\(\int \frac {(a+b \tan (c+d x))^{3/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) [852]

Optimal result

Integrand size = 25, antiderivative size = 286 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {(i a-b)^{3/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {\left (3 a^2-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{4 \sqrt {b} d}+\frac {(i a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {3 a \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}}+\frac {(a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {4326, 3651, 3728, 3736, 6857, 65, 223, 212, 95, 211, 214} \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\left (3 a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{4 \sqrt {b} d}+\frac {(-b+i a)^{3/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(b+i a)^{3/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(a+b \tan (c+d x))^{3/2}}{2 d \sqrt {\cot (c+d x)}}+\frac {3 a \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\cot (c+d x)}} \]

[In]

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

[Out]

((I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[T
an[c + d*x]])/d + ((3*a^2 - 8*b^2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c +
d*x]]*Sqrt[Tan[c + d*x]])/(4*Sqrt[b]*d) + ((I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a
+ b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (3*a*Sqrt[a + b*Tan[c + d*x]])/(4*d*Sqrt[Cot[c +
d*x]]) + (a + b*Tan[c + d*x])^(3/2)/(2*d*Sqrt[Cot[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 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 211

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

Rule 212

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

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 3651

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^n/(f*(m + n - 1))), x] + Dist[1/(m + n - 1), Int[(a +
b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a^2*c*(m + n - 1) - b*(b*c*(m - 1) + a*d*n) + (2*a*b
*c + a^2*d - b^2*d)*(m + n - 1)*Tan[e + f*x] + b*(b*c*n + a*d*(2*m + n - 2))*Tan[e + f*x]^2, x], x], x] /; Fre
eQ[{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] && GtQ[m, 1] && GtQ[n
, 0] && IntegerQ[2*n]

Rule 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3736

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] :> With[{ff = FreeFactors[Tan[e + f*x], x
]}, Dist[ff/f, Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^2)/(1 + ff^2*x^2)), x], x, Tan[
e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] &&
NeQ[c^2 + d^2, 0]

Rule 4326

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ
[u, x]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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

Mathematica [A] (verified)

Time = 3.80 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (\left (3 a^2-8 b^2\right ) \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) (a+b \tan (c+d x))+\sqrt {a} \sqrt {b} \sqrt {1+\frac {b \tan (c+d x)}{a}} \left (-4 \sqrt [4]{-1} \sqrt {-a+i b} (i a+b) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {a+b \tan (c+d x)}+4 (-1)^{3/4} (a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {a+b \tan (c+d x)}+\sqrt {\tan (c+d x)} \left (5 a^2+7 a b \tan (c+d x)+2 b^2 \tan ^2(c+d x)\right )\right )\right )}{4 \sqrt {a} \sqrt {b} d \sqrt {a+b \tan (c+d x)} \sqrt {1+\frac {b \tan (c+d x)}{a}}} \]

[In]

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

[Out]

(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((3*a^2 - 8*b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*(a + b*T
an[c + d*x]) + Sqrt[a]*Sqrt[b]*Sqrt[1 + (b*Tan[c + d*x])/a]*(-4*(-1)^(1/4)*Sqrt[-a + I*b]*(I*a + b)*ArcTan[((-
1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + 4*(-1)^(3/4)*
(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[
c + d*x]] + Sqrt[Tan[c + d*x]]*(5*a^2 + 7*a*b*Tan[c + d*x] + 2*b^2*Tan[c + d*x]^2))))/(4*Sqrt[a]*Sqrt[b]*d*Sqr
t[a + b*Tan[c + d*x]]*Sqrt[1 + (b*Tan[c + d*x])/a])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2274\) vs. \(2(232)=464\).

Time = 39.34 (sec) , antiderivative size = 2275, normalized size of antiderivative = 7.95

method	result	size
default	\(\text {Expression too large to display}\)	\(2275\)

[In]

int((a+b*tan(d*x+c))^(3/2)/cot(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/8/d*csc(d*x+c)*(2*2^(1/2)*b^(3/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*((sin(d*x+c)*cos(d*x+c)*a-cos(d*x+c)^2*b+b)/(co
s(d*x+c)+1)^2)^(1/2)*cos(d*x+c)*sin(d*x+c)+4*cos(d*x+c)^2*b^(5/2)*arctan((2^(1/2)*((sin(d*x+c)*cos(d*x+c)*a-co
s(d*x+c)^2*b+b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)-(b+(a^2+b^2)^(1/2))^(1/2)*cos(d*x+c)+(b+(a^2+b^2)^(1/2))^(1
/2))/(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))^(1/2))+4*cos(d*x+c)^2*b^(5/2)*arctan((2^(1/2)*((sin(d*x+c)*cos(d*x+c)
*a-cos(d*x+c)^2*b+b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)+(b+(a^2+b^2)^(1/2))^(1/2)*cos(d*x+c)-(b+(a^2+b^2)^(1/2
))^(1/2))/(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))^(1/2))+4*cos(d*x+c)^2*b^(3/2)*ln(-(-cos(d*x+c)*cot(d*x+c)*a+2*si
n(d*x+c)*(csc(d*x+c)*(cot(d*x+c)^2*a-2*cot(d*x+c)*csc(d*x+c)*a+csc(d*x+c)^2*a-2*b*csc(d*x+c)+2*cot(d*x+c)*b-a)
*(cos(d*x+c)-1))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+2*a*cot(d*x+c)+sin(d*x+c)*a-2*(a^2+b^2)^(1/2)*cos(d*x+c)-2*b*
cos(d*x+c)-csc(d*x+c)*a+2*(a^2+b^2)^(1/2)+2*b)/(cos(d*x+c)-1))*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^
(1/2)-4*cos(d*x+c)^2*b^(3/2)*ln((cos(d*x+c)*cot(d*x+c)*a-2*a*cot(d*x+c)+2*sin(d*x+c)*(csc(d*x+c)*(cot(d*x+c)^2
*a-2*cot(d*x+c)*csc(d*x+c)*a+csc(d*x+c)^2*a-2*b*csc(d*x+c)+2*cot(d*x+c)*b-a)*(cos(d*x+c)-1))^(1/2)*(b+(a^2+b^2
)^(1/2))^(1/2)+2*(a^2+b^2)^(1/2)*cos(d*x+c)+2*b*cos(d*x+c)-sin(d*x+c)*a+csc(d*x+c)*a-2*(a^2+b^2)^(1/2)-2*b)/(c
os(d*x+c)-1))*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)+2*((sin(d*x+c)*cos(d*x+c)*a-cos(d*x+c)^2*b+
b)/(cos(d*x+c)+1)^2)^(1/2)*2^(1/2)*b^(3/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)+5*((sin(d*x+c)*cos(d*x+c)*a-c
os(d*x+c)^2*b+b)/(cos(d*x+c)+1)^2)^(1/2)*2^(1/2)*b^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*a*cos(d*x+c)^2+3*cos(d*x+c
)^2*2^(1/2)*arctanh(((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(csc(d*x+c)+cot(d*x+c))/b^
(1/2))*(-b+(a^2+b^2)^(1/2))^(1/2)*a^2-8*cos(d*x+c)^2*2^(1/2)*arctanh(((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c)/(
cos(d*x+c)+1)^2)^(1/2)*(csc(d*x+c)+cot(d*x+c))/b^(1/2))*(-b+(a^2+b^2)^(1/2))^(1/2)*b^2-4*cos(d*x+c)^2*(a^2+b^2
)^(1/2)*b^(3/2)*arctan((2^(1/2)*((sin(d*x+c)*cos(d*x+c)*a-cos(d*x+c)^2*b+b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)
-(b+(a^2+b^2)^(1/2))^(1/2)*cos(d*x+c)+(b+(a^2+b^2)^(1/2))^(1/2))/(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))^(1/2))-4*
cos(d*x+c)^2*(a^2+b^2)^(1/2)*b^(3/2)*arctan((2^(1/2)*((sin(d*x+c)*cos(d*x+c)*a-cos(d*x+c)^2*b+b)/(cos(d*x+c)+1
)^2)^(1/2)*sin(d*x+c)+(b+(a^2+b^2)^(1/2))^(1/2)*cos(d*x+c)-(b+(a^2+b^2)^(1/2))^(1/2))/(cos(d*x+c)-1)/(-b+(a^2+
b^2)^(1/2))^(1/2))-2*cos(d*x+c)^2*(a^2+b^2)^(1/2)*b^(1/2)*ln(-(-cos(d*x+c)*cot(d*x+c)*a+2*sin(d*x+c)*(csc(d*x+
c)*(cot(d*x+c)^2*a-2*cot(d*x+c)*csc(d*x+c)*a+csc(d*x+c)^2*a-2*b*csc(d*x+c)+2*cot(d*x+c)*b-a)*(cos(d*x+c)-1))^(
1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+2*a*cot(d*x+c)+sin(d*x+c)*a-2*(a^2+b^2)^(1/2)*cos(d*x+c)-2*b*cos(d*x+c)-csc(d*x
+c)*a+2*(a^2+b^2)^(1/2)+2*b)/(cos(d*x+c)-1))*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)+2*cos(d*x+c)
^2*(a^2+b^2)^(1/2)*b^(1/2)*ln((cos(d*x+c)*cot(d*x+c)*a-2*a*cot(d*x+c)+2*sin(d*x+c)*(csc(d*x+c)*(cot(d*x+c)^2*a
-2*cot(d*x+c)*csc(d*x+c)*a+csc(d*x+c)^2*a-2*b*csc(d*x+c)+2*cot(d*x+c)*b-a)*(cos(d*x+c)-1))^(1/2)*(b+(a^2+b^2)^
(1/2))^(1/2)+2*(a^2+b^2)^(1/2)*cos(d*x+c)+2*b*cos(d*x+c)-sin(d*x+c)*a+csc(d*x+c)*a-2*(a^2+b^2)^(1/2)-2*b)/(cos
(d*x+c)-1))*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)+5*((sin(d*x+c)*cos(d*x+c)*a-cos(d*x+c)^2*b+b)
/(cos(d*x+c)+1)^2)^(1/2)*2^(1/2)*b^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*a*cos(d*x+c)-4*cos(d*x+c)^2*b^(1/2)*arctan
((2^(1/2)*((sin(d*x+c)*cos(d*x+c)*a-cos(d*x+c)^2*b+b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)-(b+(a^2+b^2)^(1/2))^(
1/2)*cos(d*x+c)+(b+(a^2+b^2)^(1/2))^(1/2))/(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))^(1/2))*a^2-4*cos(d*x+c)^2*b^(1/
2)*arctan((2^(1/2)*((sin(d*x+c)*cos(d*x+c)*a-cos(d*x+c)^2*b+b)/(cos(d*x+c)+1)^2)^(1/2)*sin(d*x+c)+(b+(a^2+b^2)
^(1/2))^(1/2)*cos(d*x+c)-(b+(a^2+b^2)^(1/2))^(1/2))/(cos(d*x+c)-1)/(-b+(a^2+b^2)^(1/2))^(1/2))*a^2)*(a+b*tan(d
*x+c))^(1/2)/(cos(d*x+c)+1)/((sin(d*x+c)*cos(d*x+c)*a-cos(d*x+c)^2*b+b)/(cos(d*x+c)+1)^2)^(1/2)/cot(d*x+c)^(3/
2)*2^(1/2)/b^(1/2)/(-b+(a^2+b^2)^(1/2))^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3781 vs. \(2 (228) = 456\).

Time = 1.13 (sec) , antiderivative size = 7599, normalized size of antiderivative = 26.57 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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