\(\int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) [847]

Optimal result

Integrand size = 25, antiderivative size = 264 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=-\frac {i (i a-b)^{3/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {i (i a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (5 a^2-b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{5 a d}-\frac {4 b \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {2 a \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d} \]

[Out]

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

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4326, 3648, 3730, 3697, 3696, 95, 209, 212} \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {2 \left (5 a^2-b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{5 a d}-\frac {i (-b+i a)^{3/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {i (b+i a)^{3/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {4 b \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d} \]

[In]

Int[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(3/2),x]

[Out]

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

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 209

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

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 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 3696

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[A^2/f, Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e
+ f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 +
B^2, 0]

Rule 3697

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(A + I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 -
I*Tan[e + f*x]), x], x] + Dist[(A - I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 + I*Tan[e +
f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2
+ B^2, 0]

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 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]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {7}{2}}(c+d x)} \, dx \\ & = -\frac {2 a \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {1}{5} \left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-3 a b+\frac {5}{2} \left (a^2-b^2\right ) \tan (c+d x)+2 a b \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {4 b \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {2 a \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}+\frac {\left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {3}{4} a \left (5 a^2-b^2\right )-\frac {15}{2} a^2 b \tan (c+d x)-3 a b^2 \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{15 a} \\ & = \frac {2 \left (5 a^2-b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{5 a d}-\frac {4 b \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {2 a \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {15 a^3 b}{4}-\frac {15}{8} a^2 \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{15 a^2} \\ & = \frac {2 \left (5 a^2-b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{5 a d}-\frac {4 b \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {2 a \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {1}{2} \left (i (a-i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} \left (i (a+i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {2 \left (5 a^2-b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{5 a d}-\frac {4 b \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {2 a \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 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}{(1-i x) \sqrt {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}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {2 \left (5 a^2-b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{5 a d}-\frac {4 b \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {2 a \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 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}{1-(i a+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}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \\ & = -\frac {i (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 {i (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 {2 \left (5 a^2-b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{5 a d}-\frac {4 b \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {2 a \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.65 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.83 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {\cot ^{\frac {5}{2}}(c+d x) \left (-5 \sqrt [4]{-1} a (-a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {5}{2}}(c+d x)+5 \sqrt [4]{-1} a (a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {5}{2}}(c+d x)+2 \sqrt {a+b \tan (c+d x)} \left (-a^2-2 a b \tan (c+d x)+\left (5 a^2-b^2\right ) \tan ^2(c+d x)\right )\right )}{5 a d} \]

[In]

Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(3/2),x]

[Out]

(Cot[c + d*x]^(5/2)*(-5*(-1)^(1/4)*a*(-a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sq
rt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(5/2) + 5*(-1)^(1/4)*a*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*S
qrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(5/2) + 2*Sqrt[a + b*Tan[c + d*x]]*(-a^2 - 2*a*b*Tan
[c + d*x] + (5*a^2 - b^2)*Tan[c + d*x]^2)))/(5*a*d)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2794\) vs. \(2(216)=432\).

Time = 39.87 (sec) , antiderivative size = 2795, normalized size of antiderivative = 10.59

method	result	size
default	\(\text {Expression too large to display}\)	\(2795\)

[In]

int(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/20/d*csc(d*x+c)*(-1/(1-cos(d*x+c))*(csc(d*x+c)*(1-cos(d*x+c))^2-sin(d*x+c)))^(7/2)*(1-cos(d*x+c))*((csc(d*x+
c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)/(csc(d*x+c)^2*(1-cos(d*x+c))^2-1))^(1/2)*(csc(d*x+c)^4*
a^2*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(1-cos(
d*x+c))^4*(-b+(a^2+b^2)^(1/2))^(1/2)-5*csc(d*x+c)^3*ln(-1/(1-cos(d*x+c))*(csc(d*x+c)*a*(1-cos(d*x+c))^2+2*sin(
d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(
a^2+b^2)^(1/2))^(1/2)-2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*b*(1-cos(d*x+c))-sin(d*x+c)*a))*(a^2+b^2)^(1/2)*(b+(a
^2+b^2)^(1/2))^(1/2)*b*(1-cos(d*x+c))^3*(-b+(a^2+b^2)^(1/2))^(1/2)+5*csc(d*x+c)^3*ln(1/(1-cos(d*x+c))*(-csc(d*
x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+
c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+2*b*(1-cos(d*x+c))+sin(d*
x+c)*a))*(a^2+b^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*b*(1-cos(d*x+c))^3*(-b+(a^2+b^2)^(1/2))^(1/2)-5*csc(d*x+c)^
3*a^2*ln(-1/(1-cos(d*x+c))*(csc(d*x+c)*a*(1-cos(d*x+c))^2+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x
+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-2*(a^2+b^2)^(1/2)*(1-cos
(d*x+c))-2*b*(1-cos(d*x+c))-sin(d*x+c)*a))*(b+(a^2+b^2)^(1/2))^(1/2)*(1-cos(d*x+c))^3*(-b+(a^2+b^2)^(1/2))^(1/
2)+5*csc(d*x+c)^3*ln(-1/(1-cos(d*x+c))*(csc(d*x+c)*a*(1-cos(d*x+c))^2+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*
a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-2*(a^2+b^2)^
(1/2)*(1-cos(d*x+c))-2*b*(1-cos(d*x+c))-sin(d*x+c)*a))*(b+(a^2+b^2)^(1/2))^(1/2)*b^2*(1-cos(d*x+c))^3*(-b+(a^2
+b^2)^(1/2))^(1/2)+5*csc(d*x+c)^3*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*ln(1/(1-cos(d*x+c))*(-c
sc(d*x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos
(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+2*b*(1-cos(d*x+c))+s
in(d*x+c)*a))*a^2*(1-cos(d*x+c))^3-5*csc(d*x+c)^3*ln(1/(1-cos(d*x+c))*(-csc(d*x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b
^2)^(1/2)*(1-cos(d*x+c))+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c)
)-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+2*b*(1-cos(d*x+c))+sin(d*x+c)*a))*(b+(a^2+b^2)^(1/2))^(1/
2)*b^2*(1-cos(d*x+c))^3*(-b+(a^2+b^2)^(1/2))^(1/2)-4*csc(d*x+c)^3*a*b*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+
c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(1-cos(d*x+c))^3+10*csc(
d*x+c)^3*(a^2+b^2)^(1/2)*arctan((-(b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(-csc(d*x+c)*(csc(d*x+c)^2
*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(-b+(a^2+b
^2)^(1/2))^(1/2))*a^2*(1-cos(d*x+c))^3+10*csc(d*x+c)^3*arctan(((b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c
))+(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2))/(1-cos(
d*x+c))*sin(d*x+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*a^2*(1-cos(d*x+c))^3*(a^2+b^2)^(1/2)-20*csc(d*x+c)^3*arctan((-(
b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)
-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(-b+(a^2+b^2)^(1/2))^(1/2))*a^2*b*(1-cos(d*x+
c))^3-20*csc(d*x+c)^3*arctan(((b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+(-csc(d*x+c)*(csc(d*x+c)^2*a*(
1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(-b+(a^2+b^2)^
(1/2))^(1/2))*a^2*(1-cos(d*x+c))^3*b-22*csc(d*x+c)^2*a^2*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(cs
c(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(1-cos(d*x+c))^2*(-b+(a^2+b^2)^(1/2))^(1/2)+4*csc(d*x+c)^2*b^2*(
-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2
)^(1/2))^(1/2)*(1-cos(d*x+c))^2+4*a*b*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c)
)-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+a^2*(-csc(d*x+c)*(csc(d*x+c)^2*a
*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2))/(csc(d*x+c)
^2*(1-cos(d*x+c))^2-1)^3/(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d
*x+c)))^(1/2)*2^(1/2)/a/(-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. 3666 vs. \(2 (210) = 420\).

Time = 0.60 (sec) , antiderivative size = 3666, normalized size of antiderivative = 13.89 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

-1/40*(5*a*d*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log((((a^6 + a^4*b^2 - 1
2*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^
4)*d - 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*
a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) + 2*((a^7 - 5*a^3*b^4
- 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x +
c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x +
c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c)^2 + 5*a*d*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6
- 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(-(((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3
*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d - 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (
a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(
a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) + 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3
- 6*a^2*b^5)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x +
c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1
))*tan(d*x + c)^2 - 5*a*d*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log((((a^6
+ a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^
2 + 12*a^2*b^4)*d - 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6
*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) - 2*((a^7
- 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*
d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqr
t(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c)^2 - 5*a*d*sqrt((3*a^2*b - b^3 + d
^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(-(((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*
a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d - 2*(a^3*d^3*tan(d*x + c) + 2*
a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3
+ d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) - 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6
*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d
^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d
*x + c)^2 + 1))*tan(d*x + c)^2 + 5*a*d*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2
)*log((((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a
^6 - 7*a^4*b^2 + 12*a^2*b^4)*d + 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*s
qrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d
^2) + 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) - (2*(a^3*
b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b
^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c)^2 + 5*a*d*sqrt((3*a^
2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(-(((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x +
c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d + 2*(a^3*d^3*tan(
d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3
*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) + 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x +
c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) - (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b
^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x
+ c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c)^2 - 5*a*d*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b
^4)/d^4))/d^2)*log((((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(
d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d + 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(
d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2
*b^4)/d^4))/d^2) - 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x +
c) - (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*
b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c)^2 - 5*a
*d*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(-(((a^6 + a^4*b^2 - 12*a^2*b^4
)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d + 2*
(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/
d^4))*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) - 2*((a^7 - 5*a^3*b^4 - 12*a*b^
6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) - (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (
a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/
sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c)^2 + 16*(2*a*b*tan(d*x + c) - (5*a^2 - b^2)*tan(d*x + c)
^2 + a^2)*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(a*d*tan(d*x + c)^2)

Sympy [F(-1)]

Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)**(7/2)*(a+b*tan(d*x+c))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{\frac {7}{2}} \,d x } \]

[In]

integrate(cot(d*x+c)^(7/2)*(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)^(7/2), x)

Giac [F(-1)]

Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2} \,d x \]

[In]

int(cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^(3/2),x)

[Out]

int(cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^(3/2), x)