Integrand size = 37, antiderivative size = 779 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\frac {i b^2}{3 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 x}{3 d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b (a+b \text {arcsinh}(c x))}{3 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}}+\frac {i b x (a+b \text {arcsinh}(c x))}{3 d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {i (a+b \text {arcsinh}(c x))^2}{3 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}+\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c d f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:
1/3*I*b^2/c/d/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-1/3*b^2*x/d/f^2/(d+I *c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+1/3*b*(a+b*arcsinh(c*x))/c/d/f^2/(d+I*c*d* x)^(1/2)/(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+1/3*I*b*x*(a+b*arcsinh(c*x))/ d/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+2/3*x*(a+b*arc sinh(c*x))^2/d/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-1/3*I*(a+b*arcsinh( c*x))^2/c/d/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)/(c^2*x^2+1)+1/3*x*(a+b *arcsinh(c*x))^2/d/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)/(c^2*x^2+1)+2/3 *(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2/c/d/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f *x)^(1/2)+2/3*I*b*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2 +1)^(1/2))/c/d/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-4/3*b*(c^2*x^2+1)^( 1/2)*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)/c/d/f^2/(d+I*c*d*x )^(1/2)/(f-I*c*f*x)^(1/2)+1/3*b^2*(c^2*x^2+1)^(1/2)*polylog(2,-I*(c*x+(c^2 *x^2+1)^(1/2)))/c/d/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-1/3*b^2*(c^2*x ^2+1)^(1/2)*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/c/d/f^2/(d+I*c*d*x)^(1/2) /(f-I*c*f*x)^(1/2)-2/3*b^2*(c^2*x^2+1)^(1/2)*polylog(2,-(c*x+(c^2*x^2+1)^( 1/2))^2)/c/d/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)
Time = 9.56 (sec) , antiderivative size = 757, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)) ,x]
Output:
(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(a^2/(4*d^2*f^3*(-I + c*x)) + ((I/6)*a^2)/(d^2*f^3*(I + c*x)^2) + (5*a^2)/(12*d^2*f^3*(I + c*x))))/c - ((I/3)*a*b*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*(4*c*x*ArcSin h[c*x] - (2*I)*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] + Sqrt[1 + c^2*x^2]*(1 + (2*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 2*c*x*(ArcTan[Tanh[ArcSinh[c*x]/2]] + (2*I)*Log[Sqrt[1 + c^2*x^2]]) - 4*Log[Sqrt[1 + c^2*x^2]])))/(c*d*f^2*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[-(d*f*(1 + c^2*x^2))]) - ((I/6)*b^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[1 + c^ 2*x^2]*(-9*Pi*ArcSinh[c*x] + ((2 - I*ArcSinh[c*x])*ArcSinh[c*x])/(I + c*x) - (1 - 4*I)*ArcSinh[c*x]^2 + 3*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcS inh[c*x]] - 5*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + 16*Pi* Log[1 + E^ArcSinh[c*x]] + 5*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - 16 *Pi*Log[Cosh[ArcSinh[c*x]/2]] - 3*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] - (10*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (6*I)*PolyLog[2, I/E^ArcSinh[c *x]] - ((2*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^3 + (I*(4 - 5*ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2 ])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]) - ((3*I)*ArcSinh[c*x]^2 *Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/( c*d*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[-(d*f*(1 + c^2*x^2))] )
Time = 1.70 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.47, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6211, 27, 6253, 2009}
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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6211 |
| \(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \int \frac {d (i c x+1) (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{5/2}}dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (c^2 x^2+1\right )^{5/2} \int \frac {(i c x+1) (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{5/2}}dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {d \left (c^2 x^2+1\right )^{5/2} \int \left (\frac {i c x (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{5/2}}+\frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{5/2}}\right )dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \left (c^2 x^2+1\right )^{5/2} \left (\frac {2 i b \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 c}+\frac {i b x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )}+\frac {b (a+b \text {arcsinh}(c x))}{3 c \left (c^2 x^2+1\right )}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 \sqrt {c^2 x^2+1}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {i (a+b \text {arcsinh}(c x))^2}{3 c \left (c^2 x^2+1\right )^{3/2}}+\frac {2 (a+b \text {arcsinh}(c x))^2}{3 c}-\frac {4 b \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{3 c}+\frac {b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c}-\frac {b^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c}-\frac {2 b^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c}-\frac {b^2 x}{3 \sqrt {c^2 x^2+1}}+\frac {i b^2}{3 c \sqrt {c^2 x^2+1}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
Input:
Int[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)),x]
Output:
(d*(1 + c^2*x^2)^(5/2)*(((I/3)*b^2)/(c*Sqrt[1 + c^2*x^2]) - (b^2*x)/(3*Sqr t[1 + c^2*x^2]) + (b*(a + b*ArcSinh[c*x]))/(3*c*(1 + c^2*x^2)) + ((I/3)*b* x*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2) + (2*(a + b*ArcSinh[c*x])^2)/(3*c) - ((I/3)*(a + b*ArcSinh[c*x])^2)/(c*(1 + c^2*x^2)^(3/2)) + (x*(a + b*ArcSin h[c*x])^2)/(3*(1 + c^2*x^2)^(3/2)) + (2*x*(a + b*ArcSinh[c*x])^2)/(3*Sqrt[ 1 + c^2*x^2]) + (((2*I)/3)*b*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/ c - (4*b*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*c) + (b^2*Po lyLog[2, (-I)*E^ArcSinh[c*x]])/(3*c) - (b^2*PolyLog[2, I*E^ArcSinh[c*x]])/ (3*c) - (2*b^2*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c)))/((d + I*c*d*x)^(5/ 2)*(f - I*c*f*x)^(5/2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_ ) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 + c^2*x ^2)^q) Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^ 2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n , 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))
Time = 6.77 (sec) , antiderivative size = 1084, normalized size of antiderivative = 1.39
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1084\) |
| parts | \(\text {Expression too large to display}\) | \(1084\) |
Input:
int((a+b*arcsinh(x*c))^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(5/2),x,method=_RET URNVERBOSE)
Output:
a^2*(I/c/d/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(3/2)+2/d*(-1/3*I/d/c/f/(f-I*c* f*x)^(3/2)*(d+I*c*d*x)^(1/2)-1/3*I/c/d/f^2/(f-I*c*f*x)^(1/2)*(d+I*c*d*x)^( 1/2)))+1/3*b^2*(1-5*polylog(2,I*(x*c+(c^2*x^2+1)^(1/2)))-3*polylog(2,-I*(x *c+(c^2*x^2+1)^(1/2)))+I*arcsinh(x*c)*x^3*c^3+I*arcsinh(x*c)*x*c+2*arcsinh (x*c)^2+arcsinh(x*c)-5*arcsinh(x*c)*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))+3*(c^2 *x^2+1)^(1/2)*arcsinh(x*c)^2*x*c+I*(c^2*x^2+1)^(1/2)-3*ln(1+I*(x*c+(c^2*x^ 2+1)^(1/2)))*arcsinh(x*c)*x^4*c^4+2*arcsinh(x*c)^2*(c^2*x^2+1)^(1/2)*x^3*c ^3-6*ln(1+I*(x*c+(c^2*x^2+1)^(1/2)))*arcsinh(x*c)*x^2*c^2+c^4*x^4-5*arcsin h(x*c)*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))*x^4*c^4+I*(c^2*x^2+1)^(1/2)*x^2*c^2 -10*arcsinh(x*c)*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))*x^2*c^2-(c^2*x^2+1)^(1/2) *x*c-(c^2*x^2+1)^(1/2)*c^3*x^3+2*c^2*x^2+arcsinh(x*c)*c^2*x^2+2*arcsinh(x* c)^2*x^4*c^4-3*polylog(2,-I*(x*c+(c^2*x^2+1)^(1/2)))*x^4*c^4+4*arcsinh(x*c )^2*x^2*c^2-6*polylog(2,-I*(x*c+(c^2*x^2+1)^(1/2)))*x^2*c^2-I*(c^2*x^2+1)^ (1/2)*arcsinh(x*c)^2-10*polylog(2,I*(x*c+(c^2*x^2+1)^(1/2)))*x^2*c^2-5*pol ylog(2,I*(x*c+(c^2*x^2+1)^(1/2)))*x^4*c^4-3*arcsinh(x*c)*ln(1+I*(x*c+(c^2* x^2+1)^(1/2))))*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(c^2*x^2+1)^(1/2) /f^3/d^2/c/(c^6*x^6+3*c^4*x^4+3*c^2*x^2+1)-1/3*a*b*(5*ln(x*c+(c^2*x^2+1)^( 1/2)+I)*x^4*c^4+3*ln(x*c+(c^2*x^2+1)^(1/2)-I)*x^4*c^4-4*arcsinh(x*c)*c^4*x ^4-4*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^3*c^3-I*x*c+10*ln(x*c+(c^2*x^2+1)^(1 /2)+I)*x^2*c^2+6*ln(x*c+(c^2*x^2+1)^(1/2)-I)*x^2*c^2-8*arcsinh(x*c)*c^2...
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (-i \, c f x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(5/2),x, algo rithm="fricas")
Output:
1/3*((2*b^2*c^2*x^2 + 2*I*b^2*c*x + b^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 3*(c^4*d^2*f^3*x^3 + I*c^3*d^2*f^3*x^ 2 + c^2*d^2*f^3*x + I*c*d^2*f^3)*integral(1/3*(3*I*sqrt(I*c*d*x + d)*sqrt( -I*c*f*x + f)*a^2 - 2*(-3*I*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*a*b + (2* b^2*c^2*x^2 + 2*I*b^2*c*x + b^2)*sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x + d)*sqrt( -I*c*f*x + f))*log(c*x + sqrt(c^2*x^2 + 1)))/(c^5*d^2*f^3*x^5 + I*c^4*d^2* f^3*x^4 + 2*c^3*d^2*f^3*x^3 + 2*I*c^2*d^2*f^3*x^2 + c*d^2*f^3*x + I*d^2*f^ 3), x))/(c^4*d^2*f^3*x^3 + I*c^3*d^2*f^3*x^2 + c^2*d^2*f^3*x + I*c*d^2*f^3 )
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((a+b*asinh(c*x))**2/(d+I*c*d*x)**(3/2)/(f-I*c*f*x)**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(5/2),x, algo rithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(5/2),x, algo rithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \] Input:
int((a + b*asinh(c*x))^2/((d + c*d*x*1i)^(3/2)*(f - c*f*x*1i)^(5/2)),x)
Output:
int((a + b*asinh(c*x))^2/((d + c*d*x*1i)^(3/2)*(f - c*f*x*1i)^(5/2)), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx=\frac {-6 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{3} i \,x^{3}-\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c i x -\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) a b \,c^{3} x^{2}-6 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{3} i \,x^{3}-\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c i x -\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) a b c -3 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{3} i \,x^{3}-\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c i x -\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) b^{2} c^{3} x^{2}-3 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{3} i \,x^{3}-\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c i x -\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) b^{2} c +2 a^{2} c^{3} x^{3}+3 a^{2} c x -a^{2} i}{3 \sqrt {f}\, \sqrt {d}\, \sqrt {c i x +1}\, \sqrt {-c i x +1}\, c d \,f^{2} \left (c^{2} x^{2}+1\right )} \] Input:
int((a+b*asinh(c*x))^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(5/2),x)
Output:
( - 6*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int(asinh(c*x)/(sqrt(c*i*x + 1)*s qrt( - c*i*x + 1)*c**3*i*x**3 - sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c**2*x* *2 + sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c*i*x - sqrt(c*i*x + 1)*sqrt( - c* i*x + 1)),x)*a*b*c**3*x**2 - 6*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int(asin h(c*x)/(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c**3*i*x**3 - sqrt(c*i*x + 1)*s qrt( - c*i*x + 1)*c**2*x**2 + sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c*i*x - s qrt(c*i*x + 1)*sqrt( - c*i*x + 1)),x)*a*b*c - 3*sqrt(c*i*x + 1)*sqrt( - c* i*x + 1)*int(asinh(c*x)**2/(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c**3*i*x**3 - sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c**2*x**2 + sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c*i*x - sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)),x)*b**2*c**3*x**2 - 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int(asinh(c*x)**2/(sqrt(c*i*x + 1)*s qrt( - c*i*x + 1)*c**3*i*x**3 - sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c**2*x* *2 + sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c*i*x - sqrt(c*i*x + 1)*sqrt( - c* i*x + 1)),x)*b**2*c + 2*a**2*c**3*x**3 + 3*a**2*c*x - a**2*i)/(3*sqrt(f)*s qrt(d)*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c*d*f**2*(c**2*x**2 + 1))