3.13.35 \(\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \, dx\) [1235]

Optimal. Leaf size=256 \[ \frac {(i a+b)^3 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(i a-b)^3 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {4 b^2 (b c-8 a d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{7 d f} \]

[Out]

(I*a+b)^3*(c-I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f-(I*a-b)^3*(c+I*d)^(3/2)*arctanh((c+d*t
an(f*x+e))^(1/2)/(c+I*d)^(1/2))/f+2*(a^3*d+3*a^2*b*c-3*a*b^2*d-b^3*c)*(c+d*tan(f*x+e))^(1/2)/f+2/3*b*(3*a^2-b^
2)*(c+d*tan(f*x+e))^(3/2)/f-4/35*b^2*(-8*a*d+b*c)*(c+d*tan(f*x+e))^(5/2)/d^2/f+2/7*b^2*(a+b*tan(f*x+e))*(c+d*t
an(f*x+e))^(5/2)/d/f

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Rubi [A]
time = 0.51, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3647, 3711, 3609, 3620, 3618, 65, 214} \begin {gather*} \frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (a^3 d+3 a^2 b c-3 a b^2 d-b^3 c\right ) \sqrt {c+d \tan (e+f x)}}{f}-\frac {4 b^2 (b c-8 a d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {(-b+i a)^3 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {(b+i a)^3 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((I*a + b)^3*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f - ((I*a - b)^3*(c + I*d)^(3/2)
*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d)*Sqrt[c + d*Ta
n[e + f*x]])/f + (2*b*(3*a^2 - b^2)*(c + d*Tan[e + f*x])^(3/2))/(3*f) - (4*b^2*(b*c - 8*a*d)*(c + d*Tan[e + f*
x])^(5/2))/(35*d^2*f) + (2*b^2*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2))/(7*d*f)

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 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 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 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rubi steps

\begin {align*} \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \, dx &=\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {2 \int (c+d \tan (e+f x))^{3/2} \left (\frac {1}{2} \left (7 a^3 d-2 b^2 \left (b c+\frac {5 a d}{2}\right )\right )+\frac {7}{2} b \left (3 a^2-b^2\right ) d \tan (e+f x)-b^2 (b c-8 a d) \tan ^2(e+f x)\right ) \, dx}{7 d}\\ &=-\frac {4 b^2 (b c-8 a d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {2 \int (c+d \tan (e+f x))^{3/2} \left (\frac {7}{2} a \left (a^2-3 b^2\right ) d+\frac {7}{2} b \left (3 a^2-b^2\right ) d \tan (e+f x)\right ) \, dx}{7 d}\\ &=\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {4 b^2 (b c-8 a d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {2 \int \sqrt {c+d \tan (e+f x)} \left (\frac {7}{2} d \left (a^3 c-3 a b^2 c-3 a^2 b d+b^3 d\right )+\frac {7}{2} d \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) \tan (e+f x)\right ) \, dx}{7 d}\\ &=\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {4 b^2 (b c-8 a d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {2 \int \frac {-\frac {7}{2} d \left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right )+\frac {7}{2} d \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{7 d}\\ &=\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {4 b^2 (b c-8 a d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {1}{2} \left ((a-i b)^3 (c-i d)^2\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b)^3 (c+i d)^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {4 b^2 (b c-8 a d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {\left ((i a+b)^3 (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}+\frac {\left ((i a-b)^3 (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {4 b^2 (b c-8 a d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {\left ((a-i b)^3 (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}-\frac {\left ((a+i b)^3 (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}\\ &=\frac {(i a+b)^3 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(i a-b)^3 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {4 b^2 (b c-8 a d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{7 d f}\\ \end {align*}

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Mathematica [A]
time = 3.46, size = 247, normalized size = 0.96 \begin {gather*} \frac {\frac {4 b^2 (-b c+8 a d) (c+d \tan (e+f x))^{5/2}}{d}+10 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}-\frac {35}{3} (i a+b)^3 d \left (-3 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c-3 i d+d \tan (e+f x))\right )+\frac {35}{3} (i a-b)^3 d \left (-3 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c+3 i d+d \tan (e+f x))\right )}{35 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*b^2*(-(b*c) + 8*a*d)*(c + d*Tan[e + f*x])^(5/2))/d + 10*b^2*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2
) - (35*(I*a + b)^3*d*(-3*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + Sqrt[c + d*Tan[e +
f*x]]*(4*c - (3*I)*d + d*Tan[e + f*x])))/3 + (35*(I*a - b)^3*d*(-3*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e +
f*x]]/Sqrt[c + I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c + (3*I)*d + d*Tan[e + f*x])))/3)/(35*d*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1747\) vs. \(2(224)=448\).
time = 0.53, size = 1748, normalized size = 6.83

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1748\)
default	\(\text {Expression too large to display}\)	\(1748\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f/d^2*(1/7*b^3*(c+d*tan(f*x+e))^(7/2)+3/5*a*b^2*d*(c+d*tan(f*x+e))^(5/2)-1/5*b^3*c*(c+d*tan(f*x+e))^(5/2)+a^
2*b*d^2*(c+d*tan(f*x+e))^(3/2)-1/3*b^3*d^2*(c+d*tan(f*x+e))^(3/2)+a^3*d^3*(c+d*tan(f*x+e))^(1/2)+3*a^2*b*c*d^2
*(c+d*tan(f*x+e))^(1/2)-3*a*b^2*d^3*(c+d*tan(f*x+e))^(1/2)-b^3*c*d^2*(c+d*tan(f*x+e))^(1/2)-d^2*(1/4/d*(1/2*((
c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d-3*(
c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d-(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d^2+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c
*d+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^2-2*(2*(c^2+d^2)^(1/2)+2*
c)^(1/2)*b^3*c*d)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+2*(2
*(c^2+d^2)^(1/2)*a^3*d^2+6*(c^2+d^2)^(1/2)*a^2*b*c*d-6*(c^2+d^2)^(1/2)*a*b^2*d^2-2*(c^2+d^2)^(1/2)*b^3*c*d-1/2
*((c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d-
3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d-(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d^2+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*
b*c*d+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^2-2*(2*(c^2+d^2)^(1/2)
+2*c)^(1/2)*b^3*c*d)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(
1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))+1/4/d*(1/2*(-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*a^3*c+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*a*b^2*c-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3
*c^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d^2-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d-3*(2*(c^2+d^2)^(1/2)+2*c)
^(1/2)*a*b^2*c^2+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^2+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d)*ln(d*tan(f
*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+2*(2*(c^2+d^2)^(1/2)*a^3*d^2+6*(
c^2+d^2)^(1/2)*a^2*b*c*d-6*(c^2+d^2)^(1/2)*a*b^2*d^2-2*(c^2+d^2)^(1/2)*b^3*c*d+1/2*(-(c^2+d^2)^(1/2)*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*a^3*c+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d+3*(c^2+d^2)^(1/2)*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*a*b^2*c-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*a^3*c^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d^2-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d-3*(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*a*b^2*c^2+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^2+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d)*(2*(c^
2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c
)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))))

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Maxima [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((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [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((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 79.38, size = 2500, normalized size = 9.77 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c + d*tan(e + f*x))^(1/2)*(((6*b^3*c - 6*a*b^2*d)/(d^2*f) - (4*b^3*c)/(d^2*f))*(c^2 + d^2) - 2*c*(2*c*((6*b^3
*c - 6*a*b^2*d)/(d^2*f) - (4*b^3*c)/(d^2*f)) - (6*b*(a*d - b*c)^2)/(d^2*f) + (2*b^3*(c^2 + d^2))/(d^2*f)) + (2
*(a*d - b*c)^3)/(d^2*f)) - atan(((((8*(4*a^3*d^5*f^2 - 12*a*b^2*d^5*f^2 - 4*b^3*c*d^4*f^2 + 4*a^3*c^2*d^3*f^2
- 4*b^3*c^3*d^2*f^2 + 12*a^2*b*c*d^4*f^2 - 12*a*b^2*c^2*d^3*f^2 + 12*a^2*b*c^3*d^2*f^2))/f^3 - 64*c*d^2*(c + d
*tan(e + f*x))^(1/2)*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 + 48*a*b^5*d^3*f^2 + 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f
^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 - 160*a^3*b^3*d^3*f^2 - 144*a*b^5*c^2*d*f^2
- 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 + 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/64 - f^4*(a^1
2*c^6 + a^12*d^6 + b^12*c^6 + b^12*d^6 + 6*a^2*b^10*c^6 + 15*a^4*b^8*c^6 + 20*a^6*b^6*c^6 + 15*a^8*b^4*c^6 + 6
*a^10*b^2*c^6 + 6*a^2*b^10*d^6 + 15*a^4*b^8*d^6 + 20*a^6*b^6*d^6 + 15*a^8*b^4*d^6 + 6*a^10*b^2*d^6 + 3*a^12*c^
2*d^4 + 3*a^12*c^4*d^2 + 3*b^12*c^2*d^4 + 3*b^12*c^4*d^2 + 18*a^2*b^10*c^2*d^4 + 18*a^2*b^10*c^4*d^2 + 45*a^4*
b^8*c^2*d^4 + 45*a^4*b^8*c^4*d^2 + 60*a^6*b^6*c^2*d^4 + 60*a^6*b^6*c^4*d^2 + 45*a^8*b^4*c^2*d^4 + 45*a^8*b^4*c
^4*d^2 + 18*a^10*b^2*c^2*d^4 + 18*a^10*b^2*c^4*d^2))^(1/2) + a^6*c^3*f^2 - b^6*c^3*f^2 + 6*a*b^5*d^3*f^2 + 6*a
^5*b*d^3*f^2 - 3*a^6*c*d^2*f^2 + 3*b^6*c*d^2*f^2 + 15*a^2*b^4*c^3*f^2 - 15*a^4*b^2*c^3*f^2 - 20*a^3*b^3*d^3*f^
2 - 18*a*b^5*c^2*d*f^2 - 18*a^5*b*c^2*d*f^2 - 45*a^2*b^4*c*d^2*f^2 + 60*a^3*b^3*c^2*d*f^2 + 45*a^4*b^2*c*d^2*f
^2)/(4*f^4))^(1/2))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 + 48*a*b^5*d^3*f^2 + 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^
2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 - 160*a^3*b^3*d^3*f^2 - 144*a*b^5*c^2*d*f^2 -
144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 + 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/64 - f^4*(a^12
*c^6 + a^12*d^6 + b^12*c^6 + b^12*d^6 + 6*a^2*b^10*c^6 + 15*a^4*b^8*c^6 + 20*a^6*b^6*c^6 + 15*a^8*b^4*c^6 + 6*
a^10*b^2*c^6 + 6*a^2*b^10*d^6 + 15*a^4*b^8*d^6 + 20*a^6*b^6*d^6 + 15*a^8*b^4*d^6 + 6*a^10*b^2*d^6 + 3*a^12*c^2
*d^4 + 3*a^12*c^4*d^2 + 3*b^12*c^2*d^4 + 3*b^12*c^4*d^2 + 18*a^2*b^10*c^2*d^4 + 18*a^2*b^10*c^4*d^2 + 45*a^4*b
^8*c^2*d^4 + 45*a^4*b^8*c^4*d^2 + 60*a^6*b^6*c^2*d^4 + 60*a^6*b^6*c^4*d^2 + 45*a^8*b^4*c^2*d^4 + 45*a^8*b^4*c^
4*d^2 + 18*a^10*b^2*c^2*d^4 + 18*a^10*b^2*c^4*d^2))^(1/2) + a^6*c^3*f^2 - b^6*c^3*f^2 + 6*a*b^5*d^3*f^2 + 6*a^
5*b*d^3*f^2 - 3*a^6*c*d^2*f^2 + 3*b^6*c*d^2*f^2 + 15*a^2*b^4*c^3*f^2 - 15*a^4*b^2*c^3*f^2 - 20*a^3*b^3*d^3*f^2
- 18*a*b^5*c^2*d*f^2 - 18*a^5*b*c^2*d*f^2 - 45*a^2*b^4*c*d^2*f^2 + 60*a^3*b^3*c^2*d*f^2 + 45*a^4*b^2*c*d^2*f^
2)/(4*f^4))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^6 - b^6*d^6 + 15*a^2*b^4*d^6 - 15*a^4*b^2*d^6 - 6*a^
6*c^2*d^4 + a^6*c^4*d^2 + 6*b^6*c^2*d^4 - b^6*c^4*d^2 - 24*a*b^5*c^3*d^3 - 80*a^3*b^3*c*d^5 - 24*a^5*b*c^3*d^3
- 90*a^2*b^4*c^2*d^4 + 15*a^2*b^4*c^4*d^2 + 80*a^3*b^3*c^3*d^3 + 90*a^4*b^2*c^2*d^4 - 15*a^4*b^2*c^4*d^2 + 24
*a*b^5*c*d^5 + 24*a^5*b*c*d^5))/f^2)*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 + 48*a*b^5*d^3*f^2 + 48*a^5*b*d^3*f^2
- 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 - 160*a^3*b^3*d^3*f^2 - 144*
a*b^5*c^2*d*f^2 - 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 + 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)
^2/64 - f^4*(a^12*c^6 + a^12*d^6 + b^12*c^6 + b^12*d^6 + 6*a^2*b^10*c^6 + 15*a^4*b^8*c^6 + 20*a^6*b^6*c^6 + 15
*a^8*b^4*c^6 + 6*a^10*b^2*c^6 + 6*a^2*b^10*d^6 + 15*a^4*b^8*d^6 + 20*a^6*b^6*d^6 + 15*a^8*b^4*d^6 + 6*a^10*b^2
*d^6 + 3*a^12*c^2*d^4 + 3*a^12*c^4*d^2 + 3*b^12*c^2*d^4 + 3*b^12*c^4*d^2 + 18*a^2*b^10*c^2*d^4 + 18*a^2*b^10*c
^4*d^2 + 45*a^4*b^8*c^2*d^4 + 45*a^4*b^8*c^4*d^2 + 60*a^6*b^6*c^2*d^4 + 60*a^6*b^6*c^4*d^2 + 45*a^8*b^4*c^2*d^
4 + 45*a^8*b^4*c^4*d^2 + 18*a^10*b^2*c^2*d^4 + 18*a^10*b^2*c^4*d^2))^(1/2) + a^6*c^3*f^2 - b^6*c^3*f^2 + 6*a*b
^5*d^3*f^2 + 6*a^5*b*d^3*f^2 - 3*a^6*c*d^2*f^2 + 3*b^6*c*d^2*f^2 + 15*a^2*b^4*c^3*f^2 - 15*a^4*b^2*c^3*f^2 - 2
0*a^3*b^3*d^3*f^2 - 18*a*b^5*c^2*d*f^2 - 18*a^5*b*c^2*d*f^2 - 45*a^2*b^4*c*d^2*f^2 + 60*a^3*b^3*c^2*d*f^2 + 45
*a^4*b^2*c*d^2*f^2)/(4*f^4))^(1/2)*1i - (((8*(4*a^3*d^5*f^2 - 12*a*b^2*d^5*f^2 - 4*b^3*c*d^4*f^2 + 4*a^3*c^2*d
^3*f^2 - 4*b^3*c^3*d^2*f^2 + 12*a^2*b*c*d^4*f^2 - 12*a*b^2*c^2*d^3*f^2 + 12*a^2*b*c^3*d^2*f^2))/f^3 + 64*c*d^2
*(c + d*tan(e + f*x))^(1/2)*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 + 48*a*b^5*d^3*f^2 + 48*a^5*b*d^3*f^2 - 24*a^6*
c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 - 160*a^3*b^3*d^3*f^2 - 144*a*b^5*c^2
*d*f^2 - 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 + 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/64 - f
^4*(a^12*c^6 + a^12*d^6 + b^12*c^6 + b^12*d^6 + 6*a^2*b^10*c^6 + 15*a^4*b^8*c^6 + 20*a^6*b^6*c^6 + 15*a^8*b^4*
c^6 + 6*a^10*b^2*c^6 + 6*a^2*b^10*d^6 + 15*a^4*b^8*d^6 + 20*a^6*b^6*d^6 + 15*a^8*b^4*d^6 + 6*a^10*b^2*d^6 + 3*
a^12*c^2*d^4 + 3*a^12*c^4*d^2 + 3*b^12*c^2*d^4 + 3*b^12*c^4*d^2 + 18*a^2*b^10*c^2*d^4 + 18*a^2*b^10*c^4*d^2 +
45*a^4*b^8*c^2*d^4 + 45*a^4*b^8*c^4*d^2 + 60*a^...

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