3.345 \(\int \frac{\tan (c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{(A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d \sqrt{a-i b}}-\frac{(A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d \sqrt{a+i b}}+\frac{2 B \sqrt{a+b \tan (c+d x)}}{b d} \]

[Out]

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

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Rubi [A]  time = 0.222946, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3592, 3539, 3537, 63, 208} \[ -\frac{(A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d \sqrt{a-i b}}-\frac{(A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d \sqrt{a+i b}}+\frac{2 B \sqrt{a+b \tan (c+d x)}}{b d} \]

Antiderivative was successfully verified.

[In]

Int[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

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

Rule 3592

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

Rule 3539

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

Rule 3537

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.504041, size = 118, normalized size = 0.95 \[ -\frac{\frac{(A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{\sqrt{a-i b}}+\frac{(A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{\sqrt{a+i b}}-\frac{2 B \sqrt{a+b \tan (c+d x)}}{b}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Tan[c + d*x]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

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

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Maple [B]  time = 0.106, size = 3997, normalized size = 32.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d/(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*a+1/4/d/(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))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-2/d/(a^2+b^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*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))*B+1/4/d*b^2/(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))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+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))*B-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))*B+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))*B-1/4/d*b^2/(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))*A*(2*(a^2+b^2)^(1/2)+2*a)^(
1/2)-1/d/b^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*a^2+1/d*b^2/(a^2+b^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-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))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/4/d/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))*A*(2*(a^2+b^2)^(1/2)+2*a)
^(1/2)*a-1/d*b^2/(a^2+b^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+1/d/b^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*a^2-1/4/d/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))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+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))*B*(2*(a^2+b^2)^(1/2
)+2*a)^(1/2)-1/4/d/(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))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2-1/d/(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*a^3-1/4/d/(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))*A*(2*(a^2+b^2)^(1/2)+2*a)^
(1/2)*a+2/d/(a^2+b^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*a^2+1/4/d/(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))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+1/d/(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*a+1/d/b^2/(a^2+b^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*a^4-1/4/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))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2-1/4/d/b^2/(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))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a
^3+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))*B*a^4+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))*B*a^2-1/d/b^2/(a^2+b^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*a^4+1/4/d/b^2/(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))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+1/d*b^2/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arct
an(((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*a+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))*B*a^2+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))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3-1/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))*B*a^2+1
/d/(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*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))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+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))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^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))*B*a^4-1/4/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))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+1/d/b^2*(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*a-1/d/b^2/(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*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))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/d/b^2*(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*a+1/d/b^2/(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*a^3-1/d*b^2/(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*a-3/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))*B*a^2+2*B*(a+b*tan(d*x+c))^(1/2)/b/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )}{\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)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((B*tan(d*x + c) + A)*tan(d*x + c)/sqrt(b*tan(d*x + c) + a), x)

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Fricas [B]  time = 20.9122, size = 17095, normalized size = 137.86 \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)*(A+B*tan(d*x+c))/(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(-((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sq
rt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) - (A^4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*
A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt((4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*
b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))^(3
/4)*arctan(-((2*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^5 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^4*b + 4*(A
^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^3*b^2 - 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^2*b^3 + 2*(A^7*B + 3*A
^5*B^3 + 3*A^3*B^5 + A*B^7)*a*b^4 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*b^5)*d^4*sqrt((4*A^2*B^2*a^2 - 4*(A^3*
B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^
2 + b^2)*d^4)) + (2*(A^9*B + 4*A^7*B^3 + 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*a^4 - (A^10 + 3*A^8*B^2 + 2*A^6*B^4 -
2*A^4*B^6 - 3*A^2*B^8 - B^10)*a^3*b + 2*(A^9*B + 4*A^7*B^3 + 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*a^2*b^2 - (A^10 +
3*A^8*B^2 + 2*A^6*B^4 - 2*A^4*B^6 - 3*A^2*B^8 - B^10)*a*b^3)*d^2*sqrt((4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b +
(A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((A*a^5 + B*a^4*b + 2*A*a^3*b^2 + 2*B*a
^2*b^3 + A*a*b^4 + B*b^5)*d^7*sqrt((4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4
+ 2*a^2*b^2 + b^4)*d^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) + ((A^3 + A*B^2)*a^4 + 2*(A^3 + A*B^
2)*a^2*b^2 + (A^3 + A*B^2)*b^4)*d^5*sqrt((4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)
/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sq
rt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) - (A^4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*
A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt(((4*(A^4*B^2 + A^2*B^4)*a^4 - 4*(A^5*
B - A*B^5)*a^3*b + (A^6 + 3*A^4*B^2 + 3*A^2*B^4 + B^6)*a^2*b^2 - 4*(A^5*B - A*B^5)*a*b^3 + (A^6 - A^4*B^2 - A^
2*B^4 + B^6)*b^4)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*((4*A^3*B^2*a^4 -
4*(A^4*B - A^2*B^3)*a^3*b + (A^5 + 2*A^3*B^2 + A*B^4)*a^2*b^2 - 4*(A^4*B - A^2*B^3)*a*b^3 + (A^5 - 2*A^3*B^2
+ A*B^4)*b^4)*d^3*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) + (4*(A^5*B^2 + A^3*B^4)*a^3 -
4*(A^6*B - A^4*B^3 - 2*A^2*B^5)*a^2*b + (A^7 - 5*A^5*B^2 - A^3*B^4 + 5*A*B^6)*a*b^2 + (A^6*B - A^4*B^3 - A^2*B
^5 + B^7)*b^3)*d*cos(d*x + c))*sqrt(-((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt
((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) - (A^4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^
2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d
*x + c))*((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^3 - 4*(A^7*B
+ A^5*B^3 - A^3*B^5 - A*B^7)*a^2*b + (A^8 - 2*A^4*B^4 + B^8)*a*b^2)*cos(d*x + c) + (4*(A^6*B^2 + 2*A^4*B^4 +
A^2*B^6)*a^2*b - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^2 + (A^8 - 2*A^4*B^4 + B^8)*b^3)*sin(d*x + c))/cos(
d*x + c))*((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))^(3/4) + sqrt(2)*((2*(A^4*B + A^2*B^3)*a^6 - (A^5 - 2*A^3
*B^2 - 3*A*B^4)*a^5*b + (3*A^4*B + 4*A^2*B^3 + B^5)*a^4*b^2 - 2*(A^5 - 2*A^3*B^2 - 3*A*B^4)*a^3*b^3 + 2*(A^2*B
^3 + B^5)*a^2*b^4 - (A^5 - 2*A^3*B^2 - 3*A*B^4)*a*b^5 - (A^4*B - B^5)*b^6)*d^7*sqrt((4*A^2*B^2*a^2 - 4*(A^3*B
- A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2
+ b^2)*d^4)) + (2*(A^6*B + 2*A^4*B^3 + A^2*B^5)*a^5 - (A^7 + A^5*B^2 - A^3*B^4 - A*B^6)*a^4*b + 4*(A^6*B + 2*A
^4*B^3 + A^2*B^5)*a^3*b^2 - 2*(A^7 + A^5*B^2 - A^3*B^4 - A*B^6)*a^2*b^3 + 2*(A^6*B + 2*A^4*B^3 + A^2*B^5)*a*b^
4 - (A^7 + A^5*B^2 - A^3*B^4 - A*B^6)*b^5)*d^5*sqrt((4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2
+ B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*
b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) - (A^4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*B^2 + B^
4)*b^2)/(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt((a*cos(d*x + c) + b*sin(d*
x + c))/cos(d*x + c))*((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))^(3/4))/(4*(A^10*B^2 + 4*A^8*B^4 + 6*A^6*B^6
+ 4*A^4*B^8 + A^2*B^10)*a^2*b - 4*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a*b^2 + (A
^12 + 2*A^10*B^2 - A^8*B^4 - 4*A^6*B^6 - A^4*B^8 + 2*A^2*B^10 + B^12)*b^3)) + 4*sqrt(2)*(a^2*b + b^3)*d^5*sqrt
(-((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^
2)*d^4)) - (A^4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b +
(A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt((4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^
4 + 2*a^2*b^2 + b^4)*d^4))*((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))^(3/4)*arctan(((2*(A^7*B + 3*A^5*B^3 + 3
*A^3*B^5 + A*B^7)*a^5 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^4*b + 4*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*
a^3*b^2 - 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^2*b^3 + 2*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a*b^4 - (A
^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*b^5)*d^4*sqrt((4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^
4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) + (2*(A^9*B + 4*A^7*B^3
+ 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*a^4 - (A^10 + 3*A^8*B^2 + 2*A^6*B^4 - 2*A^4*B^6 - 3*A^2*B^8 - B^10)*a^3*b +
2*(A^9*B + 4*A^7*B^3 + 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*a^2*b^2 - (A^10 + 3*A^8*B^2 + 2*A^6*B^4 - 2*A^4*B^6 - 3*
A^2*B^8 - B^10)*a*b^3)*d^2*sqrt((4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 +
2*a^2*b^2 + b^4)*d^4)) + sqrt(2)*((A*a^5 + B*a^4*b + 2*A*a^3*b^2 + 2*B*a^2*b^3 + A*a*b^4 + B*b^5)*d^7*sqrt((4*
A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((A^4 +
2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) + ((A^3 + A*B^2)*a^4 + 2*(A^3 + A*B^2)*a^2*b^2 + (A^3 + A*B^2)*b^4)*d^5*sq
rt((4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(
-((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2
)*d^4)) - (A^4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b +
(A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt(((4*(A^4*B^2 + A^2*B^4)*a^4 - 4*(A^5*B - A*B^5)*a^3*b + (A^6 + 3*A^4*B^2 +
3*A^2*B^4 + B^6)*a^2*b^2 - 4*(A^5*B - A*B^5)*a*b^3 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^4)*d^2*sqrt((A^4 + 2*A^
2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*((4*A^3*B^2*a^4 - 4*(A^4*B - A^2*B^3)*a^3*b + (A^5 + 2*
A^3*B^2 + A*B^4)*a^2*b^2 - 4*(A^4*B - A^2*B^3)*a*b^3 + (A^5 - 2*A^3*B^2 + A*B^4)*b^4)*d^3*sqrt((A^4 + 2*A^2*B^
2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) + (4*(A^5*B^2 + A^3*B^4)*a^3 - 4*(A^6*B - A^4*B^3 - 2*A^2*B^5)*a^2*b
+ (A^7 - 5*A^5*B^2 - A^3*B^4 + 5*A*B^6)*a*b^2 + (A^6*B - A^4*B^3 - A^2*B^5 + B^7)*b^3)*d*cos(d*x + c))*sqrt(-(
(2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*
d^4)) - (A^4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A
^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((A^4 + 2*A^2*B^2 + B^4)/((a^
2 + b^2)*d^4))^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^3 - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^2*b +
(A^8 - 2*A^4*B^4 + B^8)*a*b^2)*cos(d*x + c) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^2*b - 4*(A^7*B + A^5*B^3 -
A^3*B^5 - A*B^7)*a*b^2 + (A^8 - 2*A^4*B^4 + B^8)*b^3)*sin(d*x + c))/cos(d*x + c))*((A^4 + 2*A^2*B^2 + B^4)/((a
^2 + b^2)*d^4))^(3/4) - sqrt(2)*((2*(A^4*B + A^2*B^3)*a^6 - (A^5 - 2*A^3*B^2 - 3*A*B^4)*a^5*b + (3*A^4*B + 4*A
^2*B^3 + B^5)*a^4*b^2 - 2*(A^5 - 2*A^3*B^2 - 3*A*B^4)*a^3*b^3 + 2*(A^2*B^3 + B^5)*a^2*b^4 - (A^5 - 2*A^3*B^2 -
3*A*B^4)*a*b^5 - (A^4*B - B^5)*b^6)*d^7*sqrt((4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)
*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) + (2*(A^6*B + 2*A^4*B^3 +
A^2*B^5)*a^5 - (A^7 + A^5*B^2 - A^3*B^4 - A*B^6)*a^4*b + 4*(A^6*B + 2*A^4*B^3 + A^2*B^5)*a^3*b^2 - 2*(A^7 + A
^5*B^2 - A^3*B^4 - A*B^6)*a^2*b^3 + 2*(A^6*B + 2*A^4*B^3 + A^2*B^5)*a*b^4 - (A^7 + A^5*B^2 - A^3*B^4 - A*B^6)*
b^5)*d^5*sqrt((4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d
^4)))*sqrt(-((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/
((a^2 + b^2)*d^4)) - (A^4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 - 4*(A^3*B - A*
B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((A^4 + 2*A^2*B^
2 + B^4)/((a^2 + b^2)*d^4))^(3/4))/(4*(A^10*B^2 + 4*A^8*B^4 + 6*A^6*B^6 + 4*A^4*B^8 + A^2*B^10)*a^2*b - 4*(A^1
1*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a*b^2 + (A^12 + 2*A^10*B^2 - A^8*B^4 - 4*A^6*B^6
- A^4*B^8 + 2*A^2*B^10 + B^12)*b^3)) - sqrt(2)*((2*A*B*b^2 + (A^2 - B^2)*a*b)*d^3*sqrt((A^4 + 2*A^2*B^2 + B^4
)/((a^2 + b^2)*d^4)) + (A^4 + 2*A^2*B^2 + B^4)*b*d)*sqrt(-((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 -
B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) - (A^4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*
B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*((A^4 + 2*A^2*B^2 + B^4
)/((a^2 + b^2)*d^4))^(1/4)*log(((4*(A^4*B^2 + A^2*B^4)*a^4 - 4*(A^5*B - A*B^5)*a^3*b + (A^6 + 3*A^4*B^2 + 3*A^
2*B^4 + B^6)*a^2*b^2 - 4*(A^5*B - A*B^5)*a*b^3 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^4)*d^2*sqrt((A^4 + 2*A^2*B^
2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*((4*A^3*B^2*a^4 - 4*(A^4*B - A^2*B^3)*a^3*b + (A^5 + 2*A^3*
B^2 + A*B^4)*a^2*b^2 - 4*(A^4*B - A^2*B^3)*a*b^3 + (A^5 - 2*A^3*B^2 + A*B^4)*b^4)*d^3*sqrt((A^4 + 2*A^2*B^2 +
B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) + (4*(A^5*B^2 + A^3*B^4)*a^3 - 4*(A^6*B - A^4*B^3 - 2*A^2*B^5)*a^2*b + (A
^7 - 5*A^5*B^2 - A^3*B^4 + 5*A*B^6)*a*b^2 + (A^6*B - A^4*B^3 - A^2*B^5 + B^7)*b^3)*d*cos(d*x + c))*sqrt(-((2*A
*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)
) - (A^4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 -
2*A^2*B^2 + B^4)*b^2))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((A^4 + 2*A^2*B^2 + B^4)/((a^2 +
b^2)*d^4))^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^3 - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^2*b + (A^8
- 2*A^4*B^4 + B^8)*a*b^2)*cos(d*x + c) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^2*b - 4*(A^7*B + A^5*B^3 - A^3*
B^5 - A*B^7)*a*b^2 + (A^8 - 2*A^4*B^4 + B^8)*b^3)*sin(d*x + c))/cos(d*x + c)) + sqrt(2)*((2*A*B*b^2 + (A^2 - B
^2)*a*b)*d^3*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) + (A^4 + 2*A^2*B^2 + B^4)*b*d)*sqrt(-((2*A*B*a^2*
b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) - (A^
4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*
B^2 + B^4)*b^2))*((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))^(1/4)*log(((4*(A^4*B^2 + A^2*B^4)*a^4 - 4*(A^5*B
- A*B^5)*a^3*b + (A^6 + 3*A^4*B^2 + 3*A^2*B^4 + B^6)*a^2*b^2 - 4*(A^5*B - A*B^5)*a*b^3 + (A^6 - A^4*B^2 - A^2*
B^4 + B^6)*b^4)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*((4*A^3*B^2*a^4 - 4
*(A^4*B - A^2*B^3)*a^3*b + (A^5 + 2*A^3*B^2 + A*B^4)*a^2*b^2 - 4*(A^4*B - A^2*B^3)*a*b^3 + (A^5 - 2*A^3*B^2 +
A*B^4)*b^4)*d^3*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) + (4*(A^5*B^2 + A^3*B^4)*a^3 - 4*
(A^6*B - A^4*B^3 - 2*A^2*B^5)*a^2*b + (A^7 - 5*A^5*B^2 - A^3*B^4 + 5*A*B^6)*a*b^2 + (A^6*B - A^4*B^3 - A^2*B^5
+ B^7)*b^3)*d*cos(d*x + c))*sqrt(-((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((
A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) - (A^4 + 2*A^2*B^2 + B^4)*a^2 - (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*
B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x
+ c))*((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^3 - 4*(A^7*B +
A^5*B^3 - A^3*B^5 - A*B^7)*a^2*b + (A^8 - 2*A^4*B^4 + B^8)*a*b^2)*cos(d*x + c) + (4*(A^6*B^2 + 2*A^4*B^4 + A^
2*B^6)*a^2*b - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^2 + (A^8 - 2*A^4*B^4 + B^8)*b^3)*sin(d*x + c))/cos(d*
x + c)) + 8*(A^4*B + 2*A^2*B^3 + B^5)*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/((A^4 + 2*A^2*B^2
+ B^4)*b*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \tan{\left (c + d x \right )}\right ) \tan{\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)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**(1/2),x)

[Out]

Integral((A + B*tan(c + d*x))*tan(c + d*x)/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{{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )}{\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)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*tan(d*x + c)/sqrt(b*tan(d*x + c) + a), x)