3.531 \(\int \frac{\tan ^2(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx\)

Optimal. Leaf size=424 \[ \frac{b \log \left (-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \log \left (\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{2 \sqrt{a+b \tan (c+d x)}}{b d} \]

[Out]

-((b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[
2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d)) + (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*
Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*Log[a +
Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqr
t[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqr
t[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (2*Sqrt[a +
b*Tan[c + d*x]])/(b*d)

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Rubi [A]  time = 0.334532, antiderivative size = 424, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {3543, 3485, 708, 1094, 634, 618, 206, 628} \[ \frac{b \log \left (-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \log \left (\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{2 \sqrt{a+b \tan (c+d x)}}{b d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^2/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

-((b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[
2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d)) + (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*
Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*Log[a +
Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqr
t[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqr
t[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (2*Sqrt[a +
b*Tan[c + d*x]])/(b*d)

Rule 3543

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[
(d^2*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e
+ f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] &&  !LeQ[m, -1] &&  !(EqQ[m, 2] && EqQ
[a, 0])

Rule 3485

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x
, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]

Rule 708

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^2 + a*e^2 - 2*c
*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1094

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [C]  time = 0.190288, size = 108, normalized size = 0.25 \[ \frac{\frac{2 \sqrt{a+b \tan (c+d x)}}{b}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{\sqrt{a-i b}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{\sqrt{a+i b}}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^2/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

((I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[
a + I*b]])/Sqrt[a + I*b] + (2*Sqrt[a + b*Tan[c + d*x]])/b)/d

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Maple [B]  time = 0.053, size = 1577, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a+b*tan(d*x+c))^(1/2)/b/d-1/4/d/b/(a^2+b^2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a
)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2-1/4/d*b/(a^2+b^2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c)
)^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/4/d/b/(a^2+b^2)^(3/2)*l
n(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)
^(1/2)*a^3+1/4/d*b/(a^2+b^2)^(3/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2
+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/d/b/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+
b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^2+1/d*b/(a^2+b^2)^(1/2)/(2
*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)
-2*a)^(1/2))-1/d/b/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)
^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^4-3/d*b/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arct
an((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^2-2/d*b^3/(a^2+b^
2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2
+b^2)^(1/2)-2*a)^(1/2))+1/4/d/b/(a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)
-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+1/4/d*b/(a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)
^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-1/4/d/b/(a^2+b^2)^(3/2)*ln((a+
b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2
)*a^3-1/4/d*b/(a^2+b^2)^(3/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)
^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/d/b/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^
2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^2-1/d*b/(a^2+b^2)^(1/2)/(2*(a^2
+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)
^(1/2))+1/d/b/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d
*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^4+3/d*b/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((
2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^2+2/d*b^3/(a^2+b^2)^(3
/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)
^(1/2)-2*a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (d x + c\right )^{2}}{\sqrt{b \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.94813, size = 4091, normalized size = 9.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*sqrt(2)*(a^2*b + b^3)*d^5*sqrt(((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) + a^2 + b^2)/b^2)*sqrt(b^2/
((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d^4))^(3/4)*arctan((sqrt(2)*(a^4 + 2*a^2*b^2 + b^4)*d^7*sqrt((s
qrt(2)*b^3*d*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*
d^4)) + a^2 + b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4)*cos(d*x + c) + (a^2*b^2 + b^4)*d^2*sqrt(1/((a^2 + b^2)*d^4
))*cos(d*x + c) + a*b^2*cos(d*x + c) + b^3*sin(d*x + c))/cos(d*x + c))*sqrt(((a^3 + a*b^2)*d^2*sqrt(1/((a^2 +
b^2)*d^4)) + a^2 + b^2)/b^2)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d^4))^(5/4) - sqrt(2)*(a^
4*b + 2*a^2*b^3 + b^5)*d^7*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(((a^3 + a*b^2)*d^2*sqrt(1
/((a^2 + b^2)*d^4)) + a^2 + b^2)/b^2)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d^4))^(5/4) - (a
^4 + 2*a^2*b^2 + b^4)*d^4*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) - (a^3 + a*b^2)*d^
2*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/b^2) + 4*sqrt(2)*(a^2*b + b^3)*d^5*sqrt(((a^3 + a*b^2)*d^2*sqrt(1/(
(a^2 + b^2)*d^4)) + a^2 + b^2)/b^2)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d^4))^(3/4)*arctan
((sqrt(2)*(a^4 + 2*a^2*b^2 + b^4)*d^7*sqrt(-(sqrt(2)*b^3*d*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)
)*sqrt(((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) + a^2 + b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4)*cos(d*x + c)
- (a^2*b^2 + b^4)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) - a*b^2*cos(d*x + c) - b^3*sin(d*x + c))/cos(d*x
+ c))*sqrt(((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) + a^2 + b^2)/b^2)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d
^4))*(1/((a^2 + b^2)*d^4))^(5/4) - sqrt(2)*(a^4*b + 2*a^2*b^3 + b^5)*d^7*sqrt((a*cos(d*x + c) + b*sin(d*x + c)
)/cos(d*x + c))*sqrt(((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) + a^2 + b^2)/b^2)*sqrt(b^2/((a^4 + 2*a^2*b^2
+ b^4)*d^4))*(1/((a^2 + b^2)*d^4))^(5/4) + (a^4 + 2*a^2*b^2 + b^4)*d^4*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)
)*sqrt(1/((a^2 + b^2)*d^4)) + (a^3 + a*b^2)*d^2*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/b^2) + sqrt(2)*(a*b*d
^3*sqrt(1/((a^2 + b^2)*d^4)) - b*d)*sqrt(((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) + a^2 + b^2)/b^2)*(1/((a
^2 + b^2)*d^4))^(1/4)*log((sqrt(2)*b^3*d*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(((a^3 + a*b
^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) + a^2 + b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4)*cos(d*x + c) + (a^2*b^2 + b^4)
*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + a*b^2*cos(d*x + c) + b^3*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*(
a*b*d^3*sqrt(1/((a^2 + b^2)*d^4)) - b*d)*sqrt(((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) + a^2 + b^2)/b^2)*(
1/((a^2 + b^2)*d^4))^(1/4)*log(-(sqrt(2)*b^3*d*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(((a^3
+ a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) + a^2 + b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4)*cos(d*x + c) - (a^2*b^2
+ b^4)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) - a*b^2*cos(d*x + c) - b^3*sin(d*x + c))/cos(d*x + c)) + 8*s
qrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/(b*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (d x + c\right )^{2}}{\sqrt{b \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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