Integrand size = 37, antiderivative size = 779 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=-\frac {i b^2}{3 c d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 x}{3 d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b (a+b \text {arcsinh}(c x))}{3 c d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}}-\frac {i b x (a+b \text {arcsinh}(c x))}{3 d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i (a+b \text {arcsinh}(c x))^2}{3 c d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}+\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:
-1/3*I*b^2/c/d^2/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-1/3*b^2*x/d^2/f/(d+ I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+1/3*b*(a+b*arcsinh(c*x))/c/d^2/f/(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^2/f/(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*ar csinh(c*x))^2/d^2/f/(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^2/f/(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^2/f/(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^2/f/(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^2/f/(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^2/f/(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^2/f/(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^2/f/(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^2/f/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)
Time = 9.59 (sec) , antiderivative size = 754, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(3/2)) ,x]
Output:
(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(((-1/6*I)*a^2)/(d^3*f^2*(-I + c*x)^2) + (5*a^2)/(12*d^3*f^2*(-I + c*x)) + a^2/(4*d^3*f^2*(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*Ar cSinh[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^2*f* (-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]*(7*Pi*ArcSinh[c*x] + ((2 + I*ArcSinh[c*x])*ArcSinh[c*x])/(-I + c*x) - (1 + 4*I)*ArcSinh[c*x]^2 - 5*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E ^ArcSinh[c*x]] + 3*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] - 1 6*Pi*Log[1 + E^ArcSinh[c*x]] - 3*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 16*Pi*Log[Cosh[ArcSinh[c*x]/2]] + 5*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x] )/4]] + (6*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (10*I)*PolyLog[2, I/E^ArcS inh[c*x]] + ((3*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x] /2] - I*Sinh[ArcSinh[c*x]/2]) + ((2*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2] )/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3 + ((-4 + 5*ArcSinh[c*x ]^2)*Sinh[ArcSinh[c*x]/2])/((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/ 2])))/(c*d^2*f*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[-(d*f*(1 + c^2 *x^2))])
Time = 1.17 (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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6211 |
| \(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \int \frac {f (1-i c x) (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{5/2}}dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f \left (c^2 x^2+1\right )^{5/2} \int \frac {(1-i c x) (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{5/2}}dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {f \left (c^2 x^2+1\right )^{5/2} \int \left (\frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{5/2}}-\frac {i c x (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{5/2}}\right )dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f \left (c^2 x^2+1\right )^{5/2} \left (-\frac {2 i b \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 c}-\frac {i b x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )}+\frac {b (a+b \text {arcsinh}(c x))}{3 c \left (c^2 x^2+1\right )}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 \sqrt {c^2 x^2+1}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {i (a+b \text {arcsinh}(c x))^2}{3 c \left (c^2 x^2+1\right )^{3/2}}+\frac {2 (a+b \text {arcsinh}(c x))^2}{3 c}-\frac {4 b \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{3 c}-\frac {b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c}+\frac {b^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c}-\frac {2 b^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c}-\frac {b^2 x}{3 \sqrt {c^2 x^2+1}}-\frac {i b^2}{3 c \sqrt {c^2 x^2+1}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
Input:
Int[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(3/2)),x]
Output:
(f*(1 + c^2*x^2)^(5/2)*(((-1/3*I)*b^2)/(c*Sqrt[1 + c^2*x^2]) - (b^2*x)/(3* Sqrt[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*Arc Sinh[c*x])^2)/(3*(1 + c^2*x^2)^(3/2)) + (2*x*(a + b*ArcSinh[c*x])^2)/(3*Sq rt[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 *PolyLog[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 = 6.21 (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)^(5/2)/(f-I*c*f*x)^(3/2),x,method=_RET URNVERBOSE)
Output:
a^2*(1/3*I/c/d/f/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(1/2)+2/3/d*(I/c/d/f/(d+I*c *d*x)^(1/2)/(f-I*c*f*x)^(1/2)-I/c/f/d^2/(f-I*c*f*x)^(1/2)*(d+I*c*d*x)^(1/2 )))+1/3*b^2*(1-3*polylog(2,I*(x*c+(c^2*x^2+1)^(1/2)))-5*polylog(2,-I*(x*c+ (c^2*x^2+1)^(1/2)))+2*arcsinh(x*c)^2+arcsinh(x*c)-3*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)-5*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^3-10*ln(1+I*(x*c+(c^2*x^2+1)^(1/2)))*arcsinh (x*c)*x^2*c^2+c^4*x^4-3*arcsinh(x*c)*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))*x^4*c ^4-I*arcsinh(x*c)*x^3*c^3-I*arcsinh(x*c)*x*c-I*(c^2*x^2+1)^(1/2)*x^2*c^2-6 *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^4-5*polylog(2,-I*(x*c+(c^2*x^2+1)^(1/2)))*x^4*c^4+4*arcsinh(x*c)^2 *x^2*c^2-10*polylog(2,-I*(x*c+(c^2*x^2+1)^(1/2)))*x^2*c^2-6*polylog(2,I*(x *c+(c^2*x^2+1)^(1/2)))*x^2*c^2-3*polylog(2,I*(x*c+(c^2*x^2+1)^(1/2)))*x^4* c^4+I*(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2-5*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^ 2/d^3/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*x^...
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}} {\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(3/2),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^3*f^2*x^3 - I*c^3*d^3*f^2*x^ 2 + c^2*d^3*f^2*x - I*c*d^3*f^2)*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^3*f^2*x^5 - I*c^4*d^3* f^2*x^4 + 2*c^3*d^3*f^2*x^3 - 2*I*c^2*d^3*f^2*x^2 + c*d^3*f^2*x - I*d^3*f^ 2), x))/(c^4*d^3*f^2*x^3 - I*c^3*d^3*f^2*x^2 + c^2*d^3*f^2*x - I*c*d^3*f^2 )
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((a+b*asinh(c*x))**2/(d+I*c*d*x)**(5/2)/(f-I*c*f*x)**(3/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(3/2),x, algo rithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(3/2),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. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \] Input:
int((a + b*asinh(c*x))^2/((d + c*d*x*1i)^(5/2)*(f - c*f*x*1i)^(3/2)),x)
Output:
int((a + b*asinh(c*x))^2/((d + c*d*x*1i)^(5/2)*(f - c*f*x*1i)^(3/2)), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\frac {6 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{3} i \,x^{3}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c i x +\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) a b \,c^{3} x^{2}+6 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{3} i \,x^{3}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c i x +\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) a b c +3 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{3} i \,x^{3}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c i x +\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) b^{2} c^{3} x^{2}+3 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{3} i \,x^{3}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c i x +\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) b^{2} c +2 a^{2} c^{3} x^{3}+3 a^{2} c x +a^{2} i}{3 \sqrt {f}\, \sqrt {d}\, \sqrt {c i x +1}\, \sqrt {-c i x +1}\, c \,d^{2} f \left (c^{2} x^{2}+1\right )} \] Input:
int((a+b*asinh(c*x))^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(3/2),x)
Output:
(6*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int(asinh(c*x)/(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)*a*b*c**3*x**2 + 6*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int(asinh(c *x)/(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)*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)*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 + 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**2*f*(c**2*x**2 + 1))