3.535 \(\int \frac{\cot ^2(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx\)

Optimal. Leaf size=461 \[ \frac{b \log \left (-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \log \left (\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{a^{3/2} d}-\frac{\cot (c+d x) \sqrt{a+b \tan (c+d x)}}{a d} \]

[Out]

(b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sq
rt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b
*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*S
qrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + S
qrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*Log[a
+ Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*S
qrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(a*d)

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Rubi [A]  time = 0.613181, antiderivative size = 461, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {3569, 3653, 12, 3485, 708, 1094, 634, 618, 206, 628, 3634, 63, 208} \[ \frac{b \log \left (-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \log \left (\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{a^{3/2} d}-\frac{\cot (c+d x) \sqrt{a+b \tan (c+d x)}}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sq
rt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b
*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*S
qrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + S
qrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*Log[a
+ Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*S
qrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(a*d)

Rule 3569

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 3653

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]

Rule 12

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

Rule 3485

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x
, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]

Rule 708

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^2 + a*e^2 - 2*c
*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1094

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 3634

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 63

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 208

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

Rubi steps

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

Mathematica [C]  time = 0.77299, size = 142, normalized size = 0.31 \[ \frac{\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{\sqrt{a-i b}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{\sqrt{a+i b}}-\frac{\cot (c+d x) \sqrt{a+b \tan (c+d x)}}{a}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.971, size = 39033, normalized size = 84.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2/(a+b*tan(d*x+c))^(1/2),x)

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 10.1692, size = 9311, normalized size = 20.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[1/4*(4*sqrt(2)*((a^4 + a^2*b^2)*d^5*cos(d*x + c)^2 - (a^4 + a^2*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((s
qrt(2)*(a^4 + 2*a^2*b^2 + b^4)*d^7*sqrt((sqrt(2)*b^3*d*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sq
rt(((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)*cos(d*x + c) + (
a^2*b^2 + b^4)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + a*b^2*cos(d*x + c) + b^3*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)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))
*(1/((a^2 + b^2)*d^4))^(5/4) - sqrt(2)*(a^4*b + 2*a^2*b^3 + b^5)*d^7*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/co
s(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)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b
^4)*d^4))*(1/((a^2 + b^2)*d^4))^(5/4) - (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))*sq
rt(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)))/b^2) + 4*sqrt(2)*((a^4 +
a^2*b^2)*d^5*cos(d*x + c)^2 - (a^4 + a^2*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((sqrt(2)*(a^4 + 2*a^2*b^2
+ b^4)*d^7*sqrt(-(sqrt(2)*b^3*d*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(((a^3 + a*b^2)*d^2*s
qrt(1/((a^2 + b^2)*d^4)) + a^2 + b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4)*cos(d*x + c) - (a^2*b^2 + b^4)*d^2*sqrt
(1/((a^2 + b^2)*d^4))*cos(d*x + c) - a*b^2*cos(d*x + c) - b^3*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)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d^4))^
(5/4) - sqrt(2)*(a^4*b + 2*a^2*b^3 + b^5)*d^7*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)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^
2)*d^4))^(5/4) + (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)))/b^2) + 4*a*sqrt((a*cos(d*x + c) + b*sin(d*x + c)
)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - sqrt(2)*(a^2*d*cos(d*x + c)^2 - a^2*d - (a^3*d^3*cos(d*x + c)^2 -
a^3*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((sqrt(2)*b^3*d*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)*cos(d*x + c) + (a^2*b^2 + b^4)
*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + a*b^2*cos(d*x + c) + b^3*sin(d*x + c))/cos(d*x + c)) + sqrt(2)*(
a^2*d*cos(d*x + c)^2 - a^2*d - (a^3*d^3*cos(d*x + c)^2 - a^3*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(-(sqrt(2)*b^3*d*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)*cos(d*x + c) - (a^2*b^2 + b^4)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) - a*b^2*
cos(d*x + c) - b^3*sin(d*x + c))/cos(d*x + c)) + (b*cos(d*x + c)^2 - b)*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)*sq
rt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/(cos(d*x + c)^2 - 1)))/(a^2*d*cos(d*x + c)^2 - a^2*d), 1/4
*(4*sqrt(2)*((a^4 + a^2*b^2)*d^5*cos(d*x + c)^2 - (a^4 + a^2*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((sqrt(
2)*(a^4 + 2*a^2*b^2 + b^4)*d^7*sqrt((sqrt(2)*b^3*d*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)*cos(d*x + c) + (a^2*
b^2 + b^4)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + a*b^2*cos(d*x + c) + b^3*sin(d*x + c))/cos(d*x + c))*s
qrt(((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))^(5/4) - sqrt(2)*(a^4*b + 2*a^2*b^3 + b^5)*d^7*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)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*
d^4))*(1/((a^2 + b^2)*d^4))^(5/4) - (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)))/b^2) + 4*sqrt(2)*((a^4 + a^2*
b^2)*d^5*cos(d*x + c)^2 - (a^4 + a^2*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((sqrt(2)*(a^4 + 2*a^2*b^2 + b^
4)*d^7*sqrt(-(sqrt(2)*b^3*d*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)*cos(d*x + c) - (a^2*b^2 + b^4)*d^2*sqrt(1/(
(a^2 + b^2)*d^4))*cos(d*x + c) - a*b^2*cos(d*x + c) - b^3*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)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d^4))^(5/4
) - sqrt(2)*(a^4*b + 2*a^2*b^3 + b^5)*d^7*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)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d
^4))^(5/4) + (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)))/b^2) + 4*a*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/co
s(d*x + c))*cos(d*x + c)*sin(d*x + c) - sqrt(2)*(a^2*d*cos(d*x + c)^2 - a^2*d - (a^3*d^3*cos(d*x + c)^2 - a^3*
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((sqrt(2)*b^3*d*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)*cos(d*x + c) + (a^2*b^2 + b^4)*d^2
*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + a*b^2*cos(d*x + c) + b^3*sin(d*x + c))/cos(d*x + c)) + sqrt(2)*(a^2*
d*cos(d*x + c)^2 - a^2*d - (a^3*d^3*cos(d*x + c)^2 - a^3*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(-(sqrt(2)*b^3*d*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)*cos(d*x + c) - (a^2*b^2 + b^4)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) - a*b^2*cos(
d*x + c) - b^3*sin(d*x + c))/cos(d*x + c)) - 4*(b*cos(d*x + c)^2 - b)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*cos(d*x
+ c) + b*sin(d*x + c))/cos(d*x + c))/a))/(a^2*d*cos(d*x + c)^2 - a^2*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2/(a+b*tan(d*x+c))**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (d x + c\right )^{2}}{\sqrt{b \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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