Integrand size = 16, antiderivative size = 204 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {\left (b+2 a d \sqrt {x}\right )^2}{2 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac {x}{a^2+b^2}+\frac {2 b \left (b+2 a d \sqrt {x}\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {2 i a b \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {2 b \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \]
-x/(a^2+b^2)-2*I*a*b*polylog(2,-(a^2+b^2)*exp(2*I*(c+d*x^(1/2)))/(a+I*b)^2 )/(a^2+b^2)^2/d^2+2*b*ln(1+(a^2+b^2)*exp(2*I*(c+d*x^(1/2)))/(a+I*b)^2)*(b+ 2*a*d*x^(1/2))/(a^2+b^2)^2/d^2+1/2*(b+2*a*d*x^(1/2))^2/a/(a+I*b)/(a^2+b^2) /d^2-2*b*x^(1/2)/(a^2+b^2)/d/(a+b*tan(c+d*x^(1/2)))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(517\) vs. \(2(204)=408\).
Time = 5.67 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.53 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {\sec ^2\left (c+d \sqrt {x}\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right ) \left (2 b^2 \left (a^2+b^2\right ) d \sqrt {x} \sin \left (c+d \sqrt {x}\right )-a \left (a^2+b^2\right ) \left (c-d \sqrt {x}\right ) \left (c+d \sqrt {x}\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )-2 b^2 \left (b \left (c+d \sqrt {x}\right )-a \log \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )+4 a b c \left (b \left (c+d \sqrt {x}\right )-a \log \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )-2 a b \left (\sqrt {1+\frac {a^2}{b^2}} b e^{i \arctan \left (\frac {a}{b}\right )} \left (c+d \sqrt {x}\right )^2+a \left (-i \left (c+d \sqrt {x}\right ) \left (\pi -2 \arctan \left (\frac {a}{b}\right )\right )-\pi \log \left (1+e^{-2 i \left (c+d \sqrt {x}\right )}\right )-2 \left (c+d \sqrt {x}+\arctan \left (\frac {a}{b}\right )\right ) \log \left (1-e^{2 i \left (c+d \sqrt {x}+\arctan \left (\frac {a}{b}\right )\right )}\right )+\pi \log \left (\cos \left (c+d \sqrt {x}\right )\right )+2 \arctan \left (\frac {a}{b}\right ) \log \left (\sin \left (c+d \sqrt {x}+\arctan \left (\frac {a}{b}\right )\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}+\arctan \left (\frac {a}{b}\right )\right )}\right )\right )\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )\right )}{a \left (a^2+b^2\right )^2 d^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \]
(Sec[c + d*Sqrt[x]]^2*(a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]])*(2*b^2 *(a^2 + b^2)*d*Sqrt[x]*Sin[c + d*Sqrt[x]] - a*(a^2 + b^2)*(c - d*Sqrt[x])* (c + d*Sqrt[x])*(a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]]) - 2*b^2*(b*( c + d*Sqrt[x]) - a*Log[a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]]])*(a*Co s[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]]) + 4*a*b*c*(b*(c + d*Sqrt[x]) - a* Log[a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]]])*(a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]]) - 2*a*b*(Sqrt[1 + a^2/b^2]*b*E^(I*ArcTan[a/b])*(c + d*Sqrt[x])^2 + a*((-I)*(c + d*Sqrt[x])*(Pi - 2*ArcTan[a/b]) - Pi*Log[1 + E ^((-2*I)*(c + d*Sqrt[x]))] - 2*(c + d*Sqrt[x] + ArcTan[a/b])*Log[1 - E^((2 *I)*(c + d*Sqrt[x] + ArcTan[a/b]))] + Pi*Log[Cos[c + d*Sqrt[x]]] + 2*ArcTa n[a/b]*Log[Sin[c + d*Sqrt[x] + ArcTan[a/b]]] + I*PolyLog[2, E^((2*I)*(c + d*Sqrt[x] + ArcTan[a/b]))]))*(a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]]) ))/(a*(a^2 + b^2)^2*d^2*(a + b*Tan[c + d*Sqrt[x]])^2)
Time = 0.74 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4226, 3042, 4216, 3042, 4215, 2620, 2715, 2838}
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.
\(\displaystyle \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx\) |
\(\Big \downarrow \) 4226 |
\(\displaystyle 2 \int \frac {\sqrt {x}}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 \int \frac {\sqrt {x}}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\) |
\(\Big \downarrow \) 4216 |
\(\displaystyle 2 \left (\frac {\int \frac {b+2 a d \sqrt {x}}{a+b \tan \left (c+d \sqrt {x}\right )}d\sqrt {x}}{d \left (a^2+b^2\right )}-\frac {b \sqrt {x}}{d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}-\frac {x}{2 \left (a^2+b^2\right )}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 \left (\frac {\int \frac {b+2 a d \sqrt {x}}{a+b \tan \left (c+d \sqrt {x}\right )}d\sqrt {x}}{d \left (a^2+b^2\right )}-\frac {b \sqrt {x}}{d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}-\frac {x}{2 \left (a^2+b^2\right )}\right )\) |
\(\Big \downarrow \) 4215 |
\(\displaystyle 2 \left (\frac {2 i b \int \frac {e^{2 i \left (c+d \sqrt {x}\right )} \left (b+2 a d \sqrt {x}\right )}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}d\sqrt {x}+\frac {\left (2 a d \sqrt {x}+b\right )^2}{4 a d (a+i b)}}{d \left (a^2+b^2\right )}-\frac {b \sqrt {x}}{d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}-\frac {x}{2 \left (a^2+b^2\right )}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle 2 \left (\frac {2 i b \left (\frac {i a \int \log \left (\frac {e^{2 i \left (c+d \sqrt {x}\right )} \left (a^2+b^2\right )}{(a+i b)^2}+1\right )d\sqrt {x}}{a^2+b^2}-\frac {i \left (2 a d \sqrt {x}+b\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {\left (2 a d \sqrt {x}+b\right )^2}{4 a d (a+i b)}}{d \left (a^2+b^2\right )}-\frac {b \sqrt {x}}{d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}-\frac {x}{2 \left (a^2+b^2\right )}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle 2 \left (\frac {2 i b \left (\frac {a \int \frac {\log \left (\frac {e^{2 i \left (c+d \sqrt {x}\right )} \left (a^2+b^2\right )}{(a+i b)^2}+1\right )}{\sqrt {x}}de^{2 i \left (c+d \sqrt {x}\right )}}{2 d \left (a^2+b^2\right )}-\frac {i \left (2 a d \sqrt {x}+b\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {\left (2 a d \sqrt {x}+b\right )^2}{4 a d (a+i b)}}{d \left (a^2+b^2\right )}-\frac {b \sqrt {x}}{d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}-\frac {x}{2 \left (a^2+b^2\right )}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle 2 \left (\frac {2 i b \left (-\frac {a \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}-\frac {i \left (2 a d \sqrt {x}+b\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {\left (2 a d \sqrt {x}+b\right )^2}{4 a d (a+i b)}}{d \left (a^2+b^2\right )}-\frac {b \sqrt {x}}{d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}-\frac {x}{2 \left (a^2+b^2\right )}\right )\) |
2*(-1/2*x/(a^2 + b^2) + ((b + 2*a*d*Sqrt[x])^2/(4*a*(a + I*b)*d) + (2*I)*b *(((-1/2*I)*(b + 2*a*d*Sqrt[x])*Log[1 + ((a^2 + b^2)*E^((2*I)*(c + d*Sqrt[ x])))/(a + I*b)^2])/((a^2 + b^2)*d) - (a*PolyLog[2, -(((a^2 + b^2)*E^((2*I )*(c + d*Sqrt[x])))/(a + I*b)^2)])/(2*(a^2 + b^2)*d)))/((a^2 + b^2)*d) - ( b*Sqrt[x])/((a^2 + b^2)*d*(a + b*Tan[c + d*Sqrt[x]])))
3.1.44.3.1 Defintions of rubi rules used
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Sy mbol] :> Simp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + I*b)), x] + Simp[2*I*b In t[(c + d*x)^m*(E^Simp[2*I*(e + f*x), x]/((a + I*b)^2 + (a^2 + b^2)*E^Simp[2 *I*(e + f*x), x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2 , 0] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol ] :> Simp[-(c + d*x)^2/(2*d*(a^2 + b^2)), x] + (Simp[1/(f*(a^2 + b^2)) In t[(b*d + 2*a*c*f + 2*a*d*f*x)/(a + b*Tan[e + f*x]), x], x] - Simp[b*((c + d *x)/(f*(a^2 + b^2)*(a + b*Tan[e + f*x]))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]
Int[((a_.) + (b_.)*Tan[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Simp[1 /n Subst[Int[x^(1/n - 1)*(a + b*Tan[c + d*x])^p, x], x, x^n], x] /; FreeQ [{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
\[\int \frac {1}{\left (a +b \tan \left (c +d \sqrt {x}\right )\right )^{2}}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (177) = 354\).
Time = 0.28 (sec) , antiderivative size = 828, normalized size of antiderivative = 4.06 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 \, b^{3} d \sqrt {x} - {\left (a^{3} - a b^{2}\right )} d^{2} x + {\left (a^{3} - a b^{2}\right )} d^{2} - {\left (i \, a b^{2} \tan \left (d \sqrt {x} + c\right ) + i \, a^{2} b\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) - {\left (-i \, a b^{2} \tan \left (d \sqrt {x} + c\right ) - i \, a^{2} b\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) - 2 \, {\left (a^{2} b d \sqrt {x} + a^{2} b c + {\left (a b^{2} d \sqrt {x} + a b^{2} c\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}}\right ) - 2 \, {\left (a^{2} b d \sqrt {x} + a^{2} b c + {\left (a b^{2} d \sqrt {x} + a b^{2} c\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}}\right ) + {\left (2 \, a^{2} b c - a b^{2} + {\left (2 \, a b^{2} c - b^{3}\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (\frac {{\left (i \, a b + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{\tan \left (d \sqrt {x} + c\right )^{2} + 1}\right ) + {\left (2 \, a^{2} b c - a b^{2} + {\left (2 \, a b^{2} c - b^{3}\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (\frac {{\left (i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{\tan \left (d \sqrt {x} + c\right )^{2} + 1}\right ) - {\left (2 \, a b^{2} d \sqrt {x} + {\left (a^{2} b - b^{3}\right )} d^{2} x - {\left (a^{2} b - b^{3}\right )} d^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d^{2} \tan \left (d \sqrt {x} + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d^{2}} \]
-(2*b^3*d*sqrt(x) - (a^3 - a*b^2)*d^2*x + (a^3 - a*b^2)*d^2 - (I*a*b^2*tan (d*sqrt(x) + c) + I*a^2*b)*dilog(2*((I*a*b - b^2)*tan(d*sqrt(x) + c)^2 - a ^2 - I*a*b + (I*a^2 - 2*a*b - I*b^2)*tan(d*sqrt(x) + c))/((a^2 + b^2)*tan( d*sqrt(x) + c)^2 + a^2 + b^2) + 1) - (-I*a*b^2*tan(d*sqrt(x) + c) - I*a^2* b)*dilog(2*((-I*a*b - b^2)*tan(d*sqrt(x) + c)^2 - a^2 + I*a*b + (-I*a^2 - 2*a*b + I*b^2)*tan(d*sqrt(x) + c))/((a^2 + b^2)*tan(d*sqrt(x) + c)^2 + a^2 + b^2) + 1) - 2*(a^2*b*d*sqrt(x) + a^2*b*c + (a*b^2*d*sqrt(x) + a*b^2*c)* tan(d*sqrt(x) + c))*log(-2*((I*a*b - b^2)*tan(d*sqrt(x) + c)^2 - a^2 - I*a *b + (I*a^2 - 2*a*b - I*b^2)*tan(d*sqrt(x) + c))/((a^2 + b^2)*tan(d*sqrt(x ) + c)^2 + a^2 + b^2)) - 2*(a^2*b*d*sqrt(x) + a^2*b*c + (a*b^2*d*sqrt(x) + a*b^2*c)*tan(d*sqrt(x) + c))*log(-2*((-I*a*b - b^2)*tan(d*sqrt(x) + c)^2 - a^2 + I*a*b + (-I*a^2 - 2*a*b + I*b^2)*tan(d*sqrt(x) + c))/((a^2 + b^2)* tan(d*sqrt(x) + c)^2 + a^2 + b^2)) + (2*a^2*b*c - a*b^2 + (2*a*b^2*c - b^3 )*tan(d*sqrt(x) + c))*log(((I*a*b + b^2)*tan(d*sqrt(x) + c)^2 - a^2 + I*a* b + (I*a^2 + I*b^2)*tan(d*sqrt(x) + c))/(tan(d*sqrt(x) + c)^2 + 1)) + (2*a ^2*b*c - a*b^2 + (2*a*b^2*c - b^3)*tan(d*sqrt(x) + c))*log(((I*a*b - b^2)* tan(d*sqrt(x) + c)^2 + a^2 + I*a*b + (I*a^2 + I*b^2)*tan(d*sqrt(x) + c))/( tan(d*sqrt(x) + c)^2 + 1)) - (2*a*b^2*d*sqrt(x) + (a^2*b - b^3)*d^2*x - (a ^2*b - b^3)*d^2)*tan(d*sqrt(x) + c))/((a^4*b + 2*a^2*b^3 + b^5)*d^2*tan(d* sqrt(x) + c) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2)
\[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (177) = 354\).
Time = 0.60 (sec) , antiderivative size = 994, normalized size of antiderivative = 4.87 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Too large to display} \]
((a^3 - I*a^2*b + a*b^2 - I*b^3)*d^2*x - 2*(-I*a*b^2 + b^3 + (-I*a*b^2 - b ^3)*cos(2*d*sqrt(x) + 2*c) + (a*b^2 - I*b^3)*sin(2*d*sqrt(x) + 2*c))*arcta n2(-b*cos(2*d*sqrt(x) + 2*c) + a*sin(2*d*sqrt(x) + 2*c) + b, a*cos(2*d*sqr t(x) + 2*c) + b*sin(2*d*sqrt(x) + 2*c) + a) - 4*((I*a^2*b + a*b^2)*d*sqrt( x)*cos(2*d*sqrt(x) + 2*c) - (a^2*b - I*a*b^2)*d*sqrt(x)*sin(2*d*sqrt(x) + 2*c) + (I*a^2*b - a*b^2)*d*sqrt(x))*arctan2((2*a*b*cos(2*d*sqrt(x) + 2*c) - (a^2 - b^2)*sin(2*d*sqrt(x) + 2*c))/(a^2 + b^2), (2*a*b*sin(2*d*sqrt(x) + 2*c) + a^2 + b^2 + (a^2 - b^2)*cos(2*d*sqrt(x) + 2*c))/(a^2 + b^2)) + (( a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*d^2*x - 4*(I*a*b^2 + b^3)*d*sqrt(x))*co s(2*d*sqrt(x) + 2*c) - 2*(I*a^2*b - a*b^2 + (I*a^2*b + a*b^2)*cos(2*d*sqrt (x) + 2*c) - (a^2*b - I*a*b^2)*sin(2*d*sqrt(x) + 2*c))*dilog((I*a + b)*e^( 2*I*d*sqrt(x) + 2*I*c)/(-I*a + b)) + (a*b^2 + I*b^3 + (a*b^2 - I*b^3)*cos( 2*d*sqrt(x) + 2*c) + (I*a*b^2 + b^3)*sin(2*d*sqrt(x) + 2*c))*log((a^2 + b^ 2)*cos(2*d*sqrt(x) + 2*c)^2 + 4*a*b*sin(2*d*sqrt(x) + 2*c) + (a^2 + b^2)*s in(2*d*sqrt(x) + 2*c)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*d*sqrt(x) + 2*c) ) + 2*((a^2*b - I*a*b^2)*d*sqrt(x)*cos(2*d*sqrt(x) + 2*c) - (-I*a^2*b - a* b^2)*d*sqrt(x)*sin(2*d*sqrt(x) + 2*c) + (a^2*b + I*a*b^2)*d*sqrt(x))*log(( (a^2 + b^2)*cos(2*d*sqrt(x) + 2*c)^2 + 4*a*b*sin(2*d*sqrt(x) + 2*c) + (a^2 + b^2)*sin(2*d*sqrt(x) + 2*c)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*d*sqrt( x) + 2*c))/(a^2 + b^2)) + ((I*a^3 + 3*a^2*b - 3*I*a*b^2 - b^3)*d^2*x + ...
\[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )\right )}^2} \,d x \]