3.1.44 \(\int \frac {1}{(a+b \tan (c+d \sqrt {x}))^2} \, dx\) [44]

Optimal. Leaf size=204 \[ \frac {\left (b+2 a d \sqrt {x}\right )^2}{2 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac {x}{a^2+b^2}+\frac {2 b \left (b+2 a d \sqrt {x}\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {2 i a b \text {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {2 b \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \]

[Out]

-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)))

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Rubi [A]
time = 0.19, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3824, 3814, 3813, 2221, 2317, 2438} \begin {gather*} -\frac {2 i a b \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {2 b \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 )}{d^2 \left (a^2+b^2\right )^2}-\frac {2 b \sqrt {x}}{d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}+\frac {\left (2 a d \sqrt {x}+b\right )^2}{2 a d^2 (a+i b) \left (a^2+b^2\right )}-\frac {x}{a^2+b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*Sqrt[x]])^(-2),x]

[Out]

(b + 2*a*d*Sqrt[x])^2/(2*a*(a + I*b)*(a^2 + b^2)*d^2) - x/(a^2 + b^2) + (2*b*(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)^2*d^2) - ((2*I)*a*b*PolyLog[2, -(((a^2 + b^2)*E^
((2*I)*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)^2*d^2) - (2*b*Sqrt[x])/((a^2 + b^2)*d*(a + b*Tan[c + d*Sq
rt[x]]))

Rule 2221

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]
- Dist[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]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2438

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]

Rule 3813

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(d*
(m + 1)*(a + I*b)), x] + Dist[2*I*b, Int[(c + d*x)^m*(E^Simp[2*I*(e + f*x), x]/((a + I*b)^2 + (a^2 + b^2)*E^Si
mp[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]

Rule 3814

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] + (Dist[1/(f*(a^2 + b^2)), Int[(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]

Rule 3824

Int[((a_.) + (b_.)*Tan[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*Ta
n[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {x}{(a+b \tan (c+d x))^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {x}{a^2+b^2}-\frac {2 b \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}+\frac {2 \text {Subst}\left (\int \frac {b+2 a d x}{a+b \tan (c+d x)} \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right ) d}\\ &=\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 \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}+\frac {(4 i b) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} (b+2 a d x)}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right ) d}\\ &=\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 b \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}-\frac {(4 a b) \text {Subst}\left (\int \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right )^2 d}\\ &=\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 b \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}+\frac {(2 i a b) \text {Subst}\left (\int \frac {\log \left (1+\frac {\left (a^2+b^2\right ) x}{(a+i b)^2}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{\left (a^2+b^2\right )^2 d^2}\\ &=\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 \text {Li}_2\left (-\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 )}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(772\) vs. \(2(204)=408\).
time = 6.59, size = 772, normalized size = 3.78 \begin {gather*} \frac {\left (-c+d \sqrt {x}\right ) \left (c+d \sqrt {x}\right ) \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 )^2}{(a-i b) (a+i b) d^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2}+\frac {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 ) \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 )^2}{a (a-i b) (a+i b) \left (a^2+b^2\right ) d^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2}-\frac {4 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 ) \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 )^2}{(a-i b) (a+i b) \left (a^2+b^2\right ) d^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2}-\frac {2 \left (e^{i \text {ArcTan}\left (\frac {a}{b}\right )} \left (c+d \sqrt {x}\right )^2+\frac {a \left (i \left (c+d \sqrt {x}\right ) \left (-\pi +2 \text {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}+\text {ArcTan}\left (\frac {a}{b}\right )\right ) \log \left (1-e^{2 i \left (c+d \sqrt {x}+\text {ArcTan}\left (\frac {a}{b}\right )\right )}\right )+\pi \log \left (\cos \left (c+d \sqrt {x}\right )\right )+2 \text {ArcTan}\left (\frac {a}{b}\right ) \log \left (\sin \left (c+d \sqrt {x}+\text {ArcTan}\left (\frac {a}{b}\right )\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}+\text {ArcTan}\left (\frac {a}{b}\right )\right )}\right )\right )}{\sqrt {1+\frac {a^2}{b^2}} b}\right ) \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 )^2}{(a-i b) (a+i b) \sqrt {\frac {a^2+b^2}{b^2}} d^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2}+\frac {2 \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 (-b^2 c \sin \left (c+d \sqrt {x}\right )+b^2 \left (c+d \sqrt {x}\right ) \sin \left (c+d \sqrt {x}\right )\right )}{a (a-i b) (a+i b) d^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Tan[c + d*Sqrt[x]])^(-2),x]

[Out]

((-c + d*Sqrt[x])*(c + d*Sqrt[x])*Sec[c + d*Sqrt[x]]^2*(a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]])^2)/((a -
I*b)*(a + I*b)*d^2*(a + b*Tan[c + d*Sqrt[x]])^2) + (2*b^2*(-(b*(c + d*Sqrt[x])) + a*Log[a*Cos[c + d*Sqrt[x]] +
b*Sin[c + d*Sqrt[x]]])*Sec[c + d*Sqrt[x]]^2*(a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]])^2)/(a*(a - I*b)*(a
+ I*b)*(a^2 + b^2)*d^2*(a + b*Tan[c + d*Sqrt[x]])^2) - (4*b*c*(-(b*(c + d*Sqrt[x])) + a*Log[a*Cos[c + d*Sqrt[x
]] + b*Sin[c + d*Sqrt[x]]])*Sec[c + d*Sqrt[x]]^2*(a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]])^2)/((a - I*b)*(
a + I*b)*(a^2 + b^2)*d^2*(a + b*Tan[c + d*Sqrt[x]])^2) - (2*(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])*Lo
g[1 - E^((2*I)*(c + d*Sqrt[x] + ArcTan[a/b]))] + Pi*Log[Cos[c + d*Sqrt[x]]] + 2*ArcTan[a/b]*Log[Sin[c + d*Sqrt
[x] + ArcTan[a/b]]] + I*PolyLog[2, E^((2*I)*(c + d*Sqrt[x] + ArcTan[a/b]))]))/(Sqrt[1 + a^2/b^2]*b))*Sec[c + d
*Sqrt[x]]^2*(a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]])^2)/((a - I*b)*(a + I*b)*Sqrt[(a^2 + b^2)/b^2]*d^2*(a
+ b*Tan[c + d*Sqrt[x]])^2) + (2*Sec[c + d*Sqrt[x]]^2*(a*Cos[c + d*Sqrt[x]] + b*Sin[c + d*Sqrt[x]])*(-(b^2*c*S
in[c + d*Sqrt[x]]) + b^2*(c + d*Sqrt[x])*Sin[c + d*Sqrt[x]]))/(a*(a - I*b)*(a + I*b)*d^2*(a + b*Tan[c + d*Sqrt
[x]])^2)

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Maple [F]
time = 0.95, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \tan \left (c +d \sqrt {x}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(c+d*x^(1/2)))^2,x)

[Out]

int(1/(a+b*tan(c+d*x^(1/2)))^2,x)

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Maxima [B] 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 = 0.74, size = 994, normalized size = 4.87 \begin {gather*} \frac {{\left (a^{3} - i \, a^{2} b + a b^{2} - i \, b^{3}\right )} d^{2} x - 2 \, {\left (-i \, a b^{2} + b^{3} + {\left (-i \, a b^{2} - b^{3}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a b^{2} - i \, b^{3}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )\right )} \arctan \left (-b \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + a \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + b, a \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + a\right ) - 4 \, {\left ({\left (i \, a^{2} b + a b^{2}\right )} d \sqrt {x} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - {\left (a^{2} b - i \, a b^{2}\right )} d \sqrt {x} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (i \, a^{2} b - a b^{2}\right )} d \sqrt {x}\right )} \arctan \left (\frac {2 \, a b \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - {\left (a^{2} - b^{2}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )}{a^{2} + b^{2}}, \frac {2 \, a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right )}{a^{2} + b^{2}}\right ) + {\left ({\left (a^{3} - 3 i \, a^{2} b - 3 \, a b^{2} + i \, b^{3}\right )} d^{2} x - 4 \, {\left (i \, a b^{2} + b^{3}\right )} d \sqrt {x}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - 2 \, {\left (i \, a^{2} b - a b^{2} + {\left (i \, a^{2} b + a b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - {\left (a^{2} b - i \, a b^{2}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )\right )} {\rm Li}_2\left (\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d \sqrt {x} + 2 i \, c\right )}}{-i \, a + b}\right ) + {\left (a b^{2} + i \, b^{3} + {\left (a b^{2} - i \, b^{3}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (i \, a b^{2} + b^{3}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )\right )} \log \left ({\left (a^{2} + b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 4 \, a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right )\right ) + 2 \, {\left ({\left (a^{2} b - i \, a b^{2}\right )} d \sqrt {x} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - {\left (-i \, a^{2} b - a b^{2}\right )} d \sqrt {x} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a^{2} b + i \, a b^{2}\right )} d \sqrt {x}\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 4 \, a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right )}{a^{2} + b^{2}}\right ) + {\left ({\left (i \, a^{3} + 3 \, a^{2} b - 3 i \, a b^{2} - b^{3}\right )} d^{2} x + 4 \, {\left (a b^{2} - i \, b^{3}\right )} d \sqrt {x}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )}{{\left (a^{5} - i \, a^{4} b + 2 \, a^{3} b^{2} - 2 i \, a^{2} b^{3} + a b^{4} - i \, b^{5}\right )} d^{2} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - {\left (-i \, a^{5} - a^{4} b - 2 i \, a^{3} b^{2} - 2 \, a^{2} b^{3} - i \, a b^{4} - b^{5}\right )} d^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a^{5} + i \, a^{4} b + 2 \, a^{3} b^{2} + 2 i \, a^{2} b^{3} + a b^{4} + i \, b^{5}\right )} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(c+d*x^(1/2)))^2,x, algorithm="maxima")

[Out]

((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))*arctan2(-b*cos(2*d*sqrt(x) + 2*c) + a*sin(2*d*sqrt(x) + 2*c) + b, a*cos(2*d*s
qrt(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))*cos(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)*sin(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*sqr
t(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 + 4*(a*b^2 - I*b^3)*d*sqrt(x))*sin(2*d*sqrt(x) + 2*c))/((a^5 - I*
a^4*b + 2*a^3*b^2 - 2*I*a^2*b^3 + a*b^4 - I*b^5)*d^2*cos(2*d*sqrt(x) + 2*c) - (-I*a^5 - a^4*b - 2*I*a^3*b^2 -
2*a^2*b^3 - I*a*b^4 - b^5)*d^2*sin(2*d*sqrt(x) + 2*c) + (a^5 + I*a^4*b + 2*a^3*b^2 + 2*I*a^2*b^3 + a*b^4 + I*b
^5)*d^2)

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Fricas [B] 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 = 0.43, size = 828, normalized size = 4.06 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

-(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)*ta
n(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*s
qrt(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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \tan {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(c+d*x**(1/2)))**2,x)

[Out]

Integral((a + b*tan(c + d*sqrt(x)))**(-2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

integrate((b*tan(d*sqrt(x) + c) + a)^(-2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*tan(c + d*x^(1/2)))^2,x)

[Out]

int(1/(a + b*tan(c + d*x^(1/2)))^2, x)

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