3.6.44 \(\int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\) [544]

Optimal. Leaf size=192 \[ \frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}-\frac {b \left (a^2+3 b^2\right )}{a^2 \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {\cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.45, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3650, 3730, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} \frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {b \left (a^2+3 b^2\right )}{a^2 d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{3/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{3/2}}-\frac {\cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 3.51, size = 184, normalized size = 0.96 \begin {gather*} -\frac {-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {i a^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2}}+\frac {i a^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2}}+\frac {b \left (a^2+3 b^2\right )}{\left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {a \cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}}{a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.55, size = 60231, normalized size = 313.70

method	result	size
default	\(\text {Expression too large to display}\)	\(60231\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6454 vs. \(2 (158) = 316\).
time = 2.55, size = 12983, normalized size = 67.62 \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)^2/(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

[-1/4*(4*sqrt(2)*((a^13 + 3*a^11*b^2 + 2*a^9*b^4 - 2*a^7*b^6 - 3*a^5*b^8 - a^3*b^10)*d^5*cos(d*x + c)^4 - (a^1
3 + 2*a^11*b^2 - 2*a^9*b^4 - 8*a^7*b^6 - 7*a^5*b^8 - 2*a^3*b^10)*d^5*cos(d*x + c)^2 - (a^11*b^2 + 4*a^9*b^4 +
6*a^7*b^6 + 4*a^5*b^8 + a^3*b^10)*d^5 + 2*((a^12*b + 4*a^10*b^3 + 6*a^8*b^5 + 4*a^6*b^7 + a^4*b^9)*d^5*cos(d*x
+ c)^3 - (a^12*b + 4*a^10*b^3 + 6*a^8*b^5 + 4*a^6*b^7 + a^4*b^9)*d^5*cos(d*x + c))*sin(d*x + c))*sqrt((a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*
b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*a
rctan(((3*a^12 + 14*a^10*b^2 + 25*a^8*b^4 + 20*a^6*b^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12)*d^4*sqrt((9*a^4*b^2 -
6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/
((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (3*a^9 + 8*a^7*b^2 + 6*a^5*b^4 - a*b^8)*d^2*sqrt((9*a^4*b^2 - 6*a
^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + sqrt(2)*
(2*(a^13 + 6*a^11*b^2 + 15*a^9*b^4 + 20*a^7*b^6 + 15*a^5*b^8 + 6*a^3*b^10 + a*b^12)*d^7*sqrt((9*a^4*b^2 - 6*a^
2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^
6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (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^5*s
qrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^
12)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6
+ 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt(((9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6
- 4*a^2*b^8 + b^10)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*((9*a^8*b^3
+ 12*a^6*b^5 - 2*a^4*b^7 - 4*a^2*b^9 + b^11)*d^3*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c
) + 2*(9*a^5*b^3 - 6*a^3*b^5 + a*b^7)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 - 6*a^5*b
^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6)
)*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (
9*a^5*b^2 - 6*a^3*b^4 + a*b^6)*cos(d*x + c) + (9*a^4*b^3 - 6*a^2*b^5 + b^7)*sin(d*x + c))/cos(d*x + c))*(1/((a
^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4) + sqrt(2)*(2*(3*a^15*b + 17*a^13*b^3 + 39*a^11*b^5 + 45*a^9*b^7
+ 25*a^7*b^9 + 3*a^5*b^11 - 3*a^3*b^13 - a*b^15)*d^7*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 +
15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^
4)) + (3*a^12*b + 14*a^10*b^3 + 25*a^8*b^5 + 20*a^6*b^7 + 5*a^4*b^9 - 2*a^2*b^11 - b^13)*d^5*sqrt((9*a^4*b^2 -
6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt(
(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*
a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/(
(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6)) + 4*sqrt(2)*((a^13 + 3*a^11*b^
2 + 2*a^9*b^4 - 2*a^7*b^6 - 3*a^5*b^8 - a^3*b^10)*d^5*cos(d*x + c)^4 - (a^13 + 2*a^11*b^2 - 2*a^9*b^4 - 8*a^7*
b^6 - 7*a^5*b^8 - 2*a^3*b^10)*d^5*cos(d*x + c)^2 - (a^11*b^2 + 4*a^9*b^4 + 6*a^7*b^6 + 4*a^5*b^8 + a^3*b^10)*d
^5 + 2*((a^12*b + 4*a^10*b^3 + 6*a^8*b^5 + 4*a^6*b^7 + a^4*b^9)*d^5*cos(d*x + c)^3 - (a^12*b + 4*a^10*b^3 + 6*
a^8*b^5 + 4*a^6*b^7 + a^4*b^9)*d^5*cos(d*x + c))*sin(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9
- 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b
^4 + b^6))*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a
^2*b^10 + b^12)*d^4))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*arctan(-((3*a^12 + 14*a^10*b^2 + 25*
a^8*b^4 + 20*a^6*b^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12)*d^4*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b
^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^
6)*d^4)) + (3*a^9 + 8*a^7*b^2 + 6*a^5*b^4 - a*b^8)*d^2*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2
+ 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - sqrt(2)*(2*(a^13 + 6*a^11*b^2 + 15*a^9*b^4
+ 20*a^7*b^6 + 15*a^5*b^8 + 6*a^3*b^10 + a*b^12)*d^7*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 +
15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d
^4)) + (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^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/
((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 +...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Mupad [B]
time = 4.56, size = 2500, normalized size = 13.02 \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)^2/(a + b*tan(c + d*x))^(3/2),x)

[Out]

log(72*a^14*b^25*d^4 - ((a + b*tan(c + d*x))^(1/2)*(144*a^14*b^26*d^5 + 864*a^16*b^24*d^5 + 2048*a^18*b^22*d^5
+ 2240*a^20*b^20*d^5 + 672*a^22*b^18*d^5 - 896*a^24*b^16*d^5 - 896*a^26*b^14*d^5 - 192*a^28*b^12*d^5 + 80*a^3
0*b^10*d^5 + 32*a^32*b^8*d^5) + (-(((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^
4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(576*
a^15*b^27*d^6 - ((a + b*tan(c + d*x))^(1/2)*(576*a^15*b^28*d^7 + 5184*a^17*b^26*d^7 + 21568*a^19*b^24*d^7 + 53
888*a^21*b^22*d^7 + 87808*a^23*b^20*d^7 + 94976*a^25*b^18*d^7 + 66304*a^27*b^16*d^7 + 27008*a^29*b^14*d^7 + 42
88*a^31*b^12*d^7 - 832*a^33*b^10*d^7 - 320*a^35*b^8*d^7) - (-(((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6
*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3
*a^4*b^2*d^4)))^(1/2)*((a + b*tan(c + d*x))^(1/2)*(-(((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*
a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*
d^4)))^(1/2)*(512*a^18*b^28*d^9 + 5376*a^20*b^26*d^9 + 25344*a^22*b^24*d^9 + 70656*a^24*b^22*d^9 + 129024*a^26
*b^20*d^9 + 161280*a^28*b^18*d^9 + 139776*a^30*b^16*d^9 + 82944*a^32*b^14*d^9 + 32256*a^34*b^12*d^9 + 7424*a^3
6*b^10*d^9 + 768*a^38*b^8*d^9) + 768*a^16*b^29*d^8 + 7680*a^18*b^27*d^8 + 34304*a^20*b^25*d^8 + 90112*a^22*b^2
3*d^8 + 154112*a^24*b^21*d^8 + 179200*a^26*b^19*d^8 + 143360*a^28*b^17*d^8 + 77824*a^30*b^15*d^8 + 27392*a^32*
b^13*d^8 + 5632*a^34*b^11*d^8 + 512*a^36*b^9*d^8))*(-(((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3
*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2
*d^4)))^(1/2) + 3456*a^17*b^25*d^6 + 8480*a^19*b^23*d^6 + 10976*a^21*b^21*d^6 + 8736*a^23*b^19*d^6 + 6496*a^25
*b^17*d^6 + 6496*a^27*b^15*d^6 + 5280*a^29*b^13*d^6 + 2336*a^31*b^11*d^6 + 416*a^33*b^9*d^6))*(-(((8*a^3*d^2 -
24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^
6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2) + 456*a^16*b^23*d^4 + 1176*a^18*b^21*d^4 + 1512*a^20*
b^19*d^4 + 840*a^22*b^17*d^4 - 168*a^24*b^15*d^4 - 504*a^26*b^13*d^4 - 264*a^28*b^11*d^4 - 48*a^30*b^9*d^4)*(-
(((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b
^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2) + log(72*a^14*b^25*d^4 - ((a + b*tan(c
+ d*x))^(1/2)*(144*a^14*b^26*d^5 + 864*a^16*b^24*d^5 + 2048*a^18*b^22*d^5 + 2240*a^20*b^20*d^5 + 672*a^22*b^18
*d^5 - 896*a^24*b^16*d^5 - 896*a^26*b^14*d^5 - 192*a^28*b^12*d^5 + 80*a^30*b^10*d^5 + 32*a^32*b^8*d^5) + ((((8
*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d
^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(576*a^15*b^27*d^6 - ((a + b*tan(c + d*x))^
(1/2)*(576*a^15*b^28*d^7 + 5184*a^17*b^26*d^7 + 21568*a^19*b^24*d^7 + 53888*a^21*b^22*d^7 + 87808*a^23*b^20*d^
7 + 94976*a^25*b^18*d^7 + 66304*a^27*b^16*d^7 + 27008*a^29*b^14*d^7 + 4288*a^31*b^12*d^7 - 832*a^33*b^10*d^7 -
320*a^35*b^8*d^7) - ((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(
1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*((a + b*tan(c + d
*x))^(1/2)*((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) - a^3
*d^2 + 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(512*a^18*b^28*d^9 + 5376*a
^20*b^26*d^9 + 25344*a^22*b^24*d^9 + 70656*a^24*b^22*d^9 + 129024*a^26*b^20*d^9 + 161280*a^28*b^18*d^9 + 13977
6*a^30*b^16*d^9 + 82944*a^32*b^14*d^9 + 32256*a^34*b^12*d^9 + 7424*a^36*b^10*d^9 + 768*a^38*b^8*d^9) + 768*a^1
6*b^29*d^8 + 7680*a^18*b^27*d^8 + 34304*a^20*b^25*d^8 + 90112*a^22*b^23*d^8 + 154112*a^24*b^21*d^8 + 179200*a^
26*b^19*d^8 + 143360*a^28*b^17*d^8 + 77824*a^30*b^15*d^8 + 27392*a^32*b^13*d^8 + 5632*a^34*b^11*d^8 + 512*a^36
*b^9*d^8))*((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) - a^3
*d^2 + 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2) + 3456*a^17*b^25*d^6 + 8480
*a^19*b^23*d^6 + 10976*a^21*b^21*d^6 + 8736*a^23*b^19*d^6 + 6496*a^25*b^17*d^6 + 6496*a^27*b^15*d^6 + 5280*a^2
9*b^13*d^6 + 2336*a^31*b^11*d^6 + 416*a^33*b^9*d^6))*((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 -
3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^
2*d^4)))^(1/2) + 456*a^16*b^23*d^4 + 1176*a^18*b^21*d^4 + 1512*a^20*b^19*d^4 + 840*a^22*b^17*d^4 - 168*a^24*b^
15*d^4 - 504*a^26*b^13*d^4 - 264*a^28*b^11*d^4 - 48*a^30*b^9*d^4)*((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4
- a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d
^4 + 3*a^4*b^2*d^4)))^(1/2) - log(72*a^14*b^25*...

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