3.6.36 \(\int \frac {\cot ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) [536]

Optimal. Leaf size=194 \[ \frac {\left (8 a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b} d}+\frac {3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d} \]

[Out]

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

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Rubi [A]
time = 0.39, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3650, 3730, 3735, 12, 3620, 3618, 65, 214, 3715} \begin {gather*} \frac {3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 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 a^{5/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 3735

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*ta
n[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*
C)*Tan[e + f*x], x], x], x] + Dist[(A*b^2 + 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, 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 ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx &=-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}-\frac {\int \frac {\cot ^2(c+d x) \left (\frac {3 b}{2}+2 a \tan (c+d x)+\frac {3}{2} b \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a}\\ &=\frac {3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}+\frac {\int \frac {\cot (c+d x) \left (\frac {1}{4} \left (-8 a^2+3 b^2\right )+\frac {3}{4} b^2 \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a^2}\\ &=\frac {3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}+\frac {\int \frac {2 a^2 \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a^2}+\frac {1}{8} \left (-8+\frac {3 b^2}{a^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 {3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}-\frac {\left (8-\frac {3 b^2}{a^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{8 d}+\int \frac {\tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}+\frac {1}{2} i \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {1}{2} i \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {\left (8-\frac {3 b^2}{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 )}{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 a^{5/2} d}+\frac {3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}+\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{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 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 a^{5/2} d}+\frac {3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a 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 )}{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 )}{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 a^{5/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b} d}+\frac {3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}\\ \end {align*}

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Mathematica [A]
time = 3.28, size = 203, normalized size = 1.05 \begin {gather*} \frac {\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}}-\frac {4 a^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {a-\sqrt {-b^2}}}-\frac {4 a^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {a+\sqrt {-b^2}}}+3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}-2 a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.27, size = 58397, normalized size = 301.02

method	result	size
default	\(\text {Expression too large to display}\)	\(58397\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3/(a+b*tan(d*x+c))^(1/2),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))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2416 vs. \(2 (158) = 316\).
time = 1.67, size = 4908, normalized size = 25.30 \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))^(1/2),x, algorithm="fricas")

[Out]

[-1/16*(16*sqrt(2)*((a^5 + a^3*b^2)*d^5*cos(d*x + c)^2 - (a^5 + a^3*b^2)*d^5)*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/
((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d^4))^(3/4)*arcta
n(-((a^4 + 2*a^2*b^2 + b^4)*d^4*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) + (a^3 + a*b
^2)*d^2*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*d^7*sqrt(b^2/((a^4 + 2*a^
2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) + (a^4 + 2*a^2*b^2 + b^4)*d^5*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d
^4)))*sqrt(((a^2 + b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*((a^2 + b^2)*d^3*sqrt(1/((a^2 + b
^2)*d^4))*cos(d*x + c) + a*d*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 +
a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4) + a*cos(d*x + c) + b*sin(d*
x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4
))^(3/4) + sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*d^7*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*
d^4)) + (a^4 + 2*a^2*b^2 + b^4)*d^5*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a*cos(d*x + c) + b*sin(d*x
+ c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))
^(3/4))/b^2) + 16*sqrt(2)*((a^5 + a^3*b^2)*d^5*cos(d*x + c)^2 - (a^5 + a^3*b^2)*d^5)*sqrt(-((a^3 + a*b^2)*d^2*
sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d^4))^(3/4
)*arctan(((a^4 + 2*a^2*b^2 + b^4)*d^4*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) + (a^3
+ a*b^2)*d^2*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*d^7*sqrt(b^2/((a^4
+ 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) + (a^4 + 2*a^2*b^2 + b^4)*d^5*sqrt(b^2/((a^4 + 2*a^2*b^2 +
b^4)*d^4)))*sqrt(((a^2 + b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*((a^2 + b^2)*d^3*sqrt(1/((a
^2 + b^2)*d^4))*cos(d*x + c) + a*d*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-((
a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4) + a*cos(d*x + c) + b*
sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^
2)*d^4))^(3/4) - sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*d^7*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 +
b^2)*d^4)) + (a^4 + 2*a^2*b^2 + b^4)*d^5*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a*cos(d*x + c) + b*si
n(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)
*d^4))^(3/4))/b^2) + 4*sqrt(2)*(a^3*d*cos(d*x + c)^2 - a^3*d + (a^4*d^3*cos(d*x + c)^2 - a^4*d^3)*sqrt(1/((a^2
+ b^2)*d^4)))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4
)*log(((a^2 + b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*((a^2 + b^2)*d^3*sqrt(1/((a^2 + b^2)*d
^4))*cos(d*x + c) + a*d*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2
)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4) + a*cos(d*x + c) + b*sin(d*x + c
))/cos(d*x + c)) - 4*sqrt(2)*(a^3*d*cos(d*x + c)^2 - a^3*d + (a^4*d^3*cos(d*x + c)^2 - a^4*d^3)*sqrt(1/((a^2 +
b^2)*d^4)))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4)*
log(((a^2 + b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*((a^2 + b^2)*d^3*sqrt(1/((a^2 + b^2)*d^4
))*cos(d*x + c) + a*d*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*
d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4) + a*cos(d*x + c) + b*sin(d*x + c))
/cos(d*x + c)) + ((8*a^2 - 3*b^2)*cos(d*x + c)^2 - 8*a^2 + 3*b^2)*sqrt(a)*log(-(8*a*b*cos(d*x + c)*sin(d*x + c
) + (8*a^2 - b^2)*cos(d*x + c)^2 + b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c)*sin(d*x + c))*sqrt(a)*sqrt((a*
cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/(cos(d*x + c)^2 - 1)) - 4*(2*a^2*cos(d*x + c)^2 - 3*a*b*cos(d*x
+ c)*sin(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/(a^3*d*cos(d*x + c)^2 - a^3*d), -1/4*
(4*sqrt(2)*((a^5 + a^3*b^2)*d^5*cos(d*x + c)^2 - (a^5 + a^3*b^2)*d^5)*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 +
b^2)*d^4)) - a^2 - b^2)/b^2)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d^4))^(3/4)*arctan(-((a^4
+ 2*a^2*b^2 + b^4)*d^4*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) + (a^3 + a*b^2)*d^2*
sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*d^7*sqrt(b^2/((a^4 + 2*a^2*b^2 +
b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) + (a^4 + 2*a^2*b^2 + b^4)*d^5*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sq
rt(((a^2 + b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*((a^2 + b^2)*d^3*sqrt(1/((a^2 + b^2)*d^4)
)*cos(d*x + c) + a*d*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d
^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{3}{\left (c + d x \right )}}{\sqrt {a + b \tan {\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))**(1/2),x)

[Out]

Integral(cot(c + d*x)**3/sqrt(a + b*tan(c + d*x)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 0.59, size = 2500, normalized size = 12.89 \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))^(1/2),x)

[Out]

atan((((((((((320*a^4*b^10*d^4 - 192*a^2*b^12*d^4 + 384*a^6*b^8*d^4)/(2*a^4*d^5) - ((512*a^4*b^10*d^4 + 768*a^
6*b^8*d^4)*(1/(a*d^2 - b*d^2*1i))^(1/2)*(a + b*tan(c + d*x))^(1/2))/(4*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2))
/2 + ((a + b*tan(c + d*x))^(1/2)*(36*a*b^12*d^2 - 192*a^3*b^10*d^2 + 576*a^5*b^8*d^2))/(2*a^4*d^4))*(1/(a*d^2
- b*d^2*1i))^(1/2))/2 + (18*a*b^12*d^2 - 96*a^5*b^8*d^2)/(2*a^4*d^5))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - ((a +
b*tan(c + d*x))^(1/2)*(9*b^12 - 48*a^2*b^10 + 96*a^4*b^8))/(2*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2)*1i - ((((
((((320*a^4*b^10*d^4 - 192*a^2*b^12*d^4 + 384*a^6*b^8*d^4)/(2*a^4*d^5) + ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)
*(1/(a*d^2 - b*d^2*1i))^(1/2)*(a + b*tan(c + d*x))^(1/2))/(4*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - ((a +
b*tan(c + d*x))^(1/2)*(36*a*b^12*d^2 - 192*a^3*b^10*d^2 + 576*a^5*b^8*d^2))/(2*a^4*d^4))*(1/(a*d^2 - b*d^2*1i
))^(1/2))/2 + (18*a*b^12*d^2 - 96*a^5*b^8*d^2)/(2*a^4*d^5))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + ((a + b*tan(c +
d*x))^(1/2)*(9*b^12 - 48*a^2*b^10 + 96*a^4*b^8))/(2*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2)*1i)/(((((((((320*a^
4*b^10*d^4 - 192*a^2*b^12*d^4 + 384*a^6*b^8*d^4)/(2*a^4*d^5) - ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(1/(a*d^2
- b*d^2*1i))^(1/2)*(a + b*tan(c + d*x))^(1/2))/(4*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + ((a + b*tan(c +
d*x))^(1/2)*(36*a*b^12*d^2 - 192*a^3*b^10*d^2 + 576*a^5*b^8*d^2))/(2*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2))/
2 + (18*a*b^12*d^2 - 96*a^5*b^8*d^2)/(2*a^4*d^5))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - ((a + b*tan(c + d*x))^(1/2
)*(9*b^12 - 48*a^2*b^10 + 96*a^4*b^8))/(2*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2) + ((((((((320*a^4*b^10*d^4 -
192*a^2*b^12*d^4 + 384*a^6*b^8*d^4)/(2*a^4*d^5) + ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(1/(a*d^2 - b*d^2*1i))
^(1/2)*(a + b*tan(c + d*x))^(1/2))/(4*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - ((a + b*tan(c + d*x))^(1/2)*
(36*a*b^12*d^2 - 192*a^3*b^10*d^2 + 576*a^5*b^8*d^2))/(2*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + (18*a*b^1
2*d^2 - 96*a^5*b^8*d^2)/(2*a^4*d^5))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + ((a + b*tan(c + d*x))^(1/2)*(9*b^12 - 4
8*a^2*b^10 + 96*a^4*b^8))/(2*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2) - (9*b^12 - 24*a^2*b^10)/(a^4*d^5)))*(1/(a
*d^2 - b*d^2*1i))^(1/2)*1i - atan((((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4*a^2*d^2 + 4*b^2*
d^2))^(1/2)*(((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((320*a^4*b^10*d^4 - 192*a^2*b^12*d^4 + 384*a^6*b^8*d
^4)/(a^4*d^5) + (((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c
+ d*x))^(1/2))/(a^4*d^4))*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(36*a*b^12*
d^2 - 192*a^3*b^10*d^2 + 576*a^5*b^8*d^2))/(a^4*d^4)) + (18*a*b^12*d^2 - 96*a^5*b^8*d^2)/(a^4*d^5)) + ((a + b*
tan(c + d*x))^(1/2)*(9*b^12 - 48*a^2*b^10 + 96*a^4*b^8))/(a^4*d^4))*1i - ((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^
(1/2)*(((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((320*a^4*b^10*
d^4 - 192*a^2*b^12*d^4 + 384*a^6*b^8*d^4)/(a^4*d^5) - (((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(512*a^4*b^1
0*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(a^4*d^4))*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) + (
(a + b*tan(c + d*x))^(1/2)*(36*a*b^12*d^2 - 192*a^3*b^10*d^2 + 576*a^5*b^8*d^2))/(a^4*d^4)) + (18*a*b^12*d^2 -
96*a^5*b^8*d^2)/(a^4*d^5)) - ((a + b*tan(c + d*x))^(1/2)*(9*b^12 - 48*a^2*b^10 + 96*a^4*b^8))/(a^4*d^4))*1i)/
(((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4*a^2*d^
2 + 4*b^2*d^2))^(1/2)*(((320*a^4*b^10*d^4 - 192*a^2*b^12*d^4 + 384*a^6*b^8*d^4)/(a^4*d^5) + (((a - b*1i)/(4*a^
2*d^2 + 4*b^2*d^2))^(1/2)*(512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(a^4*d^4))*((a - b*
1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(36*a*b^12*d^2 - 192*a^3*b^10*d^2 + 576*a^5*b
^8*d^2))/(a^4*d^4)) + (18*a*b^12*d^2 - 96*a^5*b^8*d^2)/(a^4*d^5)) + ((a + b*tan(c + d*x))^(1/2)*(9*b^12 - 48*a
^2*b^10 + 96*a^4*b^8))/(a^4*d^4)) + ((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4*a^2*d^2 + 4*b^2
*d^2))^(1/2)*(((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((320*a^4*b^10*d^4 - 192*a^2*b^12*d^4 + 384*a^6*b^8*
d^4)/(a^4*d^5) - (((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c
+ d*x))^(1/2))/(a^4*d^4))*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(36*a*b^12
*d^2 - 192*a^3*b^10*d^2 + 576*a^5*b^8*d^2))/(a^4*d^4)) + (18*a*b^12*d^2 - 96*a^5*b^8*d^2)/(a^4*d^5)) - ((a + b
*tan(c + d*x))^(1/2)*(9*b^12 - 48*a^2*b^10 + 96*a^4*b^8))/(a^4*d^4)) - (9*b^12 - 24*a^2*b^10)/(a^4*d^5)))*((a
- b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*2i - ((5*b^2*(a + b*tan(c + d*x))^(1/2))/(4*a) - (3*b^2*(a + b*tan(c +
d*x))^(3/2))/(4*a^2))/(d*(a + b*tan(c + d*x))^2 + a^2*d - 2*a*d*(a + b*tan(c + d*x))) - (atan((b^16*(a + b*tan
(c + d*x))^(1/2)*1755i)/(4*(a^5)^(1/2)*(696*a^2*b^12 - 927*b^14 + 344*a^4*b^10 - 576*a^6*b^8 + (1755*b^16)/(4*
a^2) - (2997*b^18)/(32*a^4) + (243*b^20)/(32*a^...

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