\(\int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx\) [524]

Optimal result

Integrand size = 23, antiderivative size = 151 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=-\frac {5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {i (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \]

[Out]

-5*a^(3/2)*b*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d+I*(a-I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^
(1/2))/d-I*(a+I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/d-a^2*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)
/d

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3646, 3734, 3620, 3618, 65, 214, 3715} \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=-\frac {5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {i (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \]

[In]

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

[Out]

(-5*a^(3/2)*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d + (I*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x
]]/Sqrt[a - I*b]])/d - (I*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (a^2*Cot[c + d*
x]*Sqrt[a + b*Tan[c + d*x]])/d

Rule 65

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 214

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

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^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 3620

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 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 3715

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]

Rule 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*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] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\int \frac {\cot (c+d x) \left (\frac {5 a^2 b}{2}-a \left (a^2-3 b^2\right ) \tan (c+d x)-\frac {1}{2} b \left (a^2-2 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (5 a^2 b\right ) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx+\int \frac {-a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {1}{2} (a-i b)^3 \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {1}{2} (a+i b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {\left (5 a^2 b\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{d}-\frac {(i a-b)^3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}+\frac {(i a+b)^3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d} \\ & = -\frac {5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {(a-i b)^3 \text {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)^3 \text {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 {5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {i (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.54 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {-5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+i (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-i a^2 \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 a \sqrt {a+i b} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+i \sqrt {a+i b} b^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )-a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

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

[Out]

(-5*a^(3/2)*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + I*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/S
qrt[a - I*b]] - I*a^2*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*a*Sqrt[a + I*b]*b*ArcT
anh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + I*Sqrt[a + I*b]*b^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*
b]] - a^2*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d

Maple [F(-1)]

Timed out.

hanged

[In]

int(cot(d*x+c)^2*(a+b*tan(d*x+c))^(5/2),x)

[Out]

int(cot(d*x+c)^2*(a+b*tan(d*x+c))^(5/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1180 vs. \(2 (119) = 238\).

Time = 0.37 (sec) , antiderivative size = 2375, normalized size of antiderivative = 15.73 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/2*(5*a^(3/2)*b*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/tan(d*x + c))*tan(d*x + c) +
d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/
d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) + ((a^2 - b^2)*d^3*sqrt(-(25*
a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*d)*sqrt(-(a
^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2))
*tan(d*x + c) - d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a
^2*b^8 + b^10)/d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) - ((a^2 - b^2)
*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b
^6)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^
10)/d^4))/d^2))*tan(d*x + c) - d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110
*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a)
+ ((a^2 - b^2)*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - 2*(5*a^5*b^2 - 1
0*a^3*b^4 + a*b^6)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 -
20*a^2*b^8 + b^10)/d^4))/d^2))*tan(d*x + c) + d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 10
0*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*ta
n(d*x + c) + a) - ((a^2 - b^2)*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - 2
*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 +
110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2))*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*a^2)/(d*tan(d*x + c)),
1/2*(10*sqrt(-a)*a*b*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(-a)/a)*tan(d*x + c) + d*sqrt(-(a^5 - 10*a^3*b^2 + 5
*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)*log((5*a^8*b - 14*a
^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) + ((a^2 - b^2)*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^
4*b^6 - 20*a^2*b^8 + b^10)/d^4) + 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + d^
2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2))*tan(d*x + c) - d*sqrt(-(a^5 -
10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)*log(
(5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) - ((a^2 - b^2)*d^3*sqrt(-(25*a^8*b^2 - 100*a
^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*d)*sqrt(-(a^5 - 10*a^3*b^2
+ 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2))*tan(d*x + c) -
d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/
d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) + ((a^2 - b^2)*d^3*sqrt(-(25*
a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*d)*sqrt(-(a
^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2))
*tan(d*x + c) + d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a
^2*b^8 + b^10)/d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) - ((a^2 - b^2)
*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b
^6)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^
10)/d^4))/d^2))*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*a^2)/(d*tan(d*x + c))]

Sympy [F]

\[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot ^{2}{\left (c + d x \right )}\, dx \]

[In]

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

[Out]

Integral((a + b*tan(c + d*x))**(5/2)*cot(c + d*x)**2, x)

Maxima [F]

\[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(cot(d*x+c)^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)^2, x)

Giac [F(-1)]

Timed out. \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 12.18 (sec) , antiderivative size = 3366, normalized size of antiderivative = 22.29 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \]

[In]

int(cot(c + d*x)^2*(a + b*tan(c + d*x))^(5/2),x)

[Out]

log(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*((((
(-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d
^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a
^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*((1024*a^2*b^9*(a^2 + b^2
))/d - 128*b^8*(((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^
(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2)))/2 - (64*a*b^8*(a + b*tan(c + d*x))^(1/2)*(5*a^6 + 19*b^6 -
5*a^2*b^4 - 76*a^4*b^2))/d^2))/2 - (32*a*b^9*(b^8 - 23*a^8 - 100*a^2*b^6 + 44*a^4*b^4 + 122*a^6*b^2))/d^3))/2
- (16*b^8*(a + b*tan(c + d*x))^(1/2)*(2*a^12 + 2*b^12 + 12*a^2*b^10 + 55*a^4*b^8 - 335*a^6*b^6 + 405*a^8*b^4 -
13*a^10*b^2))/d^4))/2 + (40*a^2*b^9*(a^2 + b^2)^3*(2*a^6 + 2*b^6 + 11*a^2*b^4 - 9*a^4*b^2))/d^5)*((20*a^2*b^8
*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2)/(4*d^4) - a^5/(4*d^2) - (5*a*b^4)/
(4*d^2) + (5*a^3*b^2)/(2*d^2))^(1/2) - log(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^
4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 +
10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3
*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^
2)/d^4)^(1/2)*((1024*a^2*b^9*(a^2 + b^2))/d + 128*b^8*(((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^
2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2)))/2 + (64*a*b^8*(a + b
*tan(c + d*x))^(1/2)*(5*a^6 + 19*b^6 - 5*a^2*b^4 - 76*a^4*b^2))/d^2))/2 - (32*a*b^9*(b^8 - 23*a^8 - 100*a^2*b^
6 + 44*a^4*b^4 + 122*a^6*b^2))/d^3))/2 + (16*b^8*(a + b*tan(c + d*x))^(1/2)*(2*a^12 + 2*b^12 + 12*a^2*b^10 + 5
5*a^4*b^8 - 335*a^6*b^6 + 405*a^8*b^4 - 13*a^10*b^2))/d^4))/2 + (40*a^2*b^9*(a^2 + b^2)^3*(2*a^6 + 2*b^6 + 11*
a^2*b^4 - 9*a^4*b^2))/d^5)*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^
(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(4*d^4))^(1/2) - log(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^
2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/
2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a
^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2
+ 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*((1024*a^2*b^9*(a^2 + b^2))/d + 128*b^8*(-((-b^2*d^4*(5*a^4 + b^4
- 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x
))^(1/2)))/2 + (64*a*b^8*(a + b*tan(c + d*x))^(1/2)*(5*a^6 + 19*b^6 - 5*a^2*b^4 - 76*a^4*b^2))/d^2))/2 - (32*a
*b^9*(b^8 - 23*a^8 - 100*a^2*b^6 + 44*a^4*b^4 + 122*a^6*b^2))/d^3))/2 + (16*b^8*(a + b*tan(c + d*x))^(1/2)*(2*
a^12 + 2*b^12 + 12*a^2*b^10 + 55*a^4*b^8 - 335*a^6*b^6 + 405*a^8*b^4 - 13*a^10*b^2))/d^4))/2 + (40*a^2*b^9*(a^
2 + b^2)^3*(2*a^6 + 2*b^6 + 11*a^2*b^4 - 9*a^4*b^2))/d^5)*(-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 10
0*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(4*d^4))^(1/2) + log(((-((-b^2
*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(
5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4
+ b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4
- 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*((1024*a^2*b^9*(a^2 + b^2))/d - 12
8*b^8*(-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(3
*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2)))/2 - (64*a*b^8*(a + b*tan(c + d*x))^(1/2)*(5*a^6 + 19*b^6 - 5*a^2*b^
4 - 76*a^4*b^2))/d^2))/2 - (32*a*b^9*(b^8 - 23*a^8 - 100*a^2*b^6 + 44*a^4*b^4 + 122*a^6*b^2))/d^3))/2 - (16*b^
8*(a + b*tan(c + d*x))^(1/2)*(2*a^12 + 2*b^12 + 12*a^2*b^10 + 55*a^4*b^8 - 335*a^6*b^6 + 405*a^8*b^4 - 13*a^10
*b^2))/d^4))/2 + (40*a^2*b^9*(a^2 + b^2)^3*(2*a^6 + 2*b^6 + 11*a^2*b^4 - 9*a^4*b^2))/d^5)*((5*a^3*b^2)/(2*d^2)
- a^5/(4*d^2) - (5*a*b^4)/(4*d^2) - (20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b
^2*d^4)^(1/2)/(4*d^4))^(1/2) + (b*atan((b^21*(a^3)^(1/2)*(a + b*tan(c + d*x))^(1/2)*160i)/(160*a^2*b^21 + 960*
a^4*b^19 + 42400*a^6*b^17 + 63200*a^8*b^15 + 30400*a^10*b^13 + 8960*a^12*b^11 + 160*a^14*b^9) + (a^2*b^19*(a^3
)^(1/2)*(a + b*tan(c + d*x))^(1/2)*960i)/(160*a^2*b^21 + 960*a^4*b^19 + 42400*a^6*b^17 + 63200*a^8*b^15 + 3040
0*a^10*b^13 + 8960*a^12*b^11 + 160*a^14*b^9) + (a^4*b^17*(a^3)^(1/2)*(a + b*tan(c + d*x))^(1/2)*42400i)/(160*a
^2*b^21 + 960*a^4*b^19 + 42400*a^6*b^17 + 63200*a^8*b^15 + 30400*a^10*b^13 + 8960*a^12*b^11 + 160*a^14*b^9) +
(a^6*b^15*(a^3)^(1/2)*(a + b*tan(c + d*x))^(1/2)*63200i)/(160*a^2*b^21 + 960*a^4*b^19 + 42400*a^6*b^17 + 63200
*a^8*b^15 + 30400*a^10*b^13 + 8960*a^12*b^11 + 160*a^14*b^9) + (a^8*b^13*(a^3)^(1/2)*(a + b*tan(c + d*x))^(1/2
)*30400i)/(160*a^2*b^21 + 960*a^4*b^19 + 42400*a^6*b^17 + 63200*a^8*b^15 + 30400*a^10*b^13 + 8960*a^12*b^11 +
160*a^14*b^9) + (a^10*b^11*(a^3)^(1/2)*(a + b*tan(c + d*x))^(1/2)*8960i)/(160*a^2*b^21 + 960*a^4*b^19 + 42400*
a^6*b^17 + 63200*a^8*b^15 + 30400*a^10*b^13 + 8960*a^12*b^11 + 160*a^14*b^9) + (a^12*b^9*(a^3)^(1/2)*(a + b*ta
n(c + d*x))^(1/2)*160i)/(160*a^2*b^21 + 960*a^4*b^19 + 42400*a^6*b^17 + 63200*a^8*b^15 + 30400*a^10*b^13 + 896
0*a^12*b^11 + 160*a^14*b^9))*(a^3)^(1/2)*5i)/d + (a^2*b*(a + b*tan(c + d*x))^(1/2))/(a*d - d*(a + b*tan(c + d*
x)))