3.642 \(\int \frac{\cot ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=316 \[ \frac{2 b \left (5 a^2 A b-3 a^3 B-6 a b^2 B+8 A b^3\right )}{3 a^3 d \left (a^2+b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 (4 A b-3 a B) \sqrt{\cot (c+d x)}}{3 a^2 d \sqrt{a+b \tan (c+d x)}}-\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{3/2}}-\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{3/2}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d \sqrt{a+b \tan (c+d x)}} \]

[Out]

-(((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c
+ d*x]])/((I*a - b)^(3/2)*d)) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]
]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) + (2*b*(5*a^2*A*b + 8*A*b^3 - 3*a^3*B - 6*a*b^2*
B))/(3*a^3*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) + (2*(4*A*b - 3*a*B)*Sqrt[Cot[c + d*x]])
/(3*a^2*d*Sqrt[a + b*Tan[c + d*x]]) - (2*A*Cot[c + d*x]^(3/2))/(3*a*d*Sqrt[a + b*Tan[c + d*x]])

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Rubi [A]  time = 1.3177, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4241, 3609, 3649, 3616, 3615, 93, 203, 206} \[ \frac{2 b \left (5 a^2 A b-3 a^3 B-6 a b^2 B+8 A b^3\right )}{3 a^3 d \left (a^2+b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 (4 A b-3 a B) \sqrt{\cot (c+d x)}}{3 a^2 d \sqrt{a+b \tan (c+d x)}}-\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{3/2}}-\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{3/2}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d \sqrt{a+b \tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]

[Out]

-(((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c
+ d*x]])/((I*a - b)^(3/2)*d)) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]
]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) + (2*b*(5*a^2*A*b + 8*A*b^3 - 3*a^3*B - 6*a*b^2*
B))/(3*a^3*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) + (2*(4*A*b - 3*a*B)*Sqrt[Cot[c + d*x]])
/(3*a^2*d*Sqrt[a + b*Tan[c + d*x]]) - (2*A*Cot[c + d*x]^(3/2))/(3*a*d*Sqrt[a + b*Tan[c + d*x]])

Rule 4241

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ
[u, x]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n
+ 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e +
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ
[a, 0])))

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3616

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

Rule 3615

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{\cot ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx\\ &=-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d \sqrt{a+b \tan (c+d x)}}-\frac{\left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{1}{2} (4 A b-3 a B)+\frac{3}{2} a A \tan (c+d x)+2 A b \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx}{3 a}\\ &=\frac{2 (4 A b-3 a B) \sqrt{\cot (c+d x)}}{3 a^2 d \sqrt{a+b \tan (c+d x)}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d \sqrt{a+b \tan (c+d x)}}+\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{1}{4} \left (-3 a^2 A+8 A b^2-6 a b B\right )-\frac{3}{4} a^2 B \tan (c+d x)+\frac{1}{2} b (4 A b-3 a B) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx}{3 a^2}\\ &=\frac{2 b \left (5 a^2 A b+8 A b^3-3 a^3 B-6 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 (4 A b-3 a B) \sqrt{\cot (c+d x)}}{3 a^2 d \sqrt{a+b \tan (c+d x)}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d \sqrt{a+b \tan (c+d x)}}+\frac{\left (8 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{8} a^3 (a A+b B)+\frac{3}{8} a^3 (A b-a B) \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{3 a^3 \left (a^2+b^2\right )}\\ &=\frac{2 b \left (5 a^2 A b+8 A b^3-3 a^3 B-6 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 (4 A b-3 a B) \sqrt{\cot (c+d x)}}{3 a^2 d \sqrt{a+b \tan (c+d x)}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d \sqrt{a+b \tan (c+d x)}}-\frac{\left ((a+i b) (A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{1+i \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{2 \left (a^2+b^2\right )}-\frac{\left ((a-i b) (A+i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{1-i \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{2 \left (a^2+b^2\right )}\\ &=\frac{2 b \left (5 a^2 A b+8 A b^3-3 a^3 B-6 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 (4 A b-3 a B) \sqrt{\cot (c+d x)}}{3 a^2 d \sqrt{a+b \tan (c+d x)}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d \sqrt{a+b \tan (c+d x)}}-\frac{\left ((a+i b) (A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(1-i x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2+b^2\right ) d}-\frac{\left ((a-i b) (A+i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(1+i x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=\frac{2 b \left (5 a^2 A b+8 A b^3-3 a^3 B-6 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 (4 A b-3 a B) \sqrt{\cot (c+d x)}}{3 a^2 d \sqrt{a+b \tan (c+d x)}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d \sqrt{a+b \tan (c+d x)}}-\frac{\left ((a+i b) (A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-(i a+b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\left (a^2+b^2\right ) d}-\frac{\left ((a-i b) (A+i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-(-i a+b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{(i A-B) \tan ^{-1}\left (\frac{\sqrt{i a-b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{(i a-b)^{3/2} d}-\frac{(i A+B) \tanh ^{-1}\left (\frac{\sqrt{i a+b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{(i a+b)^{3/2} d}+\frac{2 b \left (5 a^2 A b+8 A b^3-3 a^3 B-6 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 (4 A b-3 a B) \sqrt{\cot (c+d x)}}{3 a^2 d \sqrt{a+b \tan (c+d x)}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d \sqrt{a+b \tan (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 3.95511, size = 301, normalized size = 0.95 \[ \frac{\sqrt{\cot (c+d x)} \left (\frac{2 b \left (5 a^2 A b-3 a^3 B-6 a b^2 B+8 A b^3\right ) \tan (c+d x)}{a^2 \left (a^2+b^2\right ) \sqrt{a+b \tan (c+d x)}}+\frac{3 \sqrt [4]{-1} a \sqrt{\tan (c+d x)} \left (\frac{(b+i a) (A+i B) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{-a-i b}}+\frac{(a+i b) (B+i A) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{a-i b}}\right )}{a^2+b^2}+\frac{8 A b-6 a B}{a \sqrt{a+b \tan (c+d x)}}-\frac{2 A \cot (c+d x)}{\sqrt{a+b \tan (c+d x)}}\right )}{3 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2),x]

[Out]

(Sqrt[Cot[c + d*x]]*((3*(-1)^(1/4)*a*(((I*a + b)*(A + I*B)*ArcTanh[((-1)^(1/4)*Sqrt[-a - I*b]*Sqrt[Tan[c + d*x
]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a - I*b] + ((a + I*b)*(I*A + B)*ArcTanh[((-1)^(1/4)*Sqrt[a - I*b]*Sqrt[Ta
n[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a - I*b])*Sqrt[Tan[c + d*x]])/(a^2 + b^2) + (8*A*b - 6*a*B)/(a*Sq
rt[a + b*Tan[c + d*x]]) - (2*A*Cot[c + d*x])/Sqrt[a + b*Tan[c + d*x]] + (2*b*(5*a^2*A*b + 8*A*b^3 - 3*a^3*B -
6*a*b^2*B)*Tan[c + d*x])/(a^2*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]])))/(3*a*d)

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Maple [C]  time = 1.119, size = 19553, normalized size = 61.9 \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)^(5/2)*(A+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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac{5}{2}}}{{\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)^(5/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((B*tan(d*x + c) + A)*cot(d*x + c)^(5/2)/(b*tan(d*x + c) + a)^(3/2), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(5/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**(5/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**(3/2),x)

[Out]

Timed out

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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 )} \cot \left (d x + c\right )^{\frac{5}{2}}}{{\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)^(5/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*cot(d*x + c)^(5/2)/(b*tan(d*x + c) + a)^(3/2), x)