3.6.23 \(\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} \, dx\) [523]

Optimal. Leaf size=138 \[ -\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b^2 \sqrt {a+b \tan (c+d x)}}{d} \]

[Out]

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

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Rubi [A]
time = 0.34, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3647, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 b^2 \sqrt {a+b \tan (c+d x)}}{d}+\frac {(a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*(a + b*Tan[c + d*x])^(5/2),x]

[Out]

(-2*a^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d + ((a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/S
qrt[a - I*b]])/d + ((a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*b^2*Sqrt[a + b*Tan
[c + d*x]])/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 3647

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 - 2)*((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 - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*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]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !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 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 (c+d x) (a+b \tan (c+d x))^{5/2} \, dx &=\frac {2 b^2 \sqrt {a+b \tan (c+d x)}}{d}+2 \int \frac {\cot (c+d x) \left (\frac {a^3}{2}+\frac {1}{2} b \left (3 a^2-b^2\right ) \tan (c+d x)+\frac {3}{2} a b^2 \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 b^2 \sqrt {a+b \tan (c+d x)}}{d}+2 \int \frac {\frac {1}{2} b \left (3 a^2-b^2\right )-\frac {1}{2} a \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+a^3 \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 b^2 \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} (i a-b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {1}{2} (i a+b)^3 \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {a^3 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {2 b^2 \sqrt {a+b \tan (c+d x)}}{d}-\frac {(a-i b)^3 \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)^3 \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 (2 a^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d}\\ &=-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 b^2 \sqrt {a+b \tan (c+d x)}}{d}-\frac {(i a-b)^3 \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 a+b)^3 \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 {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b^2 \sqrt {a+b \tan (c+d x)}}{d}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 220, normalized size = 1.59 \begin {gather*} \frac {-2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+(a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+a^2 \sqrt {a+i b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 i a \sqrt {a+i b} b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )-\sqrt {a+i b} b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 b^2 \sqrt {a+b \tan (c+d x)}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^(5/2),x]

[Out]

(-2*a^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + (a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[
a - I*b]] + a^2*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (2*I)*a*Sqrt[a + I*b]*b*ArcTan
h[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - Sqrt[a + I*b]*b^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]
+ 2*b^2*Sqrt[a + b*Tan[c + d*x]])/d

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.46, size = 28373, normalized size = 205.60

method	result	size
default	\(\text {Expression too large to display}\)	\(28373\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)*(a+b*tan(d*x+c))^(5/2),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)*(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^(5/2)*cot(d*x + c), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6594 vs. \(2 (110) = 220\).
time = 9.11, size = 13263, normalized size = 96.11 \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)*(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

[1/4*(4*sqrt(2)*d^5*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^5 - 10*a^3*b^2 +
5*a*b^4)*d^2*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4))/(25*a^8*b^2 - 100*a^6*
b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)^
(3/4)*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4)*arctan(((5*a^18 + 25*a^16*b^2 + 3
6*a^14*b^4 - 28*a^12*b^6 - 154*a^10*b^8 - 210*a^8*b^10 - 140*a^6*b^12 - 44*a^4*b^14 - 3*a^2*b^16 + b^18)*d^4*s
qrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*
a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + (5*a^23 + 35*a^21*b^2 + 91*a^19*b^4 + 69*a^17*b^6 - 174*a^15*b^8 - 546*a^1
3*b^10 - 714*a^11*b^12 - 534*a^9*b^14 - 231*a^7*b^16 - 49*a^5*b^18 - a^3*b^20 + a*b^22)*d^2*sqrt((25*a^8*b^2 -
100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + sqrt(2)*((5*a^10 - 5*a^8*b^2 - 14*a^6*b^4 + 6*a^4*b^6 +
9*a^2*b^8 - b^10)*d^7*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*sqrt((25*a^8*
b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + (5*a^15 - 5*a^13*b^2 - 39*a^11*b^4 - 9*a^9*b^6 + 7
9*a^7*b^8 + 81*a^5*b^10 + 19*a^3*b^12 - 3*a*b^14)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^
8 + b^10)/d^4))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^5 - 10*a^3*b^2 + 5*a*
b^4)*d^2*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4))/(25*a^8*b^2 - 100*a^6*b^4
+ 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^10 + 5*a^8*b^2 +
10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)^(3/4) - sqrt(2)*((a^2 - b^2)*d^7*sqrt((a^10 + 5*a^8*b^2 + 10*
a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10
)/d^4) + (a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 +
b^10)/d^4))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^5 - 10*a^3*b^2 + 5*a*b^4
)*d^2*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4))/(25*a^8*b^2 - 100*a^6*b^4 + 1
10*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt(((25*a^14 - 25*a^12*b^2 - 115*a^10*b^4 + 35*a^8*b^6 + 171*a^6*b^8 + 53*a
^4*b^10 - 17*a^2*b^12 + b^14)*d^2*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*co
s(d*x + c) + sqrt(2)*((25*a^11 - 175*a^9*b^2 + 410*a^7*b^4 - 350*a^5*b^6 + 61*a^3*b^8 - 3*a*b^10)*d^3*sqrt((a^
10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*cos(d*x + c) + (25*a^16 - 50*a^14*b^2 - 90*a
^12*b^4 + 150*a^10*b^6 + 136*a^8*b^8 - 118*a^6*b^10 - 70*a^4*b^12 + 18*a^2*b^14 - b^16)*d*cos(d*x + c))*sqrt((
a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^5 - 10*a^3*b^2 + 5*a*b^4)*d^2*sqrt((a^10 +
5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2
*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^
6 + 5*a^2*b^8 + b^10)/d^4)^(1/4) + (25*a^19 + 25*a^17*b^2 - 140*a^15*b^4 - 220*a^13*b^6 + 126*a^11*b^8 + 430*a
^9*b^10 + 260*a^7*b^12 + 20*a^5*b^14 - 15*a^3*b^16 + a*b^18)*cos(d*x + c) + (25*a^18*b + 25*a^16*b^3 - 140*a^1
4*b^5 - 220*a^12*b^7 + 126*a^10*b^9 + 430*a^8*b^11 + 260*a^6*b^13 + 20*a^4*b^15 - 15*a^2*b^17 + b^19)*sin(d*x
+ c))/((a^2 + b^2)*cos(d*x + c)))*((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)^(3/4))
/(25*a^26*b^2 + 125*a^24*b^4 + 110*a^22*b^6 - 530*a^20*b^8 - 1469*a^18*b^10 - 921*a^16*b^12 + 1716*a^14*b^14 +
3924*a^12*b^16 + 3471*a^10*b^18 + 1531*a^8*b^20 + 254*a^6*b^22 - 34*a^4*b^24 - 11*a^2*b^26 + b^28)) + 4*sqrt(
2)*d^5*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^5 - 10*a^3*b^2 + 5*a*b^4)*d^2*
sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4
*b^6 - 20*a^2*b^8 + b^10))*((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)^(3/4)*sqrt((2
5*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4)*arctan(-((5*a^18 + 25*a^16*b^2 + 36*a^14*b^4 -
28*a^12*b^6 - 154*a^10*b^8 - 210*a^8*b^10 - 140*a^6*b^12 - 44*a^4*b^14 - 3*a^2*b^16 + b^18)*d^4*sqrt((a^10 +
5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20
*a^2*b^8 + b^10)/d^4) + (5*a^23 + 35*a^21*b^2 + 91*a^19*b^4 + 69*a^17*b^6 - 174*a^15*b^8 - 546*a^13*b^10 - 714
*a^11*b^12 - 534*a^9*b^14 - 231*a^7*b^16 - 49*a^5*b^18 - a^3*b^20 + a*b^22)*d^2*sqrt((25*a^8*b^2 - 100*a^6*b^4
+ 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - sqrt(2)*((5*a^10 - 5*a^8*b^2 - 14*a^6*b^4 + 6*a^4*b^6 + 9*a^2*b^8 -
b^10)*d^7*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*sqrt((25*a^8*b^2 - 100*a^
6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + (5*a^15 - 5*a^13*b^2 - 39*a^11*b^4 - 9*a^9*b^6 + 79*a^7*b^8 +
81*a^5*b^10 + 19*a^3*b^12 - 3*a*b^14)*d^5*sqrt(...

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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 {5}{2}} \cot {\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+b*tan(d*x+c))**(5/2),x)

[Out]

Integral((a + b*tan(c + d*x))**(5/2)*cot(c + d*x), 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)*(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 10.74, size = 2500, normalized size = 18.12 \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)*(a + b*tan(c + d*x))^(5/2),x)

[Out]

log(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*((((
(-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d
^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a
^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*((128*a*b^8*(3*a^4 + b^4
+ 4*a^2*b^2))/d - 128*b^8*(((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2
*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2)))/2 + (64*a*b^8*(a + b*tan(c + d*x))^(1/2)*(9*a^6
+ 19*b^6 - 5*a^2*b^4 - 51*a^4*b^2))/d^2))/2 - (96*a^2*b^8*(a^8 + b^8 + 16*a^2*b^6 - 10*a^4*b^4 - 24*a^6*b^2))/
d^3))/2 - (32*b^8*(a + b*tan(c + d*x))^(1/2)*(3*a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 18*a^6*b^6 + 45*a^8*b^
4 - 24*a^10*b^2))/d^4))/2 + (32*a^3*b^10*(a^2 + b^2)^3*(6*a^4 + b^4 + 3*a^2*b^2))/d^5)*((20*a^2*b^8*d^4 - b^10
*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2)/(4*d^4) + a^5/(4*d^2) + (5*a*b^4)/(4*d^2) - (
5*a^3*b^2)/(2*d^2))^(1/2) - log(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*
a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2
*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d
^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/
2)*((128*a*b^8*(3*a^4 + b^4 + 4*a^2*b^2))/d + 128*b^8*(((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^
2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2)))/2 - (64*a*b^8*(a + b
*tan(c + d*x))^(1/2)*(9*a^6 + 19*b^6 - 5*a^2*b^4 - 51*a^4*b^2))/d^2))/2 - (96*a^2*b^8*(a^8 + b^8 + 16*a^2*b^6
- 10*a^4*b^4 - 24*a^6*b^2))/d^3))/2 + (32*b^8*(a + b*tan(c + d*x))^(1/2)*(3*a^12 + b^12 + 6*a^2*b^10 + 15*a^4*
b^8 + 18*a^6*b^6 + 45*a^8*b^4 - 24*a^10*b^2))/d^4))/2 + (32*a^3*b^10*(a^2 + b^2)^3*(6*a^4 + b^4 + 3*a^2*b^2))/
d^5)*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*
b^4*d^2 - 10*a^3*b^2*d^2)/(4*d^4))^(1/2) - log(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5
*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4
*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 +
10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^
3*b^2*d^2)/d^4)^(1/2)*((128*a*b^8*(3*a^4 + b^4 + 4*a^2*b^2))/d + 128*b^8*(-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^
2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2)))/
2 - (64*a*b^8*(a + b*tan(c + d*x))^(1/2)*(9*a^6 + 19*b^6 - 5*a^2*b^4 - 51*a^4*b^2))/d^2))/2 - (96*a^2*b^8*(a^8
+ b^8 + 16*a^2*b^6 - 10*a^4*b^4 - 24*a^6*b^2))/d^3))/2 + (32*b^8*(a + b*tan(c + d*x))^(1/2)*(3*a^12 + b^12 +
6*a^2*b^10 + 15*a^4*b^8 + 18*a^6*b^6 + 45*a^8*b^4 - 24*a^10*b^2))/d^4))/2 + (32*a^3*b^10*(a^2 + b^2)^3*(6*a^4
+ b^4 + 3*a^2*b^2))/d^5)*(-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(
1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(4*d^4))^(1/2) + log(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2
)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2
) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^
5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2
- 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*((128*a*b^8*(3*a^4 + b^4 + 4*a^2*b^2))/d - 128*b^8*(-((-b^2*d^4*(5*
a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*t
an(c + d*x))^(1/2)))/2 + (64*a*b^8*(a + b*tan(c + d*x))^(1/2)*(9*a^6 + 19*b^6 - 5*a^2*b^4 - 51*a^4*b^2))/d^2))
/2 - (96*a^2*b^8*(a^8 + b^8 + 16*a^2*b^6 - 10*a^4*b^4 - 24*a^6*b^2))/d^3))/2 - (32*b^8*(a + b*tan(c + d*x))^(1
/2)*(3*a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 18*a^6*b^6 + 45*a^8*b^4 - 24*a^10*b^2))/d^4))/2 + (32*a^3*b^10*
(a^2 + b^2)^3*(6*a^4 + b^4 + 3*a^2*b^2))/d^5)*(a^5/(4*d^2) - (20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 10
0*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2)/(4*d^4) + (5*a*b^4)/(4*d^2) - (5*a^3*b^2)/(2*d^2))^(1/2) + (2*b^2*(a + b
*tan(c + d*x))^(1/2))/d + (atan((b^20*(a^5)^(1/2)*(a + b*tan(c + d*x))^(1/2)*64i)/(64*a^3*b^20 + 384*a^5*b^18
+ 960*a^7*b^16 - 1280*a^9*b^14 + 3520*a^11*b^12 + 6016*a^13*b^10 + 576*a^15*b^8) + (a^2*b^18*(a^5)^(1/2)*(a +
b*tan(c + d*x))^(1/2)*384i)/(64*a^3*b^20 + 384*a^5*b^18 + 960*a^7*b^16 - 1280*a^9*b^14 + 3520*a^11*b^12 + 6016
*a^13*b^10 + 576*a^15*b^8) + (a^4*b^16*(a^5)^(1/2)*(a + b*tan(c + d*x))^(1/2)*960i)/(64*a^3*b^20 + 384*a^5*b^1
8 + 960*a^7*b^16 - 1280*a^9*b^14 + 3520*a^11*b^...

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