Integrand size = 19, antiderivative size = 66 \[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=-\frac {4 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {2 x \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f} \] Output:
-4*(a+I*a*sinh(f*x+e))^(1/2)/f^2+2*x*(a+I*a*sinh(f*x+e))^(1/2)*tanh(1/2*e+ 1/4*I*Pi+1/2*f*x)/f
Time = 0.34 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.32 \[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {2 \left ((-2+i f x) \cosh \left (\frac {1}{2} (e+f x)\right )+(-2 i+f x) \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a+i a \sinh (e+f x)}}{f^2 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:
Integrate[x*Sqrt[a + I*a*Sinh[e + f*x]],x]
Output:
(2*((-2 + I*f*x)*Cosh[(e + f*x)/2] + (-2*I + f*x)*Sinh[(e + f*x)/2])*Sqrt[ a + I*a*Sinh[e + f*x]])/(f^2*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2]))
Time = 0.38 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.36, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3042, 3800, 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.
\(\displaystyle \int x \sqrt {a+i a \sinh (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x \sqrt {a+a \sin (i e+i f x)}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \int x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 i \int -i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int -i \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2 i \int \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )\) |
Input:
Int[x*Sqrt[a + I*a*Sinh[e + f*x]],x]
Output:
Sech[e/2 + (I/4)*Pi + (f*x)/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)*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 = 0.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.59
method | result | size |
risch | \(\frac {i \sqrt {2}\, \sqrt {a \left (i {\mathrm e}^{2 f x +2 e}-i+2 \,{\mathrm e}^{f x +e}\right ) {\mathrm e}^{-f x -e}}\, \left (i x f +f x \,{\mathrm e}^{f x +e}+2 i-2 \,{\mathrm e}^{f x +e}\right ) \left ({\mathrm e}^{f x +e}-i\right )}{\left (i {\mathrm e}^{2 f x +2 e}-i+2 \,{\mathrm e}^{f x +e}\right ) f^{2}}\) | \(105\) |
Input:
int(x*(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*f*x+f*x*exp(f*x+e)+2*I-2*exp(f*x+e))*(exp(f*x +e)-I)/f^2
Exception generated. \[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(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)
\[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\int x \sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}\, dx \] Input:
integrate(x*(a+I*a*sinh(f*x+e))**(1/2),x)
Output:
Integral(x*sqrt(I*a*(sinh(e + f*x) - I)), x)
\[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\int { \sqrt {i \, a \sinh \left (f x + e\right ) + a} x \,d x } \] Input:
integrate(x*(a+I*a*sinh(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(I*a*sinh(f*x + e) + a)*x, x)
Time = 0.18 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76 \[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {{\left (\left (i + 1\right ) \, \sqrt {a} f x e^{\left (f x + e\right )} + \left (i - 1\right ) \, \sqrt {a} f x - \left (2 i + 2\right ) \, \sqrt {a} e^{\left (f x + e\right )} + \left (2 i - 2\right ) \, \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, f x - \frac {1}{2} \, e\right )}}{f^{2}} \] Input:
integrate(x*(a+I*a*sinh(f*x+e))^(1/2),x, algorithm="giac")
Output:
((I + 1)*sqrt(a)*f*x*e^(f*x + e) + (I - 1)*sqrt(a)*f*x - (2*I + 2)*sqrt(a) *e^(f*x + e) + (2*I - 2)*sqrt(a))*e^(-1/2*f*x - 1/2*e)/f^2
Time = 1.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.21 \[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {\sqrt {2}\,\left ({\mathrm {e}}^{e+f\,x}+1{}\mathrm {i}\right )\,\left (f\,x\,{\mathrm {e}}^{e+f\,x}+f\,x\,1{}\mathrm {i}-2\,{\mathrm {e}}^{e+f\,x}+2{}\mathrm {i}\right )\,\sqrt {a\,{\mathrm {e}}^{-e-f\,x}\,{\left ({\mathrm {e}}^{e+f\,x}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}}{f^2\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )} \] Input:
int(x*(a + a*sinh(e + f*x)*1i)^(1/2),x)
Output:
(2^(1/2)*(exp(e + f*x) + 1i)*(f*x*1i - 2*exp(e + f*x) + f*x*exp(e + f*x) + 2i)*(a*exp(- e - f*x)*(exp(e + f*x) - 1i)^2*1i)^(1/2))/(f^2*(exp(2*e + 2* f*x) + 1))
\[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sinh \left (f x +e \right ) i +1}\, x d x \right ) \] Input:
int(x*(a+I*a*sinh(f*x+e))^(1/2),x)
Output:
sqrt(a)*int(sqrt(sinh(e + f*x)*i + 1)*x,x)