3.367 \(\int \frac{\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=119 \[ -\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d \sqrt{a-i b}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d \sqrt{a+i b}} \]

[Out]

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

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Rubi [A]  time = 0.280494, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.206, Rules used = {21, 3574, 3539, 3537, 63, 208, 3634} \[ -\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d \sqrt{a-i b}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d \sqrt{a+i b}} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 3574

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

Rule 3634

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

Rubi steps

\begin{align*} \int \frac{\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx &=B \int \frac{\cot (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\left (B \int \frac{\tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx\right )+B \int \frac{\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\left (\frac{1}{2} (i B) \int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx\right )+\frac{1}{2} (i B) \int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx+\frac{B \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{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{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) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{b d}\\ &=-\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}+\frac{(i 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 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{2 B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{\sqrt{a-i b} d}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{\sqrt{a+i b} d}\\ \end{align*}

Mathematica [A]  time = 0.137833, size = 112, normalized size = 0.94 \[ \frac{B \left (-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{\sqrt{a-i b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{\sqrt{a+i b}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]

[Out]

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

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Maple [C]  time = 0.808, size = 20195, normalized size = 169.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(3/2),x)

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.84468, size = 11555, normalized size = 97.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

[1/4*(4*sqrt(2)*(a^3 + a*b^2)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^5*(B^4/((a^2 + b^2)*d^4))^(3/4)*sq
rt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2))*arctan(-((B^6*a^4 + 2*B^6*a^
2*b^2 + B^6*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^4*sqrt(B^4/((a^2 + b^2)*d^4)) + (B^8*a^3 + B^8*
a*b^2)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*sqrt(B^4*b^2/((a^4
+ 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt(B^4/((a^2 + b^2)*d^4)) + (B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4)*sqrt(B^4*b^2/(
(a^4 + 2*a^2*b^2 + b^4)*d^4))*d^5)*sqrt((B^6*a*cos(d*x + c) + B^6*b*sin(d*x + c) + (B^4*a^2 + B^4*b^2)*d^2*sqr
t(B^4/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*(B^5*a*d*cos(d*x + c) + (B^3*a^2 + B^3*b^2)*d^3*sqrt(B^4/((a^2
+ b^2)*d^4))*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(1/4)
*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)))/cos(d*x + c))*(B^4/((a^2
+ b^2)*d^4))^(3/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)) + sqrt
(2)*((B^3*a^5 + 2*B^3*a^3*b^2 + B^3*a*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt(B^4/((a^2 + b^
2)*d^4)) + (B^5*a^4 + 2*B^5*a^2*b^2 + B^5*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^5)*sqrt((a*cos(d*
x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(3/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d
^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)))/(B^10*b^2)) + 4*sqrt(2)*(a^3 + a*b^2)*sqrt(B^4*b^2/((a^4 + 2*a^2*b
^2 + b^4)*d^4))*d^5*(B^4/((a^2 + b^2)*d^4))^(3/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 +
b^2)*d^4)))/(B^2*b^2))*arctan(((B^6*a^4 + 2*B^6*a^2*b^2 + B^6*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)
)*d^4*sqrt(B^4/((a^2 + b^2)*d^4)) + (B^8*a^3 + B^8*a*b^2)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + sq
rt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt(B^4/((a^2 + b^2)*d^4)) +
(B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^5)*sqrt((B^6*a*cos(d*x + c)
+ B^6*b*sin(d*x + c) + (B^4*a^2 + B^4*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*(B^5*a*d*co
s(d*x + c) + (B^3*a^2 + B^3*b^2)*d^3*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*
x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2
+ b^2)*d^4)))/(B^2*b^2)))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(3/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)
*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)) - sqrt(2)*((B^3*a^5 + 2*B^3*a^3*b^2 + B^3*a*b^4)*sqrt(B^4*b^2/((a
^4 + 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt(B^4/((a^2 + b^2)*d^4)) + (B^5*a^4 + 2*B^5*a^2*b^2 + B^5*b^4)*sqrt(B^4*b^2
/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^5)*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^
4))^(3/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)))/(B^10*b^2)) + 2
*B^5*sqrt(a)*log(-(8*a*b*cos(d*x + c)*sin(d*x + c) + (8*a^2 - b^2)*cos(d*x + c)^2 + b^2 - 4*(2*a*cos(d*x + c)^
2 + b*cos(d*x + c)*sin(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/(cos(d*x + c)^2
- 1)) + sqrt(2)*(B^2*a^2*d^3*sqrt(B^4/((a^2 + b^2)*d^4)) + B^4*a*d)*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a
^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2))*log((B^6*a*cos(d*x + c) + B^6*b*sin(d
*x + c) + (B^4*a^2 + B^4*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*(B^5*a*d*cos(d*x + c) + (
B^3*a^2 + B^3*b^2)*d^3*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*
x + c))*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))
/(B^2*b^2)))/cos(d*x + c)) - sqrt(2)*(B^2*a^2*d^3*sqrt(B^4/((a^2 + b^2)*d^4)) + B^4*a*d)*(B^4/((a^2 + b^2)*d^4
))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2))*log((B^6*a*cos(d*
x + c) + B^6*b*sin(d*x + c) + (B^4*a^2 + B^4*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*(B^5*
a*d*cos(d*x + c) + (B^3*a^2 + B^3*b^2)*d^3*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*
sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4
/((a^2 + b^2)*d^4)))/(B^2*b^2)))/cos(d*x + c)))/(B^4*a*d), 1/4*(4*sqrt(2)*(a^3 + a*b^2)*sqrt(B^4*b^2/((a^4 + 2
*a^2*b^2 + b^4)*d^4))*d^5*(B^4/((a^2 + b^2)*d^4))^(3/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/(
(a^2 + b^2)*d^4)))/(B^2*b^2))*arctan(-((B^6*a^4 + 2*B^6*a^2*b^2 + B^6*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^
4)*d^4))*d^4*sqrt(B^4/((a^2 + b^2)*d^4)) + (B^8*a^3 + B^8*a*b^2)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d
^2 - sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt(B^4/((a^2 + b^2)*
d^4)) + (B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^5)*sqrt((B^6*a*cos(d
*x + c) + B^6*b*sin(d*x + c) + (B^4*a^2 + B^4*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*(B^5
*a*d*cos(d*x + c) + (B^3*a^2 + B^3*b^2)*d^3*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c))*sqrt((a*cos(d*x + c) + b
*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^
4/((a^2 + b^2)*d^4)))/(B^2*b^2)))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(3/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 +
a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)) + sqrt(2)*((B^3*a^5 + 2*B^3*a^3*b^2 + B^3*a*b^4)*sqrt(B^4*
b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt(B^4/((a^2 + b^2)*d^4)) + (B^5*a^4 + 2*B^5*a^2*b^2 + B^5*b^4)*sqrt(
B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^5)*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 +
b^2)*d^4))^(3/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)))/(B^10*b^
2)) + 4*sqrt(2)*(a^3 + a*b^2)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^5*(B^4/((a^2 + b^2)*d^4))^(3/4)*sq
rt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2))*arctan(((B^6*a^4 + 2*B^6*a^2
*b^2 + B^6*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^4*sqrt(B^4/((a^2 + b^2)*d^4)) + (B^8*a^3 + B^8*a
*b^2)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*sqrt(B^4*b^2/((a^4
+ 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt(B^4/((a^2 + b^2)*d^4)) + (B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4)*sqrt(B^4*b^2/((
a^4 + 2*a^2*b^2 + b^4)*d^4))*d^5)*sqrt((B^6*a*cos(d*x + c) + B^6*b*sin(d*x + c) + (B^4*a^2 + B^4*b^2)*d^2*sqrt
(B^4/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*(B^5*a*d*cos(d*x + c) + (B^3*a^2 + B^3*b^2)*d^3*sqrt(B^4/((a^2
+ b^2)*d^4))*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(1/4)*
sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)))/cos(d*x + c))*(B^4/((a^2
+ b^2)*d^4))^(3/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)) - sqrt(
2)*((B^3*a^5 + 2*B^3*a^3*b^2 + B^3*a*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt(B^4/((a^2 + b^2
)*d^4)) + (B^5*a^4 + 2*B^5*a^2*b^2 + B^5*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^5)*sqrt((a*cos(d*x
+ c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(3/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^
2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)))/(B^10*b^2)) + 8*B^5*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*cos(d*x + c) +
b*sin(d*x + c))/cos(d*x + c))/a) + sqrt(2)*(B^2*a^2*d^3*sqrt(B^4/((a^2 + b^2)*d^4)) + B^4*a*d)*(B^4/((a^2 + b
^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2))*log((B^6*a
*cos(d*x + c) + B^6*b*sin(d*x + c) + (B^4*a^2 + B^4*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2
)*(B^5*a*d*cos(d*x + c) + (B^3*a^2 + B^3*b^2)*d^3*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c))*sqrt((a*cos(d*x +
c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*s
qrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)))/cos(d*x + c)) - sqrt(2)*(B^2*a^2*d^3*sqrt(B^4/((a^2 + b^2)*d^4)) + B^4
*a*d)*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 - (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(
B^2*b^2))*log((B^6*a*cos(d*x + c) + B^6*b*sin(d*x + c) + (B^4*a^2 + B^4*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4))*c
os(d*x + c) - sqrt(2)*(B^5*a*d*cos(d*x + c) + (B^3*a^2 + B^3*b^2)*d^3*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c)
)*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 -
(a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)))/cos(d*x + c)))/(B^4*a*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} B \int \frac{\cot{\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(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))**(3/2),x)

[Out]

B*Integral(cot(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 b \tan \left (d x + c\right ) + B a\right )} \cot \left (d x + c\right )}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((B*b*tan(d*x + c) + B*a)*cot(d*x + c)/(b*tan(d*x + c) + a)^(3/2), x)