3.6.18 \(\int \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) [518]

Optimal. Leaf size=189 \[ \frac {\left (8 a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {(a-i b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d} \]

[Out]

-(a-I*b)^(3/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d-(a+I*b)^(3/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a
+I*b)^(1/2))/d+1/4*(8*a^2-3*b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)-5/4*b*cot(d*x+c)*(a+b*tan(d
*x+c))^(1/2)/d-1/2*a*cot(d*x+c)^2*(a+b*tan(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.47, antiderivative size = 189, 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 = {3648, 3730, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} \frac {\left (8 a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {(a-i b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^2 - 3*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*Sqrt[a]*d) - ((a - I*b)^(3/2)*ArcTanh[Sqrt[a +
b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (5*b
*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*d) - (a*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*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 3648

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 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 \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx &=-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {1}{2} \int \frac {\cot ^2(c+d x) \left (-\frac {5 a b}{2}+2 \left (a^2-b^2\right ) \tan (c+d x)+\frac {3}{2} a b \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {\int \frac {\cot (c+d x) \left (-\frac {1}{4} a \left (8 a^2-3 b^2\right )-4 a^2 b \tan (c+d x)-\frac {5}{4} a b^2 \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a}\\ &=-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {\int \frac {-4 a^2 b+2 a \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a}+\frac {1}{8} \left (-8 a^2+3 b^2\right ) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {1}{2} \left (i (a-i b)^2\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} \left (i (a+i b)^2\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {\left (8 a^2-3 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{8 d}\\ &=-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {(a-i b)^2 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}+\frac {(a+i b)^2 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}-\frac {\left (8 a^2-3 b^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 )}{4 b d}\\ &=\frac {\left (8 a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {\left (i (a-i b)^2\right ) \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 {\left (i (a+i b)^2\right ) \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 {\left (8 a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {(a-i b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\\ \end {align*}

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Mathematica [A]
time = 1.59, size = 168, normalized size = 0.89 \begin {gather*} \frac {\left (8 a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )-\sqrt {a} \left (4 (a-i b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+4 (a+i b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\cot (c+d x) (5 b+2 a \cot (c+d x)) \sqrt {a+b \tan (c+d x)}\right )}{4 \sqrt {a} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^2 - 3*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] - Sqrt[a]*(4*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan
[c + d*x]]/Sqrt[a - I*b]] + 4*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Cot[c + d*x]*(
5*b + 2*a*Cot[c + d*x])*Sqrt[a + b*Tan[c + d*x]]))/(4*Sqrt[a]*d)

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.19, size = 54149, normalized size = 286.50

method	result	size
default	\(\text {Expression too large to display}\)	\(54149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(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)^3*(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)^3, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4533 vs. \(2 (153) = 306\).
time = 2.61, size = 9142, normalized size = 48.37 \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)^3*(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

[-1/16*(16*sqrt(2)*(a*d^5*cos(d*x + c)^2 - a*d^5)*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^
2*sqrt((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))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)/d^4)^(3/4)*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4)*arctan(((3*a^10 + 11*a^8*b^2 + 14*a^6*b^4 + 6*a^4*b^
6 - a^2*b^8 - b^10)*d^4*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4)
+ (3*a^13 + 14*a^11*b^2 + 25*a^9*b^4 + 20*a^7*b^6 + 5*a^5*b^8 - 2*a^3*b^10 - a*b^12)*d^2*sqrt((9*a^4*b^2 - 6*a
^2*b^4 + b^6)/d^4) + sqrt(2)*((3*a^5 + 2*a^3*b^2 - a*b^4)*d^7*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*sq
rt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^8 + 2*a^6*b^2 - 4*a^4*b^4 - 2*a^2*b^6 + b^8)*d^5*sqrt((9*a^4*b^2
- 6*a^2*b^4 + b^6)/d^4))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((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))*((
a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)^(3/4) + sqrt(2)*(a*d^7*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*s
qrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (a^4 - b^4)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))*sqrt((a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((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 + 12*a^6*b^2 - 2*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b
^4 + b^6)/d^4)*cos(d*x + c) + sqrt(2)*((9*a^6 - 15*a^4*b^2 + 7*a^2*b^4 - b^6)*d^3*sqrt((a^6 + 3*a^4*b^2 + 3*a^
2*b^4 + b^6)/d^4)*cos(d*x + c) + (9*a^9 + 12*a^7*b^2 - 2*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c))*sqrt((a^
6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((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))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
)/d^4)^(1/4) + (9*a^11 + 21*a^9*b^2 + 10*a^7*b^4 - 6*a^5*b^6 - 3*a^3*b^8 + a*b^10)*cos(d*x + c) + (9*a^10*b +
21*a^8*b^3 + 10*a^6*b^5 - 6*a^4*b^7 - 3*a^2*b^9 + b^11)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c)))*((a^6 + 3*a^
4*b^2 + 3*a^2*b^4 + b^6)/d^4)^(3/4))/(9*a^14*b^2 + 39*a^12*b^4 + 61*a^10*b^6 + 35*a^8*b^8 - 5*a^6*b^10 - 11*a^
4*b^12 - a^2*b^14 + b^16)) + 16*sqrt(2)*(a*d^5*cos(d*x + c)^2 - a*d^5)*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
- (a^3 - 3*a*b^2)*d^2*sqrt((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))*((a^6 + 3
*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)^(3/4)*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4)*arctan(-((3*a^10 + 11*a^8*b^2 +
14*a^6*b^4 + 6*a^4*b^6 - a^2*b^8 - b^10)*d^4*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*sqrt((9*a^4*b^2 -
6*a^2*b^4 + b^6)/d^4) + (3*a^13 + 14*a^11*b^2 + 25*a^9*b^4 + 20*a^7*b^6 + 5*a^5*b^8 - 2*a^3*b^10 - a*b^12)*d^2
*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - sqrt(2)*((3*a^5 + 2*a^3*b^2 - a*b^4)*d^7*sqrt((a^6 + 3*a^4*b^2 + 3*
a^2*b^4 + b^6)/d^4)*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^8 + 2*a^6*b^2 - 4*a^4*b^4 - 2*a^2*b^6 + b^8
)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*s
qrt((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))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)^(3/4) - sqrt(2)*(a*d^7*sqrt((a^6 + 3*a^4*b^2 + 3
*a^2*b^4 + b^6)/d^4)*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (a^4 - b^4)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b
^6)/d^4))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((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 + 12*a^6*b^2 - 2*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2*sqrt((a^6
+ 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*cos(d*x + c) - sqrt(2)*((9*a^6 - 15*a^4*b^2 + 7*a^2*b^4 - b^6)*d^3*sqrt((
a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*cos(d*x + c) + (9*a^9 + 12*a^7*b^2 - 2*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*
cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((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))*((a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6)/d^4)^(1/4) + (9*a^11 + 21*a^9*b^2 + 10*a^7*b^4 - 6*a^5*b^6 - 3*a^3*b^8 + a*b^10)*cos(d
*x + c) + (9*a^10*b + 21*a^8*b^3 + 10*a^6*b^5 - 6*a^4*b^7 - 3*a^2*b^9 + b^11)*sin(d*x + c))/((a^2 + b^2)*cos(d
*x + c)))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)^(3/4))/(9*a^14*b^2 + 39*a^12*b^4 + 61*a^10*b^6 + 35*a^8*b^
8 - 5*a^6*b^10 - 11*a^4*b^12 - a^2*b^14 + b^16)) + 4*sqrt(2)*((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d*cos(d*x
+ c)^2 - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d + ((a^4 - 3*a^2*b^2)*d^3*cos(d*x + c)^2 - (a^4 - 3*a^2*b^2)*d
^3)*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d
^2*sqrt((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))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)/d^4)^(1/4)*log(((9*a^8 + 12*a^6*b^2 - 2*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b
^4 + b^6)/d^4)*cos(d*x + c) + sqrt(2)*((9*a^6 -...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Mupad [B]
time = 4.88, size = 2500, normalized size = 13.23 \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)^3*(a + b*tan(c + d*x))^(3/2),x)

[Out]

- atan(((((((384*a^2*b^10*d^4 - 384*b^12*d^4 + 768*a^4*b^8*d^4)/(2*d^5) + ((512*b^10*d^4 + 768*a^2*b^8*d^4)*(a
+ b*tan(c + d*x))^(1/2)*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2))/d^4)*(-(3*a*b^2 - a^2*b*3i - a^
3 + b^3*1i)/(4*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(668*a*b^12*d^2 + 1088*a^3*b^10*d^2 - 576*a^5*b^8*d^2
))/d^4)*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2) - (604*a*b^14*d^2 - 932*a^3*b^12*d^2 - 1344*a^5*b
^10*d^2 + 192*a^7*b^8*d^2)/(2*d^5))*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2) + ((a + b*tan(c + d*x
))^(1/2)*(41*b^16 + 26*a^2*b^14 + 553*a^4*b^12 - 304*a^6*b^10 + 96*a^8*b^8))/d^4)*(-(3*a*b^2 - a^2*b*3i - a^3
+ b^3*1i)/(4*d^2))^(1/2)*1i - (((((384*a^2*b^10*d^4 - 384*b^12*d^4 + 768*a^4*b^8*d^4)/(2*d^5) - ((512*b^10*d^4
+ 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2))/d^4)*(-(3
*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(668*a*b^12*d^2 + 1088*a^3*b^10
*d^2 - 576*a^5*b^8*d^2))/d^4)*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2) - (604*a*b^14*d^2 - 932*a^3
*b^12*d^2 - 1344*a^5*b^10*d^2 + 192*a^7*b^8*d^2)/(2*d^5))*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2)
- ((a + b*tan(c + d*x))^(1/2)*(41*b^16 + 26*a^2*b^14 + 553*a^4*b^12 - 304*a^6*b^10 + 96*a^8*b^8))/d^4)*(-(3*a
*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2)*1i)/((119*a^4*b^14 - 71*a^2*b^16 - 15*b^18 + 391*a^6*b^12 + 216
*a^8*b^10)/d^5 + (((((384*a^2*b^10*d^4 - 384*b^12*d^4 + 768*a^4*b^8*d^4)/(2*d^5) + ((512*b^10*d^4 + 768*a^2*b^
8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2))/d^4)*(-(3*a*b^2 - a^2*
b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(668*a*b^12*d^2 + 1088*a^3*b^10*d^2 - 576*a^
5*b^8*d^2))/d^4)*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2) - (604*a*b^14*d^2 - 932*a^3*b^12*d^2 - 1
344*a^5*b^10*d^2 + 192*a^7*b^8*d^2)/(2*d^5))*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2) + ((a + b*ta
n(c + d*x))^(1/2)*(41*b^16 + 26*a^2*b^14 + 553*a^4*b^12 - 304*a^6*b^10 + 96*a^8*b^8))/d^4)*(-(3*a*b^2 - a^2*b*
3i - a^3 + b^3*1i)/(4*d^2))^(1/2) + (((((384*a^2*b^10*d^4 - 384*b^12*d^4 + 768*a^4*b^8*d^4)/(2*d^5) - ((512*b^
10*d^4 + 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2))/d^4
)*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(668*a*b^12*d^2 + 1088*a^
3*b^10*d^2 - 576*a^5*b^8*d^2))/d^4)*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2) - (604*a*b^14*d^2 - 9
32*a^3*b^12*d^2 - 1344*a^5*b^10*d^2 + 192*a^7*b^8*d^2)/(2*d^5))*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))
^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(41*b^16 + 26*a^2*b^14 + 553*a^4*b^12 - 304*a^6*b^10 + 96*a^8*b^8))/d^4)*
(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2)))*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d^2))^(1/2)*2i
- atan(((((((384*a^2*b^10*d^4 - 384*b^12*d^4 + 768*a^4*b^8*d^4)/(2*d^5) + ((512*b^10*d^4 + 768*a^2*b^8*d^4)*(
a + b*tan(c + d*x))^(1/2)*(-(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2))/d^4)*(-(3*a*b^2 + a^2*b*3i - a
^3 - b^3*1i)/(4*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(668*a*b^12*d^2 + 1088*a^3*b^10*d^2 - 576*a^5*b^8*d^
2))/d^4)*(-(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) - (604*a*b^14*d^2 - 932*a^3*b^12*d^2 - 1344*a^5*
b^10*d^2 + 192*a^7*b^8*d^2)/(2*d^5))*(-(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) + ((a + b*tan(c + d*
x))^(1/2)*(41*b^16 + 26*a^2*b^14 + 553*a^4*b^12 - 304*a^6*b^10 + 96*a^8*b^8))/d^4)*(-(3*a*b^2 + a^2*b*3i - a^3
- b^3*1i)/(4*d^2))^(1/2)*1i - (((((384*a^2*b^10*d^4 - 384*b^12*d^4 + 768*a^4*b^8*d^4)/(2*d^5) - ((512*b^10*d^
4 + 768*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2))/d^4)*(-(
3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(668*a*b^12*d^2 + 1088*a^3*b^1
0*d^2 - 576*a^5*b^8*d^2))/d^4)*(-(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) - (604*a*b^14*d^2 - 932*a^
3*b^12*d^2 - 1344*a^5*b^10*d^2 + 192*a^7*b^8*d^2)/(2*d^5))*(-(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2
) - ((a + b*tan(c + d*x))^(1/2)*(41*b^16 + 26*a^2*b^14 + 553*a^4*b^12 - 304*a^6*b^10 + 96*a^8*b^8))/d^4)*(-(3*
a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2)*1i)/((119*a^4*b^14 - 71*a^2*b^16 - 15*b^18 + 391*a^6*b^12 + 21
6*a^8*b^10)/d^5 + (((((384*a^2*b^10*d^4 - 384*b^12*d^4 + 768*a^4*b^8*d^4)/(2*d^5) + ((512*b^10*d^4 + 768*a^2*b
^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2))/d^4)*(-(3*a*b^2 + a^2
*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(668*a*b^12*d^2 + 1088*a^3*b^10*d^2 - 576*a
^5*b^8*d^2))/d^4)*(-(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) - (604*a*b^14*d^2 - 932*a^3*b^12*d^2 -
1344*a^5*b^10*d^2 + 192*a^7*b^8*d^2)/(2*d^5))*(-(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) + ((a + b*t
an(c + d*x))^(1/2)*(41*b^16 + 26*a^2*b^14 + 553*a^4*b^12 - 304*a^6*b^10 + 96*a^8*b^8))/d^4)*(-(3*a*b^2 + a^2*b
*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) + (((((384*a...

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