3.13.47 \(\int \frac {(a+b \tan (e+f x))^4}{\sqrt {c+d \tan (e+f x)}} \, dx\) [1247]

Optimal. Leaf size=248 \[ -\frac {i (a-i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}+\frac {i (a+i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}-\frac {2 b^2 \left (40 a b c d-87 a^2 d^2-b^2 \left (8 c^2-15 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 d^3 f}-\frac {4 b^3 (2 b c-7 a d) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f} \]

[Out]

-I*(a-I*b)^4*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f/(c-I*d)^(1/2)+I*(a+I*b)^4*arctanh((c+d*tan(f*x+e)
)^(1/2)/(c+I*d)^(1/2))/f/(c+I*d)^(1/2)-2/15*b^2*(40*a*b*c*d-87*a^2*d^2-b^2*(8*c^2-15*d^2))*(c+d*tan(f*x+e))^(1
/2)/d^3/f-4/15*b^3*(-7*a*d+2*b*c)*(c+d*tan(f*x+e))^(1/2)*tan(f*x+e)/d^2/f+2/5*b^2*(c+d*tan(f*x+e))^(1/2)*(a+b*
tan(f*x+e))^2/d/f

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Rubi [A]
time = 0.49, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3647, 3718, 3711, 3620, 3618, 65, 214} \begin {gather*} -\frac {2 b^2 \left (-87 a^2 d^2+40 a b c d-\left (b^2 \left (8 c^2-15 d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 d^3 f}-\frac {4 b^3 (2 b c-7 a d) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {i (a-i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {i (a+i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[e + f*x])^4/Sqrt[c + d*Tan[e + f*x]],x]

[Out]

((-I)*(a - I*b)^4*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) + (I*(a + I*b)^4*ArcTanh[
Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) - (2*b^2*(40*a*b*c*d - 87*a^2*d^2 - b^2*(8*c^2 - 15
*d^2))*Sqrt[c + d*Tan[e + f*x]])/(15*d^3*f) - (4*b^3*(2*b*c - 7*a*d)*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(1
5*d^2*f) + (2*b^2*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(5*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 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]

Rule 3718

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

Rubi steps

\begin {align*} \int \frac {(a+b \tan (e+f x))^4}{\sqrt {c+d \tan (e+f x)}} \, dx &=\frac {2 b^2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}+\frac {2 \int \frac {(a+b \tan (e+f x)) \left (\frac {1}{2} \left (-4 b^3 c+5 a^3 d-a b^2 d\right )+\frac {5}{2} b \left (3 a^2-b^2\right ) d \tan (e+f x)-b^2 (2 b c-7 a d) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{5 d}\\ &=-\frac {4 b^3 (2 b c-7 a d) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {4 \int \frac {\frac {1}{4} \left (-8 b^4 c^2+40 a b^3 c d-15 a^4 d^2+3 a^2 b^2 d^2\right )-15 a b \left (a^2-b^2\right ) d^2 \tan (e+f x)+\frac {1}{4} b^2 \left (40 a b c d-87 a^2 d^2-b^2 \left (8 c^2-15 d^2\right )\right ) \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{15 d^2}\\ &=-\frac {2 b^2 \left (40 a b c d-87 a^2 d^2-b^2 \left (8 c^2-15 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 d^3 f}-\frac {4 b^3 (2 b c-7 a d) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {4 \int \frac {-\frac {15}{4} \left (a^4-6 a^2 b^2+b^4\right ) d^2-15 a b \left (a^2-b^2\right ) d^2 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{15 d^2}\\ &=-\frac {2 b^2 \left (40 a b c d-87 a^2 d^2-b^2 \left (8 c^2-15 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 d^3 f}-\frac {4 b^3 (2 b c-7 a d) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}+\frac {1}{2} (a-i b)^4 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} (a+i b)^4 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=-\frac {2 b^2 \left (40 a b c d-87 a^2 d^2-b^2 \left (8 c^2-15 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 d^3 f}-\frac {4 b^3 (2 b c-7 a d) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}+\frac {\left (i (a-i b)^4\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+i b)^4\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 b^2 \left (40 a b c d-87 a^2 d^2-b^2 \left (8 c^2-15 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 d^3 f}-\frac {4 b^3 (2 b c-7 a d) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {(a-i b)^4 \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 {(a+i b)^4 \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-i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}+\frac {i (a+i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}-\frac {2 b^2 \left (40 a b c d-87 a^2 d^2-b^2 \left (8 c^2-15 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 d^3 f}-\frac {4 b^3 (2 b c-7 a d) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}\\ \end {align*}

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Mathematica [A]
time = 3.45, size = 235, normalized size = 0.95 \begin {gather*} \frac {-\frac {15 i (a-i b)^4 d \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {15 i (a+i b)^4 d \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}+\frac {2 b^2 \left (-40 a b c d+87 a^2 d^2+b^2 \left (8 c^2-15 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{d^2}+\frac {4 b^3 (-2 b c+7 a d) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{d}+6 b^2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{15 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[e + f*x])^4/Sqrt[c + d*Tan[e + f*x]],x]

[Out]

(((-15*I)*(a - I*b)^4*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((15*I)*(a + I*b)^4*d
*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + (2*b^2*(-40*a*b*c*d + 87*a^2*d^2 + b^2*(8*c^
2 - 15*d^2))*Sqrt[c + d*Tan[e + f*x]])/d^2 + (4*b^3*(-2*b*c + 7*a*d)*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/d
+ 6*b^2*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(15*d*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3443\) vs. \(2(216)=432\).
time = 0.43, size = 3444, normalized size = 13.89

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3444\)
default	\(\text {Expression too large to display}\)	\(3444\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))^4/(c+d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/f/d^3*(1/5*b^4*(c+d*tan(f*x+e))^(5/2)+4/3*a*b^3*d*(c+d*tan(f*x+e))^(3/2)-2/3*b^4*c*(c+d*tan(f*x+e))^(3/2)+6*
a^2*b^2*d^2*(c+d*tan(f*x+e))^(1/2)-4*a*b^3*c*d*(c+d*tan(f*x+e))^(1/2)+b^4*c^2*(c+d*tan(f*x+e))^(1/2)-b^4*d^2*(
c+d*tan(f*x+e))^(1/2)+d^3*(1/4/d^2/(c^2+d^2)^(3/2)*(-1/2*(4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/2)*a^3*
b*c-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/2)*a*b^3*c+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^4*
c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^4*d^3-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^
3*b*c^3-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*b*c*d^2-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^
(1/2)*a^2*b^2*c^2*d-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*b^2*d^3+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2
)*(c^2+d^2)^(1/2)*a*b^3*c^3+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^3*c*d^2+(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*(c^2+d^2)^(1/2)*b^4*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^4*d^3-(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*a^4*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^4*c*d^3-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*b*c^2*d^2-4*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*b*d^4+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b^2*c^3*d+6*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a^2*b^2*c*d^3+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^3*c^2*d^2+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^3*d^4-(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*b^4*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^4*c*d^3)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2
+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))+2*(-2*a^4*c^2*d^3-2*a^4*d^5+8*a^3*b*c^3*d^2+8*a^3*b*c*d
^4+12*a^2*b^2*c^2*d^3+12*a^2*b^2*d^5-8*a*b^3*c^3*d^2-8*a*b^3*c*d^4-2*b^4*c^2*d^3-2*b^4*d^5+1/2*(4*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/2)*a^3*b*c-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/2)*a*b^3*c+(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^4*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^4*d^3-4*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*b*c^3-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*b*c*d^2-6*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*b^2*c^2*d-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*
b^2*d^3+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^3*c^3+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1
/2)*a*b^3*c*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^4*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2
)^(1/2)*b^4*d^3-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^4*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^4*c*d^3-4*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*a^3*b*c^2*d^2-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*b*d^4+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*
b^2*c^3*d+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b^2*c*d^3+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^3*c^2*d^2+4*(2*(c^
2+d^2)^(1/2)+2*c)^(1/2)*a*b^3*d^4-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^4*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^4*c*
d^3)*(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^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d
*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))+1/4/d^2/(c^2+d^2)^(3/2)*(1/2*(4*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*(c^2+d^2)^(3/2)*a^3*b*c-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/2)*a*b^3*c+(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*(c^2+d^2)^(1/2)*a^4*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^4*d^3-4*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*(c^2+d^2)^(1/2)*a^3*b*c^3-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*b*c*d^2-6*(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*b^2*c^2*d-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*b^2*d^3+4*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^3*c^3+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^3*c*d^
2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^4*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^4*d^
3-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^4*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^4*c*d^3-4*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a^3*b*c^2*d^2-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*b*d^4+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b^2*c^3*d+6*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b^2*c*d^3+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^3*c^2*d^2+4*(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*a*b^3*d^4-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^4*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^4*c*d^3)*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*a^4*c^2*d^3+2*a^4*d^5-8*a^
3*b*c^3*d^2-8*a^3*b*c*d^4-12*a^2*b^2*c^2*d^3-12*a^2*b^2*d^5+8*a*b^3*c^3*d^2+8*a*b^3*c*d^4+2*b^4*c^2*d^3+2*b^4*
d^5-1/2*(4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/2)*a^3*b*c-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/
2)*a*b^3*c+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^4*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/
2)*a^4*d^3-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*b*c^3-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)
^(1/2)*a^3*b*c*d^2-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*b^2*c^2*d-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)*(c^2+d^2)^(1/2)*a^2*b^2*d^3+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^3*c^3+4*(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^3*c*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^4*c^2*d+(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^4*d^3-(2*(c^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((a+b*tan(f*x+e))^4/(c+d*tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e) + a)^4/sqrt(d*tan(f*x + e) + c), x)

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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))^4/(c+d*tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{4}}{\sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))**4/(c+d*tan(f*x+e))**(1/2),x)

[Out]

Integral((a + b*tan(e + f*x))**4/sqrt(c + d*tan(e + f*x)), 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))^4/(c+d*tan(f*x+e))^(1/2),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 = 16.98, size = 2500, normalized size = 10.08 \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))^4/(c + d*tan(e + f*x))^(1/2),x)

[Out]

atan(((((32*(a^4*d^3*f^2 + b^4*d^3*f^2 - 6*a^2*b^2*d^3*f^2 + 4*a*b^3*c*d^2*f^2 - 4*a^3*b*c*d^2*f^2))/f^3 - 64*
c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 28*a^2*b^6 - a^3*b^5*56i + 70*a^4*b^4 +
a^5*b^3*56i - 28*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 28*a^2*b^6 - a^3
*b^5*56i + 70*a^4*b^4 + a^5*b^3*56i - 28*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/
2)*(a^8*d^2 + b^8*d^2 - 28*a^2*b^6*d^2 + 70*a^4*b^4*d^2 - 28*a^6*b^2*d^2))/f^2)*(-(a*b^7*8i - a^7*b*8i + a^8 +
b^8 - 28*a^2*b^6 - a^3*b^5*56i + 70*a^4*b^4 + a^5*b^3*56i - 28*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2)*1i - ((
(32*(a^4*d^3*f^2 + b^4*d^3*f^2 - 6*a^2*b^2*d^3*f^2 + 4*a*b^3*c*d^2*f^2 - 4*a^3*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c
+ d*tan(e + f*x))^(1/2)*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 28*a^2*b^6 - a^3*b^5*56i + 70*a^4*b^4 + a^5*b^3*
56i - 28*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 28*a^2*b^6 - a^3*b^5*56i
+ 70*a^4*b^4 + a^5*b^3*56i - 28*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^8*
d^2 + b^8*d^2 - 28*a^2*b^6*d^2 + 70*a^4*b^4*d^2 - 28*a^6*b^2*d^2))/f^2)*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 2
8*a^2*b^6 - a^3*b^5*56i + 70*a^4*b^4 + a^5*b^3*56i - 28*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2)*1i)/((((32*(a^4
*d^3*f^2 + b^4*d^3*f^2 - 6*a^2*b^2*d^3*f^2 + 4*a*b^3*c*d^2*f^2 - 4*a^3*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan
(e + f*x))^(1/2)*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 28*a^2*b^6 - a^3*b^5*56i + 70*a^4*b^4 + a^5*b^3*56i - 28
*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 28*a^2*b^6 - a^3*b^5*56i + 70*a^
4*b^4 + a^5*b^3*56i - 28*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^8*d^2 + b^
8*d^2 - 28*a^2*b^6*d^2 + 70*a^4*b^4*d^2 - 28*a^6*b^2*d^2))/f^2)*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 28*a^2*b^
6 - a^3*b^5*56i + 70*a^4*b^4 + a^5*b^3*56i - 28*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) + (((32*(a^4*d^3*f^2 +
b^4*d^3*f^2 - 6*a^2*b^2*d^3*f^2 + 4*a*b^3*c*d^2*f^2 - 4*a^3*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^
(1/2)*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 28*a^2*b^6 - a^3*b^5*56i + 70*a^4*b^4 + a^5*b^3*56i - 28*a^6*b^2)/(
4*(c*f^2 - d*f^2*1i)))^(1/2))*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 28*a^2*b^6 - a^3*b^5*56i + 70*a^4*b^4 + a^5
*b^3*56i - 28*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^8*d^2 + b^8*d^2 - 28*
a^2*b^6*d^2 + 70*a^4*b^4*d^2 - 28*a^6*b^2*d^2))/f^2)*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 28*a^2*b^6 - a^3*b^5
*56i + 70*a^4*b^4 + a^5*b^3*56i - 28*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) - (64*(a*b^11*d^2 - a^11*b*d^2 + 3
*a^3*b^9*d^2 + 2*a^5*b^7*d^2 - 2*a^7*b^5*d^2 - 3*a^9*b^3*d^2))/f^3))*(-(a*b^7*8i - a^7*b*8i + a^8 + b^8 - 28*a
^2*b^6 - a^3*b^5*56i + 70*a^4*b^4 + a^5*b^3*56i - 28*a^6*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2)*2i - (c + d*tan(e
+ f*x))^(1/2)*(2*c*((8*b^4*c - 8*a*b^3*d)/(d^3*f) - (4*b^4*c)/(d^3*f)) + (2*b^4*(c^2 + d^2))/(d^3*f) - (12*b^2
*(a*d - b*c)^2)/(d^3*f)) - ((8*b^4*c - 8*a*b^3*d)/(3*d^3*f) - (4*b^4*c)/(3*d^3*f))*(c + d*tan(e + f*x))^(3/2)
+ atan(((((32*(a^4*d^3*f^2 + b^4*d^3*f^2 - 6*a^2*b^2*d^3*f^2 + 4*a*b^3*c*d^2*f^2 - 4*a^3*b*c*d^2*f^2))/f^3 - 6
4*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(8*a*b^7 - 8*a^7*b + a^8*1i + b^8*1i - a^2*b^6*28i - 56*a^3*b^5 + a^4*b^4
*70i + 56*a^5*b^3 - a^6*b^2*28i)/(4*(c*f^2*1i - d*f^2)))^(1/2))*(-(8*a*b^7 - 8*a^7*b + a^8*1i + b^8*1i - a^2*b
^6*28i - 56*a^3*b^5 + a^4*b^4*70i + 56*a^5*b^3 - a^6*b^2*28i)/(4*(c*f^2*1i - d*f^2)))^(1/2) - (16*(c + d*tan(e
+ f*x))^(1/2)*(a^8*d^2 + b^8*d^2 - 28*a^2*b^6*d^2 + 70*a^4*b^4*d^2 - 28*a^6*b^2*d^2))/f^2)*(-(8*a*b^7 - 8*a^7
*b + a^8*1i + b^8*1i - a^2*b^6*28i - 56*a^3*b^5 + a^4*b^4*70i + 56*a^5*b^3 - a^6*b^2*28i)/(4*(c*f^2*1i - d*f^2
)))^(1/2)*1i - (((32*(a^4*d^3*f^2 + b^4*d^3*f^2 - 6*a^2*b^2*d^3*f^2 + 4*a*b^3*c*d^2*f^2 - 4*a^3*b*c*d^2*f^2))/
f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(8*a*b^7 - 8*a^7*b + a^8*1i + b^8*1i - a^2*b^6*28i - 56*a^3*b^5 +
a^4*b^4*70i + 56*a^5*b^3 - a^6*b^2*28i)/(4*(c*f^2*1i - d*f^2)))^(1/2))*(-(8*a*b^7 - 8*a^7*b + a^8*1i + b^8*1i
- a^2*b^6*28i - 56*a^3*b^5 + a^4*b^4*70i + 56*a^5*b^3 - a^6*b^2*28i)/(4*(c*f^2*1i - d*f^2)))^(1/2) + (16*(c +
d*tan(e + f*x))^(1/2)*(a^8*d^2 + b^8*d^2 - 28*a^2*b^6*d^2 + 70*a^4*b^4*d^2 - 28*a^6*b^2*d^2))/f^2)*(-(8*a*b^7
- 8*a^7*b + a^8*1i + b^8*1i - a^2*b^6*28i - 56*a^3*b^5 + a^4*b^4*70i + 56*a^5*b^3 - a^6*b^2*28i)/(4*(c*f^2*1i
- d*f^2)))^(1/2)*1i)/((((32*(a^4*d^3*f^2 + b^4*d^3*f^2 - 6*a^2*b^2*d^3*f^2 + 4*a*b^3*c*d^2*f^2 - 4*a^3*b*c*d^2
*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(8*a*b^7 - 8*a^7*b + a^8*1i + b^8*1i - a^2*b^6*28i - 56*a^3
*b^5 + a^4*b^4*70i + 56*a^5*b^3 - a^6*b^2*28i)/(4*(c*f^2*1i - d*f^2)))^(1/2))*(-(8*a*b^7 - 8*a^7*b + a^8*1i +
b^8*1i - a^2*b^6*28i - 56*a^3*b^5 + a^4*b^4*70i + 56*a^5*b^3 - a^6*b^2*28i)/(4*(c*f^2*1i - d*f^2)))^(1/2) - (1
6*(c + d*tan(e + f*x))^(1/2)*(a^8*d^2 + b^8*d^2 - 28*a^2*b^6*d^2 + 70*a^4*b^4*d^2 - 28*a^6*b^2*d^2))/f^2)*(-(8
*a*b^7 - 8*a^7*b + a^8*1i + b^8*1i - a^2*b^6*28...

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