\(\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx\) [854]
Optimal result
Integrand size = 25, antiderivative size = 310 \[
\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {i (i a-b)^{5/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {i (i a+b)^{5/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {2 b \left (49 a^2-3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{21 a d}+\frac {2 \left (7 a^2-9 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {6 a b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}
\]
[Out]
I*(I*a-b)^(5/2)*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*(I*a+b)^(5/2)*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+2/21*(7*a^2-9*b^2)*cot(d*x+c)^(3/2)*(a+b*tan(d*x+c))^(1/2)/d-6/7*a*b*cot(d*x+c)^(5/2)*(a+b*tan(d*x+c)
)^(1/2)/d-2/7*a^2*cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^(1/2)/d+2/21*b*(49*a^2-3*b^2)*cot(d*x+c)^(1/2)*(a+b*tan(d*
x+c))^(1/2)/a/d
Rubi [A] (verified)
Time = 1.54 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of
steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4326, 3646, 3730, 3697, 3696,
95, 209, 212} \[
\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {2 \left (7 a^2-9 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}+\frac {2 b \left (49 a^2-3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{21 a d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {i (-b+i a)^{5/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {i (b+i a)^{5/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {6 a b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}
\]
[In]
Int[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(5/2),x]
[Out]
(I*(I*a - b)^(5/2)*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*(I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sq
rt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*b*(49*a^2 - 3*b^2)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(2
1*a*d) + (2*(7*a^2 - 9*b^2)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(21*d) - (6*a*b*Cot[c + d*x]^(5/2)*Sq
rt[a + b*Tan[c + d*x]])/(7*d) - (2*a^2*Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]])/(7*d)
Rule 95
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 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 3646
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*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] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]
Rule 3696
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 3697
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 3730
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*Ta
n[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 4326
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*}
\text {integral}& = \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^{5/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx \\ & = -\frac {2 a^2 \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 {\frac {15 a^2 b}{2}-\frac {7}{2} a \left (a^2-3 b^2\right ) \tan (c+d x)-\frac {1}{2} b \left (6 a^2-7 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {6 a b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {2 a^2 \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 {5}{4} a^2 \left (7 a^2-9 b^2\right )+\frac {35}{4} a b \left (3 a^2-b^2\right ) \tan (c+d x)+15 a^2 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 (7 a^2-9 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {6 a b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {2 a^2 \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 {5}{8} a^2 b \left (49 a^2-3 b^2\right )+\frac {105}{8} a^3 \left (a^2-3 b^2\right ) \tan (c+d x)+\frac {5}{4} a^2 b \left (7 a^2-9 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 {2 b \left (49 a^2-3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{21 a d}+\frac {2 \left (7 a^2-9 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {6 a b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {2 a^2 \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^4 \left (a^2-3 b^2\right )-\frac {105}{16} a^3 b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^3} \\ & = \frac {2 b \left (49 a^2-3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{21 a d}+\frac {2 \left (7 a^2-9 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {6 a b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {1}{2} \left ((a-i b)^3 \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)^3 \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 {2 b \left (49 a^2-3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{21 a d}+\frac {2 \left (7 a^2-9 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {6 a b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {\left ((a-i b)^3 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {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)^3 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {2 b \left (49 a^2-3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{21 a d}+\frac {2 \left (7 a^2-9 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {6 a b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {\left ((a-i b)^3 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {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)^3 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {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 (i a-b)^{5/2} \arctan \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 (i a+b)^{5/2} \text {arctanh}\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 {2 b \left (49 a^2-3 b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{21 a d}+\frac {2 \left (7 a^2-9 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {6 a b \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d} \\
\end{align*}
Mathematica [A] (verified)
Time = 4.11 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.88
\[
\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {\cot ^{\frac {7}{2}}(c+d x) \left (21 (-1)^{3/4} a (-a+i b)^{5/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {7}{2}}(c+d x)-21 (-1)^{3/4} a (a+i b)^{5/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {7}{2}}(c+d x)-\frac {1}{2} \sec ^3(c+d x) \left (a \left (2 a^2+9 b^2\right ) \cos (c+d x)+\left (10 a^3-9 a b^2\right ) \cos (3 (c+d x))+2 b \left (-40 a^2+3 b^2+\left (58 a^2-3 b^2\right ) \cos (2 (c+d x))\right ) \sin (c+d x)\right ) \sqrt {a+b \tan (c+d x)}\right )}{21 a d}
\]
[In]
Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(5/2),x]
[Out]
(Cot[c + d*x]^(7/2)*(21*(-1)^(3/4)*a*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sq
rt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(7/2) - 21*(-1)^(3/4)*a*(a + I*b)^(5/2)*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) - (Sec[c + d*x]^3*(a*(2*a^2 + 9*b^2)*Cos[c +
d*x] + (10*a^3 - 9*a*b^2)*Cos[3*(c + d*x)] + 2*b*(-40*a^2 + 3*b^2 + (58*a^2 - 3*b^2)*Cos[2*(c + d*x)])*Sin[c +
d*x])*Sqrt[a + b*Tan[c + d*x]])/2))/(21*a*d)
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(3991\) vs. \(2(256)=512\).
Time = 39.53 (sec) , antiderivative size = 3992, normalized size of antiderivative =
12.88
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\(\text {Expression too large to display}\) |
\(3992\) |
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[In]
int(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
[Out]
1/168/d*csc(d*x+c)*(-1/(1-cos(d*x+c))*(csc(d*x+c)*(1-cos(d*x+c))^2-sin(d*x+c)))^(9/2)*(1-cos(d*x+c))*((csc(d*x
+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)/(csc(d*x+c)^2*(1-cos(d*x+c))^2-1))^(1/2)*(3*csc(d*x+c)
^6*a^3*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+
(a^2+b^2)^(1/2))^(1/2)*(1-cos(d*x+c))^6-18*csc(d*x+c)^5*a^2*b*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*
b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(1-cos(d*x+c))^5+42*csc(d*x+c)^4
*ln(1/(1-cos(d*x+c))*(-csc(d*x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*sin(d*x+c)*(-csc(d*x+c
)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1
/2)+2*b*(1-cos(d*x+c))+sin(d*x+c)*a))*(b+(a^2+b^2)^(1/2))^(1/2)*(a^2+b^2)^(1/2)*a^2*(1-cos(d*x+c))^4*(-b+(a^2+
b^2)^(1/2))^(1/2)-42*csc(d*x+c)^4*ln(1/(1-cos(d*x+c))*(-csc(d*x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b^2)^(1/2)*(1-cos
(d*x+c))+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+
c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+2*b*(1-cos(d*x+c))+sin(d*x+c)*a))*(a^2+b^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/
2)*b^2*(1-cos(d*x+c))^4*(-b+(a^2+b^2)^(1/2))^(1/2)-42*csc(d*x+c)^4*ln(-1/(1-cos(d*x+c))*(csc(d*x+c)*a*(1-cos(d
*x+c))^2+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+
c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*b*(1-cos(d*x+c))-sin(d*x+c)*a))*(b+(a^
2+b^2)^(1/2))^(1/2)*(a^2+b^2)^(1/2)*a^2*(1-cos(d*x+c))^4*(-b+(a^2+b^2)^(1/2))^(1/2)+42*csc(d*x+c)^4*ln(-1/(1-c
os(d*x+c))*(csc(d*x+c)*a*(1-cos(d*x+c))^2+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(
d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*b*(1-
cos(d*x+c))-sin(d*x+c)*a))*(a^2+b^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*b^2*(1-cos(d*x+c))^4*(-b+(a^2+b^2)^(1/2))
^(1/2)-126*csc(d*x+c)^4*ln(1/(1-cos(d*x+c))*(-csc(d*x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2
*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)
*(b+(a^2+b^2)^(1/2))^(1/2)+2*b*(1-cos(d*x+c))+sin(d*x+c)*a))*(b+(a^2+b^2)^(1/2))^(1/2)*b*a^2*(1-cos(d*x+c))^4*
(-b+(a^2+b^2)^(1/2))^(1/2)+42*csc(d*x+c)^4*ln(1/(1-cos(d*x+c))*(-csc(d*x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b^2)^(1/
2)*(1-cos(d*x+c))+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1
-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+2*b*(1-cos(d*x+c))+sin(d*x+c)*a))*(b+(a^2+b^2)^(1/2))^(1/2)*b^3*
(1-cos(d*x+c))^4*(-b+(a^2+b^2)^(1/2))^(1/2)+126*csc(d*x+c)^4*ln(-1/(1-cos(d*x+c))*(csc(d*x+c)*a*(1-cos(d*x+c))
^2+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(
1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*b*(1-cos(d*x+c))-sin(d*x+c)*a))*(b+(a^2+b^2)
^(1/2))^(1/2)*b*a^2*(1-cos(d*x+c))^4*(-b+(a^2+b^2)^(1/2))^(1/2)-42*csc(d*x+c)^4*ln(-1/(1-cos(d*x+c))*(csc(d*x+
c)*a*(1-cos(d*x+c))^2+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a
)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*b*(1-cos(d*x+c))-sin(d*x+
c)*a))*(b+(a^2+b^2)^(1/2))^(1/2)*b^3*(1-cos(d*x+c))^4*(-b+(a^2+b^2)^(1/2))^(1/2)-37*csc(d*x+c)^4*a^3*(-csc(d*x
+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))
^(1/2)*(1-cos(d*x+c))^4+36*csc(d*x+c)^4*a*b^2*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-co
t(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(1-cos(d*x+c))^4-168*csc(d*x+c)^4*arctan(((b+(a^
2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(
d*x+c))-a)*(1-cos(d*x+c)))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*(a^2+b^2)^(1/2)*b*a^2*
(1-cos(d*x+c))^4-168*csc(d*x+c)^4*arctan((-(b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(-csc(d*x+c)*(csc
(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(
-b+(a^2+b^2)^(1/2))^(1/2))*(a^2+b^2)^(1/2)*b*a^2*(1-cos(d*x+c))^4-84*csc(d*x+c)^4*a^4*arctan(((b+(a^2+b^2)^(1/
2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)
*(1-cos(d*x+c)))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*(1-cos(d*x+c))^4+252*csc(d*x+c)^
4*arctan(((b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*
(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*b^2*a^
2*(1-cos(d*x+c))^4-84*csc(d*x+c)^4*a^4*arctan((-(b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(-csc(d*x+c)
*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2))/(1-cos(d*x+c))*sin(d*x
+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*(1-cos(d*x+c))^4+252*csc(d*x+c)^4*arctan((-(b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+
c)-cot(d*x+c))+(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1
/2))/(1-cos(d*x+c))*sin(d*x+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*b^2*a^2*(1-cos(d*x+c))^4+428*csc(d*x+c)^3*b*(-csc(d
*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*a^2*(1-cos(d*x+c))
^3*(-b+(a^2+b^2)^(1/2))^(1/2)-24*csc(d*x+c)^3*b^3*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c
)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(1-cos(d*x+c))^3+37*csc(d*x+c)^2*a^3*(-csc(d
*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2
))^(1/2)*(1-cos(d*x+c))^2-36*csc(d*x+c)^2*a*b^2*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-
cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(1-cos(d*x+c))^2-18*a^2*b*(-csc(d*x+c)*(csc(d*
x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(csc
(d*x+c)-cot(d*x+c))-3*a^3*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(
d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2))/(csc(d*x+c)^2*(1-cos(d*x+c))^2-1)^4/(-csc(d*x+c)*(csc(d*x+c)^2*a*(1
-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*2^(1/2)/a/(-b+(a^2+b^2)^(1/2))^(1/2)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 5225 vs. \(2 (250) = 500\).
Time = 0.86 (sec) , antiderivative size = 5225, normalized size of antiderivative = 16.85
\[
\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display}
\]
[In]
integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")
[Out]
Too large to include
Sympy [F(-1)]
Timed out. \[
\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out}
\]
[In]
integrate(cot(d*x+c)**(9/2)*(a+b*tan(d*x+c))**(5/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{\frac {9}{2}} \,d x }
\]
[In]
integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")
[Out]
integrate((b*tan(d*x + c) + a)^(5/2)*cot(d*x + c)^(9/2), x)
Giac [F(-1)]
Timed out. \[
\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out}
\]
[In]
integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")
[Out]
Timed out
Mupad [F(-1)]
Timed out. \[
\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2} \,d x
\]
[In]
int(cot(c + d*x)^(9/2)*(a + b*tan(c + d*x))^(5/2),x)
[Out]
int(cot(c + d*x)^(9/2)*(a + b*tan(c + d*x))^(5/2), x)