3.6.51 \(\int \frac {1}{(a+b \tan (c+d x))^{5/2}} \, dx\) [551]

Optimal. Leaf size=152 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {2 b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a b}{\left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3564, 3610, 3620, 3618, 65, 214} \begin {gather*} -\frac {4 a b}{d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}-\frac {2 b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{5/2}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x])^(-5/2),x]

[Out]

((-I)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x
]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (2*b)/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (4*a*b)/((a^2 +
b^2)^2*d*Sqrt[a + b*Tan[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 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 3564

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*
(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3610

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

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]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.10, size = 108, normalized size = 0.71 \begin {gather*} \frac {i (a+i b) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {a+b \tan (c+d x)}{a-i b}\right )+(-i a-b) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {a+b \tan (c+d x)}{a+i b}\right )}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*(a + I*b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)] + ((-I)*a - b)*Hypergeometric2F1
[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)])/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(954\) vs. \(2(128)=256\).
time = 0.11, size = 955, normalized size = 6.28

method	result	size
derivativedivides	\(\frac {2 b \left (\frac {\frac {\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{5}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3} b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{4}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (6 a^{4} b^{2}+4 a^{2} b^{4}-2 b^{6}-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{5}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3} b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 b^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{5}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3} b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{4}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-6 a^{4} b^{2}-4 a^{2} b^{4}+2 b^{6}+\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{5}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3} b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 b^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2 a}{\left (a^{2}+b^{2}\right )^{2} \sqrt {a +b \tan \left (d x +c \right )}}\right )}{d}\)	\(955\)
default	\(\frac {2 b \left (\frac {\frac {\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{5}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3} b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{4}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (6 a^{4} b^{2}+4 a^{2} b^{4}-2 b^{6}-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{5}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3} b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 b^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{5}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3} b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{4}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-6 a^{4} b^{2}-4 a^{2} b^{4}+2 b^{6}+\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{4}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{5}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3} b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 b^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2 a}{\left (a^{2}+b^{2}\right )^{2} \sqrt {a +b \tan \left (d x +c \right )}}\right )}{d}\)	\(955\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/d*b*(1/(a^2+b^2)^2*(1/4/b^2/(a^2+b^2)^(3/2)*(1/2*((2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^4-(2*(a^2+
b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*b^4-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^5+2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^
3*b^2+3*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^4)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^
(1/2)+(a^2+b^2)^(1/2))+2*(6*a^4*b^2+4*a^2*b^4-2*b^6-1/2*((2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^4-(2*
(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*b^4-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^5+2*(2*(a^2+b^2)^(1/2)+2*a)^(1/
2)*a^3*b^2+3*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^4)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)
*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)))+1/4/b^2/(a^2+
b^2)^(3/2)*(-1/2*((2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^4-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1
/2)*b^4-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^5+2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3*b^2+3*(2*(a^2+b^2)^(1/2)+2*a)^(1
/2)*a*b^4)*ln(-b*tan(d*x+c)-a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-(a^2+b^2)^(1/2))+2*(-6*a^4*
b^2-4*a^2*b^4+2*b^6+1/2*((2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^4-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+
b^2)^(1/2)*b^4-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^5+2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3*b^2+3*(2*(a^2+b^2)^(1/2)+
2*a)^(1/2)*a*b^4)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((-2*(a+b*tan(d*x+c))^(1/
2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))))-1/3/(a^2+b^2)/(a+b*tan(d*x+c))^(3/2)-2/(a^2
+b^2)^2*a/(a+b*tan(d*x+c))^(1/2))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 9767 vs. \(2 (122) = 244\).
time = 2.03, size = 9767, normalized size = 64.26 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/12*(12*sqrt(2)*((a^18 + a^16*b^2 - 20*a^14*b^4 - 84*a^12*b^6 - 154*a^10*b^8 - 154*a^8*b^10 - 84*a^6*b^12 - 2
0*a^4*b^14 + a^2*b^16 + b^18)*d^5*cos(d*x + c)^4 + 2*(3*a^16*b^2 + 20*a^14*b^4 + 56*a^12*b^6 + 84*a^10*b^8 + 7
0*a^8*b^10 + 28*a^6*b^12 - 4*a^2*b^16 - b^18)*d^5*cos(d*x + c)^2 + (a^14*b^4 + 7*a^12*b^6 + 21*a^10*b^8 + 35*a
^8*b^10 + 35*a^6*b^12 + 21*a^4*b^14 + 7*a^2*b^16 + b^18)*d^5 + 4*((a^17*b + 6*a^15*b^3 + 14*a^13*b^5 + 14*a^11
*b^7 - 14*a^7*b^11 - 14*a^5*b^13 - 6*a^3*b^15 - a*b^17)*d^5*cos(d*x + c)^3 + (a^15*b^3 + 7*a^13*b^5 + 21*a^11*
b^7 + 35*a^9*b^9 + 35*a^7*b^11 + 21*a^5*b^13 + 7*a^3*b^15 + a*b^17)*d^5*cos(d*x + c))*sin(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^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45
*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((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((25*a^8*b^2 - 100*a^6*b^
4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^
10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*(1/((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)*arctan(((5*a^20 + 30*a^18*b^2 + 61*a^16*b^4 + 8*a^14*b^6 - 1
82*a^12*b^8 - 364*a^10*b^10 - 350*a^8*b^12 - 184*a^6*b^14 - 47*a^4*b^16 - 2*a^2*b^18 + b^20)*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)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a
^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((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)) + (5*a^15 + 15*a^13*b^2 + a^11*b^4 - 45*a^9*b^
6 - 65*a^7*b^8 - 35*a^5*b^10 - 5*a^3*b^12 + a*b^14)*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)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 +
120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) - sqrt(2)*((3*a^22 + 29*a^20*b^2 + 125*a^18*b^4 + 315*a
^16*b^6 + 510*a^14*b^8 + 546*a^12*b^10 + 378*a^10*b^12 + 150*a^8*b^14 + 15*a^6*b^16 - 15*a^4*b^18 - 7*a^2*b^20
- b^22)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*
b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 +
b^20)*d^4))*sqrt(1/((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)) + 2*(a^17 + 8*a^15*b
^2 + 28*a^13*b^4 + 56*a^11*b^6 + 70*a^9*b^8 + 56*a^7*b^10 + 28*a^5*b^12 + 8*a^3*b^14 + a*b^16)*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)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210
*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*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^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*
a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((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(((25*a^14*b^2 - 25*a^12*b
^4 - 115*a^10*b^6 + 35*a^8*b^8 + 171*a^6*b^10 + 53*a^4*b^12 - 17*a^2*b^14 + b^16)*d^2*sqrt(1/((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) + sqrt(2)*(2*(25*a^15*b^3 - 25*a^13*b^5 - 1
15*a^11*b^7 + 35*a^9*b^9 + 171*a^7*b^11 + 53*a^5*b^13 - 17*a^3*b^15 + a*b^17)*d^3*sqrt(1/((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) + (75*a^10*b^3 - 325*a^8*b^5 + 430*a^6*b^7 - 17
0*a^4*b^9 + 23*a^2*b^11 - b^13)*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^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sq
rt(1/((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))*(1/((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^9*b^2 - 100*a^7*b^4 + 110*a^5*b^6 - 20*a^3*b^8 + a
*b^10)*cos(d*x + c) + (25*a^8*b^3 - 100*a^6*b^5 + 110*a^4*b^7 - 20*a^2*b^9 + b^11)*sin(d*x + c))/cos(d*x + c))
*(1/((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)*((15*a^26*b + 115*a
^24*b^3 + 338*a^22*b^5 + 354*a^20*b^7 - 475*a^18*b^9 - 2055*a^16*b^11 - 3060*a^14*b^13 - 2484*a^12*b^15 - 1047
*a^10*b^17 - 75*a^8*b^19 + 130*a^6*b^21 + 50*a^4*b^23 + 3*a^2*b^25 - b^27)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4
+ 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10
*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((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)) + 2*(5*a^21*b + 30*a^19*b^3 + 61*a^17*b^5 + 8*a^15*b^7 - 182*a^1
3*b^9 - 364*a^11*b^11 - 350*a^9*b^13 - 184*a^7*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))**(5/2),x)

[Out]

Integral((a + b*tan(c + d*x))**(-5/2), 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(1/(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 8.86, size = 2500, normalized size = 16.45 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*tan(c + d*x))^(5/2),x)

[Out]

(log(16*a*b^15*d^2 - ((-1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^
2))^(1/2)*(((-1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*
(896*a^6*b^15*d^4 - 160*a^2*b^19*d^4 - 128*a^4*b^17*d^4 - 32*b^21*d^4 + 3136*a^8*b^13*d^4 + 4928*a^10*b^11*d^4
+ 4480*a^12*b^9*d^4 + 2432*a^14*b^7*d^4 + 736*a^16*b^5*d^4 + 96*a^18*b^3*d^4 + ((-1/(a^5*d^2 - b^5*d^2*1i + 5
*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5
+ 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440
*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/2))/2 - (a + b*t
an(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3
+ 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3)))/2 + 96*a^3*b^13*d^2 + 240*a^5*b^11*d^2 + 320*a^7*b^9*d^2 + 240*a^9*b^
7*d^2 + 96*a^11*b^5*d^2 + 16*a^13*b^3*d^2)*(-1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^
2*10i - 10*a^3*b^2*d^2))^(1/2))/2 - log(16*a*b^15*d^2 - ((-1/(4*(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^
2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2)))^(1/2)*(896*a^6*b^15*d^4 - 160*a^2*b^19*d^4 - 128*a^4*b^17*d^4 - 32*
b^21*d^4 + 3136*a^8*b^13*d^4 + 4928*a^10*b^11*d^4 + 4480*a^12*b^9*d^4 + 2432*a^14*b^7*d^4 + 736*a^16*b^5*d^4 +
96*a^18*b^3*d^4 - (-1/(4*(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^
2)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^
5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 6
40*a^19*b^4*d^5 + 64*a^21*b^2*d^5)) + (a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^
12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3))*(-1/(4*(a^5*d^2 - b^5*d^
2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2)))^(1/2) + 96*a^3*b^13*d^2 + 240*a^5*b^11
*d^2 + 320*a^7*b^9*d^2 + 240*a^9*b^7*d^2 + 96*a^11*b^5*d^2 + 16*a^13*b^3*d^2)*(-1/(4*(a^5*d^2 - b^5*d^2*1i + 5
*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2)))^(1/2) + atan((((-1i/(4*(a^5*d^2*1i - b^5*d^2 +
a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(896*a^6*b^15*d^4 - 160*a^2*b^19*d^4 -
128*a^4*b^17*d^4 - 32*b^21*d^4 + 3136*a^8*b^13*d^4 + 4928*a^10*b^11*d^4 + 4480*a^12*b^9*d^4 + 2432*a^14*b^7*d
^4 + 736*a^16*b^5*d^4 + 96*a^18*b^3*d^4 + (-1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*
b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^1
8*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5
+ 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5)) - (a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 1
6*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3))*
(-1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*1i - ((
-1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(896*a^6
*b^15*d^4 - 160*a^2*b^19*d^4 - 128*a^4*b^17*d^4 - 32*b^21*d^4 + 3136*a^8*b^13*d^4 + 4928*a^10*b^11*d^4 + 4480*
a^12*b^9*d^4 + 2432*a^14*b^7*d^4 + 736*a^16*b^5*d^4 + 96*a^18*b^3*d^4 - (-1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*
d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 6
40*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^1
3*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5)) + (a + b*tan(c + d*x
))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^1
2*b^6*d^3 - 16*a^16*b^2*d^3))*(-1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^
3*b^2*d^2*10i)))^(1/2)*1i)/(32*a*b^15*d^2 - ((-1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a
^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(896*a^6*b^15*d^4 - 160*a^2*b^19*d^4 - 128*a^4*b^17*d^4 - 32*b^21*d^4 +
3136*a^8*b^13*d^4 + 4928*a^10*b^11*d^4 + 4480*a^12*b^9*d^4 + 2432*a^14*b^7*d^4 + 736*a^16*b^5*d^4 + 96*a^18*b^
3*d^4 - (-1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)
*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*
a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^
4*d^5 + 64*a^21*b^2*d^5)) + (a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1
440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3))*(-1i/(4*(a^5*d^2*1i - b^5*d^2 + a*
b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3...

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