Integrand size = 21, antiderivative size = 181 \[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=-\frac {384 x \sqrt {a+i a \sinh (e+f x)}}{f^4}-\frac {16 x^3 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {768 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f^5}+\frac {96 x^2 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f^3}+\frac {2 x^4 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f} \] Output:
-384*x*(a+I*a*sinh(f*x+e))^(1/2)/f^4-16*x^3*(a+I*a*sinh(f*x+e))^(1/2)/f^2+ 768*(a+I*a*sinh(f*x+e))^(1/2)*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/f^5+96*x^2*(a+I *a*sinh(f*x+e))^(1/2)*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/f^3+2*x^4*(a+I*a*sinh(f *x+e))^(1/2)*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/f
Time = 0.25 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78 \[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {2 \left (i \left (384+192 i f x+48 f^2 x^2+8 i f^3 x^3+f^4 x^4\right ) \cosh \left (\frac {1}{2} (e+f x)\right )+\left (384-192 i f x+48 f^2 x^2-8 i f^3 x^3+f^4 x^4\right ) \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a+i a \sinh (e+f x)}}{f^5 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:
Integrate[x^4*Sqrt[a + I*a*Sinh[e + f*x]],x]
Output:
(2*(I*(384 + (192*I)*f*x + 48*f^2*x^2 + (8*I)*f^3*x^3 + f^4*x^4)*Cosh[(e + f*x)/2] + (384 - (192*I)*f*x + 48*f^2*x^2 - (8*I)*f^3*x^3 + f^4*x^4)*Sinh [(e + f*x)/2])*Sqrt[a + I*a*Sinh[e + f*x]])/(f^5*(Cosh[(e + f*x)/2] + I*Si nh[(e + f*x)/2]))
Time = 0.81 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.13, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 3800, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 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.
| \(\displaystyle \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^4 \sqrt {a+a \sin (i e+i f x)}dx\) |
| \(\Big \downarrow \) 3800 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \int x^4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \int x^4 \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {8 i \int -i x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {8 \int x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {8 \int -i x^3 \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 i \int x^3 \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 i \left (\frac {2 i x^3 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i \int x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f}+\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 i \left (\frac {2 i x^3 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i \int x^2 \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx}{f}\right )}{f}+\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 i \left (\frac {2 i x^3 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i \left (\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \int -i x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f}\right )}{f}+\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 i \left (\frac {2 i x^3 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i \left (\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \int x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f}\right )}{f}+\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 i \left (\frac {2 i x^3 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i \left (\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \int -i x \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )}{f}\right )}{f}+\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 i \left (\frac {2 i x^3 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i \left (\frac {4 i \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{f}\right )}{f}+\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 i \left (\frac {2 i x^3 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i \left (\frac {4 i \left (\frac {2 i x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 i \int \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{f}\right )}{f}+\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 i \left (\frac {2 i x^3 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i \left (\frac {4 i \left (\frac {2 i x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 i \int \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx}{f}\right )}{f}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{f}\right )}{f}+\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 i \left (\frac {2 i x^3 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i \left (\frac {4 i \left (\frac {2 i x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{f}\right )}{f}+\frac {2 x^4 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )\) |
Input:
Int[x^4*Sqrt[a + I*a*Sinh[e + f*x]],x]
Output:
Sech[e/2 + (I/4)*Pi + (f*x)/2]*((2*x^4*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/f + ((8*I)*(((2*I)*x^3*Cosh[e/2 + (I/4)*Pi + (f*x)/2])/f - ((6*I)*((2*x^2*Sin h[e/2 + (I/4)*Pi + (f*x)/2])/f + ((4*I)*(((2*I)*x*Cosh[e/2 + (I/4)*Pi + (f *x)/2])/f - ((4*I)*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/f^2))/f))/f))/f)*Sqrt[a + I*a*Sinh[e + f*x]]
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[((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_.)*((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])
Time = 0.25 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(\frac {i \sqrt {2}\, \sqrt {a \left (i {\mathrm e}^{2 f x +2 e}-i+2 \,{\mathrm e}^{f x +e}\right ) {\mathrm e}^{-f x -e}}\, \left (i x^{4} f^{4}+f^{4} x^{4} {\mathrm e}^{f x +e}+8 i x^{3} f^{3}-8 f^{3} x^{3} {\mathrm e}^{f x +e}+48 i x^{2} f^{2}+48 f^{2} x^{2} {\mathrm e}^{f x +e}+192 i x f -192 f x \,{\mathrm e}^{f x +e}+384 i+384 \,{\mathrm e}^{f x +e}\right ) \left ({\mathrm e}^{f x +e}-i\right )}{\left (i {\mathrm e}^{2 f x +2 e}-i+2 \,{\mathrm e}^{f x +e}\right ) f^{5}}\) | \(174\) |
Input:
int(x^4*(a+I*a*sinh(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
I*2^(1/2)*(a*(I*exp(2*f*x+2*e)-I+2*exp(f*x+e))*exp(-f*x-e))^(1/2)/(I*exp(2 *f*x+2*e)-I+2*exp(f*x+e))*(I*x^4*f^4+f^4*x^4*exp(f*x+e)+8*I*x^3*f^3-8*f^3* x^3*exp(f*x+e)+48*I*x^2*f^2+48*f^2*x^2*exp(f*x+e)+192*I*x*f-192*f*x*exp(f* x+e)+384*I+384*exp(f*x+e))*(exp(f*x+e)-I)/f^5
Exception generated. \[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^4*(a+I*a*sinh(f*x+e))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\int x^{4} \sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}\, dx \] Input:
integrate(x**4*(a+I*a*sinh(f*x+e))**(1/2),x)
Output:
Integral(x**4*sqrt(I*a*(sinh(e + f*x) - I)), x)
\[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\int { \sqrt {i \, a \sinh \left (f x + e\right ) + a} x^{4} \,d x } \] Input:
integrate(x^4*(a+I*a*sinh(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(I*a*sinh(f*x + e) + a)*x^4, x)
Time = 0.17 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.74 \[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {{\left (\left (i + 1\right ) \, \sqrt {a} f^{4} x^{4} e^{\left (f x + e\right )} + \left (i - 1\right ) \, \sqrt {a} f^{4} x^{4} - \left (8 i + 8\right ) \, \sqrt {a} f^{3} x^{3} e^{\left (f x + e\right )} + \left (8 i - 8\right ) \, \sqrt {a} f^{3} x^{3} + \left (48 i + 48\right ) \, \sqrt {a} f^{2} x^{2} e^{\left (f x + e\right )} + \left (48 i - 48\right ) \, \sqrt {a} f^{2} x^{2} - \left (192 i + 192\right ) \, \sqrt {a} f x e^{\left (f x + e\right )} + \left (192 i - 192\right ) \, \sqrt {a} f x + \left (384 i + 384\right ) \, \sqrt {a} e^{\left (f x + e\right )} + \left (384 i - 384\right ) \, \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, f x - \frac {1}{2} \, e\right )}}{f^{5}} \] Input:
integrate(x^4*(a+I*a*sinh(f*x+e))^(1/2),x, algorithm="giac")
Output:
((I + 1)*sqrt(a)*f^4*x^4*e^(f*x + e) + (I - 1)*sqrt(a)*f^4*x^4 - (8*I + 8) *sqrt(a)*f^3*x^3*e^(f*x + e) + (8*I - 8)*sqrt(a)*f^3*x^3 + (48*I + 48)*sqr t(a)*f^2*x^2*e^(f*x + e) + (48*I - 48)*sqrt(a)*f^2*x^2 - (192*I + 192)*sqr t(a)*f*x*e^(f*x + e) + (192*I - 192)*sqrt(a)*f*x + (384*I + 384)*sqrt(a)*e ^(f*x + e) + (384*I - 384)*sqrt(a))*e^(-1/2*f*x - 1/2*e)/f^5
Time = 0.76 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.82 \[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {\sqrt {2}\,\left ({\mathrm {e}}^{e+f\,x}+1{}\mathrm {i}\right )\,\sqrt {a\,{\mathrm {e}}^{-e-f\,x}\,{\left ({\mathrm {e}}^{e+f\,x}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}\,\left (384\,{\mathrm {e}}^{e+f\,x}+f\,x\,192{}\mathrm {i}+f^2\,x^2\,48{}\mathrm {i}+f^3\,x^3\,8{}\mathrm {i}+f^4\,x^4\,1{}\mathrm {i}+48\,f^2\,x^2\,{\mathrm {e}}^{e+f\,x}-8\,f^3\,x^3\,{\mathrm {e}}^{e+f\,x}+f^4\,x^4\,{\mathrm {e}}^{e+f\,x}-192\,f\,x\,{\mathrm {e}}^{e+f\,x}+384{}\mathrm {i}\right )}{f^5\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )} \] Input:
int(x^4*(a + a*sinh(e + f*x)*1i)^(1/2),x)
Output:
(2^(1/2)*(exp(e + f*x) + 1i)*(a*exp(- e - f*x)*(exp(e + f*x) - 1i)^2*1i)^( 1/2)*(384*exp(e + f*x) + f*x*192i + f^2*x^2*48i + f^3*x^3*8i + f^4*x^4*1i + 48*f^2*x^2*exp(e + f*x) - 8*f^3*x^3*exp(e + f*x) + f^4*x^4*exp(e + f*x) - 192*f*x*exp(e + f*x) + 384i))/(f^5*(exp(2*e + 2*f*x) + 1))
\[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sinh \left (f x +e \right ) i +1}\, x^{4}d x \right ) \] Input:
int(x^4*(a+I*a*sinh(f*x+e))^(1/2),x)
Output:
sqrt(a)*int(sqrt(sinh(e + f*x)*i + 1)*x**4,x)