*(a+I*a*sinh(d*x+c) 
)^(1/2)/d+64/15*a^2*x*(a+I*a*sinh(d*x+c))^(1/2)*tanh(1/2*c+1/4*I*Pi+1/2*d* 
x)/d
 

Mathematica [A] (verified)

Time = 9.25 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.70 \[ \int x (a+i a \sinh (c+d x))^{5/2} \, dx=\frac {a^2 (-i+\sinh (c+d x))^2 \sqrt {a+i a \sinh (c+d x)} \left (2250 (2-i d x) \cosh \left (\frac {1}{2} (c+d x)\right )+(-250-375 i d x) \cosh \left (\frac {3}{2} (c+d x)\right )-18 \cosh \left (\frac {5}{2} (c+d x)\right )+45 i d x \cosh \left (\frac {5}{2} (c+d x)\right )+4500 i \sinh \left (\frac {1}{2} (c+d x)\right )-2250 d x \sinh \left (\frac {1}{2} (c+d x)\right )+250 i \sinh \left (\frac {3}{2} (c+d x)\right )+375 d x \sinh \left (\frac {3}{2} (c+d x)\right )-18 i \sinh \left (\frac {5}{2} (c+d x)\right )+45 d x \sinh \left (\frac {5}{2} (c+d x)\right )\right )}{450 d^2 \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^5} \] Input:

Integrate[x*(a + I*a*Sinh[c + d*x])^(5/2),x]
 

Output:

(a^2*(-I + Sinh[c + d*x])^2*Sqrt[a + I*a*Sinh[c + d*x]]*(2250*(2 - I*d*x)* 
Cosh[(c + d*x)/2] + (-250 - (375*I)*d*x)*Cosh[(3*(c + d*x))/2] - 18*Cosh[( 
5*(c + d*x))/2] + (45*I)*d*x*Cosh[(5*(c + d*x))/2] + (4500*I)*Sinh[(c + d* 
x)/2] - 2250*d*x*Sinh[(c + d*x)/2] + (250*I)*Sinh[(3*(c + d*x))/2] + 375*d 
*x*Sinh[(3*(c + d*x))/2] - (18*I)*Sinh[(5*(c + d*x))/2] + 45*d*x*Sinh[(5*( 
c + d*x))/2]))/(450*d^2*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^5)
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.84, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {3042, 3800, 3042, 3791, 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.

\(\displaystyle \int x (a+i a \sinh (c+d x))^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x (a+a \sin (i c+i d x))^{5/2}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \int x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^5dx\)

\(\Big \downarrow \) 3791

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \int x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \int x \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 i \int -i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 \int \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 \int -i \sin \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}\right )-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2 i \int \sin \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

Input:

Int[x*(a + I*a*Sinh[c + d*x])^(5/2),x]
 

Output:

4*a^2*Sech[c/2 + (I/4)*Pi + (d*x)/2]*((-4*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^5 
)/(25*d^2) + (2*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^4*Sinh[c/2 + (I/4)*Pi + ( 
d*x)/2])/(5*d) + (4*((-4*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^3)/(9*d^2) + (2*x* 
Cosh[c/2 + (I/4)*Pi + (d*x)/2]^2*Sinh[c/2 + (I/4)*Pi + (d*x)/2])/(3*d) + ( 
2*((-4*Cosh[c/2 + (I/4)*Pi + (d*x)/2])/d^2 + (2*x*Sinh[c/2 + (I/4)*Pi + (d 
*x)/2])/d))/3))/5)*Sqrt[a + I*a*Sinh[c + d*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]

\[\int x \left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}d x\]

Input:

int(x*(a+I*a*sinh(d*x+c))^(5/2),x)
 

Output:

int(x*(a+I*a*sinh(d*x+c))^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int x (a+i a \sinh (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int x (a+i a \sinh (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate(x*(a+I*a*sinh(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x (a+i a \sinh (c+d x))^{5/2} \, dx=\int { {\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}} x \,d x } \] Input:

integrate(x*(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

integrate((I*a*sinh(d*x + c) + a)^(5/2)*x, x)
 

Giac [F]

\[ \int x (a+i a \sinh (c+d x))^{5/2} \, dx=\int { {\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}} x \,d x } \] Input:

integrate(x*(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

integrate((I*a*sinh(d*x + c) + a)^(5/2)*x, x)
 

Mupad [F(-1)]

Timed out. \[ \int x (a+i a \sinh (c+d x))^{5/2} \, dx=\int x\,{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \] Input:

int(x*(a + a*sinh(c + d*x)*1i)^(5/2),x)
 

Output:

int(x*(a + a*sinh(c + d*x)*1i)^(5/2), x)
 

Reduce [F]

\[ \int x (a+i a \sinh (c+d x))^{5/2} \, dx=\sqrt {a}\, a^{2} \left (-\left (\int \sqrt {\sinh \left (d x +c \right ) i +1}\, \sinh \left (d x +c \right )^{2} x d x \right )+2 \left (\int \sqrt {\sinh \left (d x +c \right ) i +1}\, \sinh \left (d x +c \right ) x d x \right ) i +\int \sqrt {\sinh \left (d x +c \right ) i +1}\, x d x \right ) \] Input:

int(x*(a+I*a*sinh(d*x+c))^(5/2),x)
                                                                                    
                                                                                    
 

Output:

sqrt(a)*a**2*( - int(sqrt(sinh(c + d*x)*i + 1)*sinh(c + d*x)**2*x,x) + 2*i 
nt(sqrt(sinh(c + d*x)*i + 1)*sinh(c + d*x)*x,x)*i + int(sqrt(sinh(c + d*x) 
*i + 1)*x,x))