3.846 \(\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=306 \[ \frac {2 \left (35 a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}+\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}-\frac {2 a \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {16 b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {(-b+i a)^{3/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(b+i a)^{3/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \]

[Out]

-(I*a-b)^(3/2)*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)
/d-(I*a+b)^(3/2)*arctanh((I*a+b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1
/2)/d+2/105*(35*a^2-3*b^2)*cot(d*x+c)^(3/2)*(a+b*tan(d*x+c))^(1/2)/a/d-16/35*b*cot(d*x+c)^(5/2)*(a+b*tan(d*x+c
))^(1/2)/d-2/7*a*cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^(1/2)/d+4/105*b*(70*a^2+3*b^2)*cot(d*x+c)^(1/2)*(a+b*tan(d*
x+c))^(1/2)/a^2/d

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Rubi [A]  time = 1.23, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4241, 3567, 3649, 3616, 3615, 93, 203, 206} \[ \frac {2 \left (35 a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}+\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}-\frac {2 a \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {16 b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {(-b+i a)^{3/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(b+i a)^{3/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(3/2),x]

[Out]

-(((I*a - b)^(3/2)*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]])/d) - ((I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqr
t[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (4*b*(70*a^2 + 3*b^2)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(10
5*a^2*d) + (2*(35*a^2 - 3*b^2)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(105*a*d) - (16*b*Cot[c + d*x]^(5/
2)*Sqrt[a + b*Tan[c + d*x]])/(35*d) - (2*a*Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]])/(7*d)

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])

Rule 3567

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

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 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 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 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]

Rubi steps

\begin {align*} \int \cot ^{\frac {9}{2}}(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))^{3/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx\\ &=-\frac {2 a \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {1}{7} \left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-4 a b+\frac {7}{2} \left (a^2-b^2\right ) \tan (c+d x)+3 a b \tan ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {16 b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {\left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {1}{4} a \left (35 a^2-3 b^2\right )-\frac {35}{2} a^2 b \tan (c+d x)-8 a b^2 \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{35 a}\\ &=\frac {2 \left (35 a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {16 b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {1}{4} a b \left (70 a^2+3 b^2\right )-\frac {105}{8} a^2 \left (a^2-b^2\right ) \tan (c+d x)-\frac {1}{4} a b \left (35 a^2-3 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^2}\\ &=\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}+\frac {2 \left (35 a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {16 b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {\left (16 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {105}{16} a^3 \left (a^2-b^2\right )+\frac {105}{8} a^4 b \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^3}\\ &=\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}+\frac {2 \left (35 a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {16 b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {1}{2} \left ((a-i b)^2 \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+\frac {1}{2} \left ((a+i b)^2 \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\\ &=\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}+\frac {2 \left (35 a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {16 b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {\left ((a-i b)^2 \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 d}+\frac {\left ((a+i b)^2 \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 d}\\ &=\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}+\frac {2 \left (35 a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {16 b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {\left ((a-i b)^2 \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 )}{d}+\frac {\left ((a+i b)^2 \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 )}{d}\\ &=-\frac {(i a-b)^{3/2} \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)}}{d}-\frac {(i a+b)^{3/2} \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)}}{d}+\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}+\frac {2 \left (35 a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {16 b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}\\ \end {align*}

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Mathematica [A]  time = 3.32, size = 255, normalized size = 0.83 \[ \frac {\cot ^{\frac {7}{2}}(c+d x) \left (105 \sqrt [4]{-1} a^2 \sqrt {-a+i b} (b+i a) \tan ^{\frac {7}{2}}(c+d x) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-105 (-1)^{3/4} a^2 (a+i b)^{3/2} \tan ^{\frac {7}{2}}(c+d x) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+2 \sqrt {a+b \tan (c+d x)} \left (-15 a^3+2 b \left (70 a^2+3 b^2\right ) \tan ^3(c+d x)+a \left (35 a^2-3 b^2\right ) \tan ^2(c+d x)-24 a^2 b \tan (c+d x)\right )\right )}{105 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(3/2),x]

[Out]

(Cot[c + d*x]^(7/2)*(105*(-1)^(1/4)*a^2*Sqrt[-a + I*b]*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c
+ d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(7/2) - 105*(-1)^(3/4)*a^2*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*
Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(7/2) + 2*Sqrt[a + b*Tan[c + d*x]]*(-
15*a^3 - 24*a^2*b*Tan[c + d*x] + a*(35*a^2 - 3*b^2)*Tan[c + d*x]^2 + 2*b*(70*a^2 + 3*b^2)*Tan[c + d*x]^3)))/(1
05*a^2*d)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 2.01, size = 25428, normalized size = 83.10 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^(3/2),x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^(3/2)*cot(d*x + c)^(9/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^(9/2)*(a + b*tan(c + d*x))^(3/2),x)

[Out]

int(cot(c + d*x)^(9/2)*(a + b*tan(c + d*x))^(3/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**(9/2)*(a+b*tan(d*x+c))**(3/2),x)

[Out]

Timed out

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