3.6.20 \(\int \tan ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx\) [520]

Optimal. Leaf size=158 \[ \frac {i (a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {4 a b \sqrt {a+b \tan (c+d x)}}{d}-\frac {2 b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 (a+b \tan (c+d x))^{7/2}}{7 b d} \]

[Out]

I*(a-I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d-I*(a+I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)
/(a+I*b)^(1/2))/d-4*a*b*(a+b*tan(d*x+c))^(1/2)/d-2/3*b*(a+b*tan(d*x+c))^(3/2)/d+2/7*(a+b*tan(d*x+c))^(7/2)/b/d

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Rubi [A]
time = 0.21, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3624, 3563, 3609, 3620, 3618, 65, 214} \begin {gather*} \frac {2 (a+b \tan (c+d x))^{7/2}}{7 b d}-\frac {2 b (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {4 a b \sqrt {a+b \tan (c+d x)}}{d}+\frac {i (a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {i (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[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2),x]

[Out]

(I*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (I*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*
Tan[c + d*x]]/Sqrt[a + I*b]])/d - (4*a*b*Sqrt[a + b*Tan[c + d*x]])/d - (2*b*(a + b*Tan[c + d*x])^(3/2))/(3*d)
+ (2*(a + b*Tan[c + d*x])^(7/2))/(7*b*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 3563

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

Rule 3609

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

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 3624

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

Rubi steps

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

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Mathematica [A]
time = 2.24, size = 205, normalized size = 1.30 \begin {gather*} \frac {21 i (a-i b)^{5/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-21 i (a+i b)^{5/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\frac {1}{2} \sec ^3(c+d x) \left (3 a \left (3 a^2-46 b^2\right ) \cos (c+d x)+\left (3 a^3-58 a b^2\right ) \cos (3 (c+d x))-2 b \left (-9 a^2+4 b^2+\left (-9 a^2+10 b^2\right ) \cos (2 (c+d x))\right ) \sin (c+d x)\right ) \sqrt {a+b \tan (c+d x)}}{21 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((21*I)*(a - I*b)^(5/2)*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - (21*I)*(a + I*b)^(5/2)*b*ArcTanh[S
qrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (Sec[c + d*x]^3*(3*a*(3*a^2 - 46*b^2)*Cos[c + d*x] + (3*a^3 - 58*a*b^
2)*Cos[3*(c + d*x)] - 2*b*(-9*a^2 + 4*b^2 + (-9*a^2 + 10*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])*Sqrt[a + b*Tan[c
+ d*x]])/2)/(21*b*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(822\) vs. \(2(130)=260\).
time = 0.15, size = 823, normalized size = 5.21

method	result	size
derivativedivides	\(\frac {\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {2 b^{2} \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-4 a \,b^{2} \sqrt {a +b \tan \left (d x +c \right )}+2 b^{2} \left (\frac {\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\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 (4 \sqrt {a^{2}+b^{2}}\, a \,b^{2}-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\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}}+\frac {-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\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 (-4 \sqrt {a^{2}+b^{2}}\, a \,b^{2}+\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\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}}\right )}{b d}\)	\(823\)
default	\(\frac {\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {2 b^{2} \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-4 a \,b^{2} \sqrt {a +b \tan \left (d x +c \right )}+2 b^{2} \left (\frac {\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\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 (4 \sqrt {a^{2}+b^{2}}\, a \,b^{2}-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\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}}+\frac {-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\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 (-4 \sqrt {a^{2}+b^{2}}\, a \,b^{2}+\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\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}}\right )}{b d}\)	\(823\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/b*(1/7*(a+b*tan(d*x+c))^(7/2)-1/3*b^2*(a+b*tan(d*x+c))^(3/2)-2*a*b^2*(a+b*tan(d*x+c))^(1/2)+b^2*(1/4/b^2*(
1/2*((2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*b^2-(2*(a
^2+b^2)^(1/2)+2*a)^(1/2)*a^3+3*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^2)*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*(4*(a^2+b^2)^(1/2)*a*b^2-1/2*((2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a
^2+b^2)^(1/2)*a^2-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*b^2-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+3*(2*(a^
2+b^2)^(1/2)+2*a)^(1/2)*a*b^2)*(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*(-1/2*((2*(a^2+b^2)^(1/2
)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*b^2-(2*(a^2+b^2)^(1/2)+2*a)^(1/
2)*a^3+3*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^2)*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*(-4*(a^2+b^2)^(1/2)*a*b^2+1/2*((2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2-(2
*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*b^2-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+3*(2*(a^2+b^2)^(1/2)+2*a)^(1
/2)*a*b^2)*(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)))))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6733 vs. \(2 (124) = 248\).
time = 5.55, size = 6733, normalized size = 42.61 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/84*(84*sqrt(2)*b*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
+ 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)*(2*(5*a^9*b - 14*a^5*b^5 - 8*a^3*b^7 + a*b^
9)*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) + (15*a^14*b + 25*a^12*b^3 - 37*a^10*b^5 - 99*a^8*b^7 - 51*a^6*b^9 +
11*a^4*b^11 + 9*a^2*b^13 - b^15)*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)*(2*a*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) + (3*a^6 + 5*a^4*
b^2 + a^2*b^4 - 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 + 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((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)*
((75*a^10*b^3 - 325*a^8*b^5 + 430*a^6*b^7 - 170*a^4*b^9 + 23*a^2*b^11 - b^13)*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) + 2*(25*a^15*b^3 - 25*a^13*b^5 - 115*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*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*b^2 + 25*a^17*b^4 - 140*a^15*b^6 - 220*a^13*b^8 + 126*a^11*b^10 + 430*a^9*b^12 + 260*a^7*b^14
+ 20*a^5*b^16 - 15*a^3*b^18 + a*b^20)*cos(d*x + c) + (25*a^18*b^3 + 25*a^16*b^5 - 140*a^14*b^7 - 220*a^12*b^9
+ 126*a^10*b^11 + 430*a^8*b^13 + 260*a^6*b^15 + 20*a^4*b^17 - 15*a^2*b^19 + b^21)*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 + 3
471*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))*cos(d*x + c)^3 + 84*sqrt(2)*
b*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*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))/(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 + 36*a^14*b^4 - 2
8*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)*(2*(5*a^9*b - 14*a^5*b^5 - 8*a^3*b^7 + a*b^9)*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) + (15*a^14*b + 25*a^12*b^3 - 37*a^10*b^5 - 99*a^8*b^7 - 51*a^6*b^9 + 11*a^4*b^11 + 9*a
^2*b^13 - b^15)*d^5*sqrt((25*a^8*b^2 - 100*a^6*...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Mupad [B]
time = 25.00, size = 2165, normalized size = 13.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((8*(8*a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 - a^4*b*5i + a^
5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10
*a^3*b^2)/(4*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5
*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*1i - (((8*(8*a*b^5*d^2 + 8*a^3*b^3
*d^2))/d^3 + 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b
^2)/(4*d^2))^(1/2))*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(a +
b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i +
a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*1i)/((((8*(8*a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c
+ d*x))^(1/2)*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 - a^
4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^
2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^
(1/2) - (16*(6*a^4*b^7 - b^11 + 8*a^6*b^5 + 3*a^8*b^3))/d^3 + (((8*(8*a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 + 64*a*b
^2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))
*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1
/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^
3*b^2)/(4*d^2))^(1/2)))*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*2i + a
tan(((((8*(8*a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 + a^4*b*5i + a^5
+ b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*
a^3*b^2)/(4*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*
a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*1i - (((8*(8*a*b^5*d^2 + 8*a^3*b^3*
d^2))/d^3 + 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^
2)/(4*d^2))^(1/2))*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(a +
b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a
^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*1i)/((((8*(8*a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c +
d*x))^(1/2)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 + a^4
*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2
*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(
1/2) - (16*(6*a^4*b^7 - b^11 + 8*a^6*b^5 + 3*a^8*b^3))/d^3 + (((8*(8*a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 + 64*a*b^
2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*
(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/
2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3
*b^2)/(4*d^2))^(1/2)))*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*2i + ((
2*a^2)/(3*b*d) - (2*(a^2 + b^2))/(3*b*d))*(a + b*tan(c + d*x))^(3/2) + (2*(a + b*tan(c + d*x))^(7/2))/(7*b*d)
+ 2*a*((2*a^2)/(b*d) - (2*(a^2 + b^2))/(b*d))*(a + b*tan(c + d*x))^(1/2)

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