3.6.52 \(\int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\) [552]

Optimal. Leaf size=195 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}+\frac {2 b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 b^2 \left (3 a^2+b^2\right )}{a^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.51, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3650, 3730, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {2 b^2 \left (3 a^2+b^2\right )}{a^2 d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}+\frac {2 b^2}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3650

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

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

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Mathematica [A]
time = 2.25, size = 237, normalized size = 1.22 \begin {gather*} \frac {2 \left (\frac {-\frac {3 \left (a^2+b^2\right )^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {3 a^2 (a+i b)^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{2 \sqrt {a-i b}}+\frac {3 a^2 (a-i b)^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{2 \sqrt {a+i b}}}{a \left (a^2+b^2\right )}+\frac {b^2}{(a+b \tan (c+d x))^{3/2}}+\frac {3 b^2 \left (3 a^2+b^2\right )}{a \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}\right )}{3 a \left (a^2+b^2\right ) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 2.85, size = 115830, normalized size = 594.00

method	result	size
default	\(\text {Expression too large to display}\)	\(115830\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 10106 vs. \(2 (163) = 326\).
time = 3.59, size = 20287, normalized size = 104.04 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(12*sqrt(2)*((a^21 + a^19*b^2 - 20*a^17*b^4 - 84*a^15*b^6 - 154*a^13*b^8 - 154*a^11*b^10 - 84*a^9*b^12
- 20*a^7*b^14 + a^5*b^16 + a^3*b^18)*d^5*cos(d*x + c)^4 + 2*(3*a^19*b^2 + 20*a^17*b^4 + 56*a^15*b^6 + 84*a^13*
b^8 + 70*a^11*b^10 + 28*a^9*b^12 - 4*a^5*b^16 - a^3*b^18)*d^5*cos(d*x + c)^2 + (a^17*b^4 + 7*a^15*b^6 + 21*a^1
3*b^8 + 35*a^11*b^10 + 35*a^9*b^12 + 21*a^7*b^14 + 7*a^5*b^16 + a^3*b^18)*d^5 + 4*((a^20*b + 6*a^18*b^3 + 14*a
^16*b^5 + 14*a^14*b^7 - 14*a^10*b^11 - 14*a^8*b^13 - 6*a^6*b^15 - a^4*b^17)*d^5*cos(d*x + c)^3 + (a^18*b^3 + 7
*a^16*b^5 + 21*a^14*b^7 + 35*a^12*b^9 + 35*a^10*b^11 + 21*a^8*b^13 + 7*a^6*b^15 + a^4*b^17)*d^5*cos(d*x + c))*
sin(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^
11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b
^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt(
(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6
+ 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*(1/((a
^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(3/4)*arctan(((5*a^20 + 30*a^18*b^2 + 61*a
^16*b^4 + 8*a^14*b^6 - 182*a^12*b^8 - 364*a^10*b^10 - 350*a^8*b^12 - 184*a^6*b^14 - 47*a^4*b^16 - 2*a^2*b^18 +
b^20)*d^4*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^
4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^
20)*d^4))*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + (5*a^15 + 15*a^13*b^
2 + a^11*b^4 - 45*a^9*b^6 - 65*a^7*b^8 - 35*a^5*b^10 - 5*a^3*b^12 + a*b^14)*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)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^1
0*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) - sqrt(2)*((a^23 + 7*a^21*b^2 +
15*a^19*b^4 - 15*a^17*b^6 - 150*a^15*b^8 - 378*a^13*b^10 - 546*a^11*b^12 - 510*a^9*b^14 - 315*a^7*b^16 - 125*
a^5*b^18 - 29*a^3*b^20 - 3*a*b^22)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^2
0 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45
*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*
d^4)) + (a^18 + 7*a^16*b^2 + 20*a^14*b^4 + 28*a^12*b^6 + 14*a^10*b^8 - 14*a^8*b^10 - 28*a^6*b^12 - 20*a^4*b^14
- 7*a^2*b^16 - b^18)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b
^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 1
0*a^2*b^18 + b^20)*d^4)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13
*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^
2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 +
b^10))*sqrt(((25*a^14 - 25*a^12*b^2 - 115*a^10*b^4 + 35*a^8*b^6 + 171*a^6*b^8 + 53*a^4*b^10 - 17*a^2*b^12 + b^
14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) + sqrt(2)*(
(25*a^16 - 50*a^14*b^2 - 90*a^12*b^4 + 150*a^10*b^6 + 136*a^8*b^8 - 118*a^6*b^10 - 70*a^4*b^12 + 18*a^2*b^14 -
b^16)*d^3*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) + (25*a^
11 - 175*a^9*b^2 + 410*a^7*b^4 - 350*a^5*b^6 + 61*a^3*b^8 - 3*a*b^10)*d*cos(d*x + c))*sqrt((a^10 + 5*a^8*b^2 +
10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5
*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^
4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(
d*x + c))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(1/4) + (25*a^9 - 100*a^7*
b^2 + 110*a^5*b^4 - 20*a^3*b^6 + a*b^8)*cos(d*x + c) + (25*a^8*b - 100*a^6*b^3 + 110*a^4*b^5 - 20*a^2*b^7 + b^
9)*sin(d*x + c))/cos(d*x + c))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(3/4)
+ sqrt(2)*((5*a^27 + 25*a^25*b^2 + 6*a^23*b^4 - 218*a^21*b^6 - 585*a^19*b^8 - 405*a^17*b^10 + 900*a^15*b^12 +
2532*a^13*b^14 + 2979*a^11*b^16 + 2015*a^9*b^18 + 790*a^7*b^20 + 150*a^5*b^22 + a^3*b^24 - 3*a*b^26)*d^7*sqrt
((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^
6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(
1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + (5*a^22 + 25*a^20*b^2 + 31*a^18*b^4
- 53*a^16*b^6 - 190*a^14*b^8 - 182*a^12*b^10 +...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot {\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [B]
time = 4.89, size = 2500, normalized size = 12.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((((a + b*tan(c + d*x))^(1/2)*(10304*a^20*b^34*d^5 - 512*a^16*b^38*d^5 - 544*a^18*b^36*d^5 - 64*a^14*b^40*
d^5 + 66976*a^22*b^32*d^5 + 221312*a^24*b^30*d^5 + 480480*a^26*b^28*d^5 + 741312*a^28*b^26*d^5 + 837408*a^30*b
^24*d^5 + 695552*a^32*b^22*d^5 + 416416*a^34*b^20*d^5 + 168896*a^36*b^18*d^5 + 37856*a^38*b^16*d^5 - 896*a^40*
b^14*d^5 - 3424*a^42*b^12*d^5 - 960*a^44*b^10*d^5 - 96*a^46*b^8*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)/(a^10*d^4 + b^10*
d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(384*a^15*b^42*d^6 - ((((((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)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(5
12*a^16*b^46*d^8 - ((((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)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^
4 + 5*a^8*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^18*b^46*d^9 + 9984*a^20*b^44*d^9 + 92160*a^22*b^42
*d^9 + 535296*a^24*b^40*d^9 + 2193408*a^26*b^38*d^9 + 6736896*a^28*b^36*d^9 + 16084992*a^30*b^34*d^9 + 3055104
0*a^32*b^32*d^9 + 46844928*a^34*b^30*d^9 + 58499584*a^36*b^28*d^9 + 59744256*a^38*b^26*d^9 + 49900032*a^40*b^2
4*d^9 + 33945600*a^42*b^22*d^9 + 18643968*a^44*b^20*d^9 + 8146944*a^46*b^18*d^9 + 2767872*a^48*b^16*d^9 + 7050
24*a^50*b^14*d^9 + 126720*a^52*b^12*d^9 + 14336*a^54*b^10*d^9 + 768*a^56*b^8*d^9))/2 + 9728*a^18*b^44*d^8 + 87
936*a^20*b^42*d^8 + 502144*a^22*b^40*d^8 + 2028544*a^24*b^38*d^8 + 6153216*a^26*b^36*d^8 + 14518784*a^28*b^34*
d^8 + 27243008*a^30*b^32*d^8 + 41213952*a^32*b^30*d^8 + 50665472*a^34*b^28*d^8 + 50775296*a^36*b^26*d^8 + 4144
3584*a^38*b^24*d^8 + 27409408*a^40*b^22*d^8 + 14543872*a^42*b^20*d^8 + 6093312*a^44*b^18*d^8 + 1966592*a^46*b^
16*d^8 + 470528*a^48*b^14*d^8 + 78336*a^50*b^12*d^8 + 8064*a^52*b^10*d^8 + 384*a^54*b^8*d^8))/2 + (a + b*tan(c
+ d*x))^(1/2)*(256*a^15*b^44*d^7 + 4608*a^17*b^42*d^7 + 40512*a^19*b^40*d^7 + 224768*a^21*b^38*d^7 + 864768*a
^23*b^36*d^7 + 2419200*a^25*b^34*d^7 + 5055232*a^27*b^32*d^7 + 8007168*a^29*b^30*d^7 + 9664512*a^31*b^28*d^7 +
8859136*a^33*b^26*d^7 + 6095232*a^35*b^24*d^7 + 3095040*a^37*b^22*d^7 + 1164800*a^39*b^20*d^7 + 376320*a^41*b
^18*d^7 + 154368*a^43*b^16*d^7 + 76288*a^45*b^14*d^7 + 28416*a^47*b^12*d^7 + 6144*a^49*b^10*d^7 + 576*a^51*b^8
*d^7))*(((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)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2
*d^4))^(1/2))/2 + 7296*a^17*b^40*d^6 + 59424*a^19*b^38*d^6 + 280992*a^21*b^36*d^6 + 866208*a^23*b^34*d^6 + 182
5824*a^25*b^32*d^6 + 2629536*a^27*b^30*d^6 + 2374944*a^29*b^28*d^6 + 727584*a^31*b^26*d^6 - 1413984*a^33*b^24*
d^6 - 2649504*a^35*b^22*d^6 - 2454816*a^37*b^20*d^6 - 1476384*a^39*b^18*d^6 - 597408*a^41*b^16*d^6 - 156192*a^
43*b^14*d^6 - 22944*a^45*b^12*d^6 - 1056*a^47*b^10*d^6 + 96*a^49*b^8*d^6))/2)*(((20*a^2*b^8*d^4 - b^10*d^4 - 1
10*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)/(a^10*d^4 +
b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2))/2 + 32*a^14*b^38*d^4 + 44
8*a^16*b^36*d^4 + 2912*a^18*b^34*d^4 + 11648*a^20*b^32*d^4 + 32032*a^22*b^30*d^4 + 64064*a^24*b^28*d^4 + 96096
*a^26*b^26*d^4 + 109824*a^28*b^24*d^4 + 96096*a^30*b^22*d^4 + 64064*a^32*b^20*d^4 + 32032*a^34*b^18*d^4 + 1164
8*a^36*b^16*d^4 + 2912*a^38*b^14*d^4 + 448*a^40*b^12*d^4 + 32*a^42*b^10*d^4)*(((20*a^2*b^8*d^4 - b^10*d^4 - 11
0*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)/(a^10*d^4 +
b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2))/2 + (log((((a + b*tan(c +
d*x))^(1/2)*(10304*a^20*b^34*d^5 - 512*a^16*b^38*d^5 - 544*a^18*b^36*d^5 - 64*a^14*b^40*d^5 + 66976*a^22*b^32*
d^5 + 221312*a^24*b^30*d^5 + 480480*a^26*b^28*d^5 + 741312*a^28*b^26*d^5 + 837408*a^30*b^24*d^5 + 695552*a^32*
b^22*d^5 + 416416*a^34*b^20*d^5 + 168896*a^36*b^18*d^5 + 37856*a^38*b^16*d^5 - 896*a^40*b^14*d^5 - 3424*a^42*b
^12*d^5 - 960*a^44*b^10*d^5 - 96*a^46*b^8*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)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 +
10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(384*a^15*b^42*d^6 - ((((-((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)/(a^10*d
^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(512*a^16*b^46*d^8 - (
(-((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)/(a^10*d^4 + b^10*d^4 + 5*...

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