3.13.34 \(\int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^3} \, dx\) [1234]

Optimal. Leaf size=342 \[ -\frac {\sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {\sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}+\frac {\sqrt {b} \left (40 a^3 b c d-24 a b^3 c d-15 a^4 d^2-6 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (8 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \left (a^2+b^2\right )^3 (b c-a d)^{3/2} f}-\frac {b \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))} \]

[Out]

1/4*(40*a^3*b*c*d-24*a*b^3*c*d-15*a^4*d^2-6*a^2*b^2*(4*c^2-3*d^2)+b^4*(8*c^2+d^2))*arctanh(b^(1/2)*(c+d*tan(f*
x+e))^(1/2)/(-a*d+b*c)^(1/2))*b^(1/2)/(a^2+b^2)^3/(-a*d+b*c)^(3/2)/f-arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1
/2))*(c-I*d)^(1/2)/(I*a+b)^3/f+arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))*(c+I*d)^(1/2)/(I*a-b)^3/f-1/2*b*(
c+d*tan(f*x+e))^(1/2)/(a^2+b^2)/f/(a+b*tan(f*x+e))^2-1/4*b*(-7*a^2*d+8*a*b*c+b^2*d)*(c+d*tan(f*x+e))^(1/2)/(a^
2+b^2)^2/(-a*d+b*c)/f/(a+b*tan(f*x+e))

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Rubi [A]
time = 1.07, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3649, 3730, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {b \left (-7 a^2 d+8 a b c+b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 f \left (a^2+b^2\right )^2 (b c-a d) (a+b \tan (e+f x))}-\frac {b \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {\sqrt {b} \left (-15 a^4 d^2+40 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-24 a b^3 c d+b^4 \left (8 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 f \left (a^2+b^2\right )^3 (b c-a d)^{3/2}}-\frac {\sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)^3}+\frac {\sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + (Sqrt[c + I*d]*ArcTanh[Sq
rt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + (Sqrt[b]*(40*a^3*b*c*d - 24*a*b^3*c*d - 15*a^4*d^2 -
6*a^2*b^2*(4*c^2 - 3*d^2) + b^4*(8*c^2 + d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4
*(a^2 + b^2)^3*(b*c - a*d)^(3/2)*f) - (b*Sqrt[c + d*Tan[e + f*x]])/(2*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) -
(b*(8*a*b*c - 7*a^2*d + b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(4*(a^2 + b^2)^2*(b*c - a*d)*f*(a + b*Tan[e + f*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 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 3649

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

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 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, 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] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

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 \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^3} \, dx &=-\frac {b \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\int \frac {\frac {1}{2} (-4 a c-b d)+2 (b c-a d) \tan (e+f x)+\frac {3}{2} b d \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )}\\ &=-\frac {b \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}+\frac {\int \frac {\frac {1}{4} \left (-8 a^3 c d+16 a b^2 c d+4 a^2 b \left (2 c^2-\frac {9 d^2}{4}\right )-b^3 \left (8 c^2+d^2\right )\right )-2 (b c-a d) \left (2 a b c-a^2 d+b^2 d\right ) \tan (e+f x)-\frac {1}{4} b d \left (8 a b c-7 a^2 d+b^2 d\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^2 (b c-a d)}\\ &=-\frac {b \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}+\frac {\int \frac {2 (b c-a d) \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )-2 (b c-a d) \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^3 (b c-a d)}-\frac {\left (b \left (40 a^3 b c d-24 a b^3 c d-15 a^4 d^2-6 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (8 c^2+d^2\right )\right )\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{8 \left (a^2+b^2\right )^3 (b c-a d)}\\ &=-\frac {b \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}+\frac {(c-i d) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^3}+\frac {(c+i d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^3}-\frac {\left (b \left (40 a^3 b c d-24 a b^3 c d-15 a^4 d^2-6 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (8 c^2+d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{8 \left (a^2+b^2\right )^3 (b c-a d) f}\\ &=-\frac {b \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}-\frac {(c+i d) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (i a-b)^3 f}+\frac {(i c+d) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (a-i b)^3 f}-\frac {\left (b \left (40 a^3 b c d-24 a b^3 c d-15 a^4 d^2-6 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (8 c^2+d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{4 \left (a^2+b^2\right )^3 d (b c-a d) f}\\ &=\frac {\sqrt {b} \left (40 a^3 b c d-24 a b^3 c d-15 a^4 d^2-6 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (8 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \left (a^2+b^2\right )^3 (b c-a d)^{3/2} f}-\frac {b \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}-\frac {(c+i d) \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 )}{(a+i b)^3 d f}+\frac {(i c+d) \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 )}{(i a+b)^3 d f}\\ &=-\frac {\sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {\sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}+\frac {\sqrt {b} \left (40 a^3 b c d-24 a b^3 c d-15 a^4 d^2-6 a^2 b^2 \left (4 c^2-3 d^2\right )+b^4 \left (8 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \left (a^2+b^2\right )^3 (b c-a d)^{3/2} f}-\frac {b \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {b \left (8 a b c-7 a^2 d+b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(747\) vs. \(2(342)=684\).
time = 6.35, size = 747, normalized size = 2.18 \begin {gather*} -\frac {b^2 (c+d \tan (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2}-\frac {-\frac {b d \sqrt {c+d \tan (e+f x)}}{f (a+b \tan (e+f x))}-\frac {2 \left (-\frac {\frac {\frac {i \sqrt {c-i d} \left (-i b (b c-a d)^2 \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right )-b (b c-a d)^2 \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(-c+i d) f}-\frac {i \sqrt {c+i d} \left (i b (b c-a d)^2 \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right )-b (b c-a d)^2 \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(-c-i d) f}}{a^2+b^2}+\frac {2 \sqrt {b c-a d} \left (\frac {1}{8} a^2 b^2 d (b c-a d) \left (8 a b c-7 a^2 d+b^2 d\right )-a b^2 (b c-a d)^2 \left (2 a b c-a^2 d+b^2 d\right )-\frac {1}{8} b^3 (b c-a d) \left (8 a^2 b c^2-8 b^3 c^2-8 a^3 c d+16 a b^2 c d-9 a^2 b d^2-b^3 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} \left (a^2+b^2\right ) (-b c+a d) f}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {\left (\frac {1}{4} b^3 (b c-a d) (4 a c+b d)-a \left (\frac {3}{4} a b^2 d (b c-a d)-b^2 (b c-a d)^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}\right )}{b}}{2 \left (a^2+b^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b^2*(c + d*Tan[e + f*x])^(3/2))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((b*d*Sqrt[c + d*
Tan[e + f*x]])/(f*(a + b*Tan[e + f*x]))) - (2*(-((((I*Sqrt[c - I*d]*((-I)*b*(b*c - a*d)^2*(3*a^2*b*c - b^3*c -
a^3*d + 3*a*b^2*d) - b*(b*c - a*d)^2*(a^3*c - 3*a*b^2*c + 3*a^2*b*d - b^3*d))*ArcTanh[Sqrt[c + d*Tan[e + f*x]
]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*(I*b*(b*c - a*d)^2*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d)
- b*(b*c - a*d)^2*(a^3*c - 3*a*b^2*c + 3*a^2*b*d - b^3*d))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(
(-c - I*d)*f))/(a^2 + b^2) + (2*Sqrt[b*c - a*d]*((a^2*b^2*d*(b*c - a*d)*(8*a*b*c - 7*a^2*d + b^2*d))/8 - a*b^2
*(b*c - a*d)^2*(2*a*b*c - a^2*d + b^2*d) - (b^3*(b*c - a*d)*(8*a^2*b*c^2 - 8*b^3*c^2 - 8*a^3*c*d + 16*a*b^2*c*
d - 9*a^2*b*d^2 - b^3*d^2))/8)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^
2)*(-(b*c) + a*d)*f))/((a^2 + b^2)*(b*c - a*d))) - (((b^3*(b*c - a*d)*(4*a*c + b*d))/4 - a*((3*a*b^2*d*(b*c -
a*d))/4 - b^2*(b*c - a*d)^2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))))/b)/
(2*(a^2 + b^2)*(b*c - a*d))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1276\) vs. \(2(304)=608\).
time = 0.54, size = 1277, normalized size = 3.73 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

2/f*d^4*(-b/d^4/(a^2+b^2)^3*((1/8*b*d*(7*a^4*d-8*a^3*b*c+6*a^2*b^2*d-8*a*b^3*c-b^4*d)/(a*d-b*c)*(c+d*tan(f*x+e
))^(3/2)+1/8*(9*a^4*d-8*a^3*b*c+10*a^2*b^2*d-8*a*b^3*c+b^4*d)*d*(c+d*tan(f*x+e))^(1/2))/((c+d*tan(f*x+e))*b+a*
d-b*c)^2+1/8*(15*a^4*d^2-40*a^3*b*c*d+24*a^2*b^2*c^2-18*a^2*b^2*d^2+24*a*b^3*c*d-8*b^4*c^2-b^4*d^2)/(a*d-b*c)/
((a*d-b*c)*b)^(1/2)*arctan(b*(c+d*tan(f*x+e))^(1/2)/((a*d-b*c)*b)^(1/2)))+1/d^4/(a^2+b^2)^3*(1/4/d*(1/2*(-(c^2
+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2+(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*a^3*c+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c-(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*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*(6*(c^2+d^2)^(1/2)*a^2*b*d-2*(c^2+d^2)^(1/2)*b^3*d-1/2*(-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*a^3+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c+3*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*a^2*b*d-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c-(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))/(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^3-3*(c
^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c-3*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*a^2*b*d+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*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*(6*(c^2+d^2)^(1/2)*a^2*b*d-2*(c^2+d^2
)^(1/2)*b^3*d+1/2*((c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)
^(1/2)*a*b^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d+3*(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*a*b^2*c+(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))/(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(-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((c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^3,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(a*d-b*c>0)', see `assume?` for
more detail

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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((c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^3,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 {\sqrt {c + d \tan {\left (e + f x \right )}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**(1/2)/(a+b*tan(f*x+e))**3,x)

[Out]

Integral(sqrt(c + d*tan(e + f*x))/(a + b*tan(e + f*x))**3, 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((c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^3,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 = 35.52, size = 2500, normalized size = 7.31 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*tan(e + f*x))^(1/2)/(a + b*tan(e + f*x))^3,x)

[Out]

(atan((((((c + d*tan(e + f*x))^(1/2)*(2*a^2*b^13*d^14 - b^15*d^14 + 49*a^4*b^11*d^14 + 2460*a^6*b^9*d^14 - 363
1*a^8*b^7*d^14 + 1922*a^10*b^5*d^14 - 225*a^12*b^3*d^14 + 17*b^15*c^2*d^12 + 16*b^15*c^4*d^10 + 96*b^15*c^6*d^
8 + 80*a*b^14*c^3*d^11 - 960*a*b^14*c^5*d^9 - 40*a^3*b^12*c*d^13 - 9264*a^5*b^10*c*d^13 + 21360*a^7*b^8*c*d^13
- 15544*a^9*b^6*c*d^13 + 3000*a^11*b^4*c*d^13 - 114*a^2*b^13*c^2*d^12 + 4848*a^2*b^13*c^4*d^10 - 640*a^2*b^13
*c^6*d^8 - 11504*a^3*b^12*c^3*d^11 + 7424*a^3*b^12*c^5*d^9 + 14319*a^4*b^11*c^2*d^12 - 27744*a^4*b^11*c^4*d^10
+ 3136*a^4*b^11*c^6*d^8 + 49824*a^5*b^10*c^3*d^11 - 21120*a^5*b^10*c^5*d^9 - 46588*a^6*b^9*c^2*d^12 + 55264*a
^6*b^9*c^4*d^10 - 3712*a^6*b^9*c^6*d^8 - 71520*a^7*b^8*c^3*d^11 + 17664*a^7*b^8*c^5*d^9 + 47871*a^8*b^7*c^2*d^
12 - 32688*a^8*b^7*c^4*d^10 + 608*a^8*b^7*c^6*d^8 + 29712*a^9*b^6*c^3*d^11 - 1984*a^9*b^6*c^5*d^9 - 13746*a^10
*b^5*c^2*d^12 + 2352*a^10*b^5*c^4*d^10 - 1200*a^11*b^4*c^3*d^11 + 225*a^12*b^3*c^2*d^12 - 24*a*b^14*c*d^13))/(
a^18*d^2*f^4 + b^18*c^2*f^4 + 8*a^2*b^16*c^2*f^4 + 28*a^4*b^14*c^2*f^4 + 56*a^6*b^12*c^2*f^4 + 70*a^8*b^10*c^2
*f^4 + 56*a^10*b^8*c^2*f^4 + 28*a^12*b^6*c^2*f^4 + 8*a^14*b^4*c^2*f^4 + a^16*b^2*c^2*f^4 + a^2*b^16*d^2*f^4 +
8*a^4*b^14*d^2*f^4 + 28*a^6*b^12*d^2*f^4 + 56*a^8*b^10*d^2*f^4 + 70*a^10*b^8*d^2*f^4 + 56*a^12*b^6*d^2*f^4 + 2
8*a^14*b^4*d^2*f^4 + 8*a^16*b^2*d^2*f^4 - 2*a*b^17*c*d*f^4 - 2*a^17*b*c*d*f^4 - 16*a^3*b^15*c*d*f^4 - 56*a^5*b
^13*c*d*f^4 - 112*a^7*b^11*c*d*f^4 - 140*a^9*b^9*c*d*f^4 - 112*a^11*b^7*c*d*f^4 - 56*a^13*b^5*c*d*f^4 - 16*a^1
5*b^3*c*d*f^4) + (((4*b^18*d^14*f^2 - 276*a^2*b^16*d^14*f^2 - 6092*a^4*b^14*d^14*f^2 + 9724*a^6*b^12*d^14*f^2
+ 18444*a^8*b^10*d^14*f^2 - 10492*a^10*b^8*d^14*f^2 - 8580*a^12*b^6*d^14*f^2 + 4884*a^14*b^4*d^14*f^2 + 64*a^1
6*b^2*d^14*f^2 + 4*b^18*c^2*d^12*f^2 - 192*b^18*c^4*d^10*f^2 - 192*b^18*c^6*d^8*f^2 - 11284*a^2*b^16*c^2*d^12*
f^2 - 5760*a^2*b^16*c^4*d^10*f^2 + 5248*a^2*b^16*c^6*d^8*f^2 - 15872*a^3*b^15*c^3*d^11*f^2 - 29696*a^3*b^15*c^
5*d^9*f^2 + 48820*a^4*b^14*c^2*d^12*f^2 + 49216*a^4*b^14*c^4*d^10*f^2 - 5696*a^4*b^14*c^6*d^8*f^2 - 37120*a^5*
b^13*c^3*d^11*f^2 + 3328*a^5*b^13*c^5*d^9*f^2 + 38780*a^6*b^12*c^2*d^12*f^2 + 11392*a^6*b^12*c^4*d^10*f^2 - 17
664*a^6*b^12*c^6*d^8*f^2 + 28416*a^7*b^11*c^3*d^11*f^2 + 73728*a^7*b^11*c^5*d^9*f^2 - 87796*a^8*b^10*c^2*d^12*
f^2 - 102464*a^8*b^10*c^4*d^10*f^2 + 3776*a^8*b^10*c^6*d^8*f^2 + 62464*a^9*b^9*c^3*d^11*f^2 + 1792*a^9*b^9*c^5
*d^9*f^2 - 35068*a^10*b^8*c^2*d^12*f^2 - 14208*a^10*b^8*c^4*d^10*f^2 + 10368*a^10*b^8*c^6*d^8*f^2 - 8192*a^11*
b^7*c^3*d^11*f^2 - 35840*a^11*b^7*c^5*d^9*f^2 + 36604*a^12*b^6*c^2*d^12*f^2 + 45248*a^12*b^6*c^4*d^10*f^2 + 64
*a^12*b^6*c^6*d^8*f^2 - 24832*a^13*b^5*c^3*d^11*f^2 - 256*a^13*b^5*c^5*d^9*f^2 + 5268*a^14*b^4*c^2*d^12*f^2 +
384*a^14*b^4*c^4*d^10*f^2 - 256*a^15*b^3*c^3*d^11*f^2 + 64*a^16*b^2*c^2*d^12*f^2 + 256*a*b^17*c*d^13*f^2 + 358
4*a*b^17*c^3*d^11*f^2 + 3328*a*b^17*c^5*d^9*f^2 + 13824*a^3*b^15*c*d^13*f^2 - 40448*a^5*b^13*c*d^13*f^2 - 4531
2*a^7*b^11*c*d^13*f^2 + 60672*a^9*b^9*c*d^13*f^2 + 27648*a^11*b^7*c*d^13*f^2 - 24576*a^13*b^5*c*d^13*f^2 - 256
*a^15*b^3*c*d^13*f^2)/(2*(a^18*d^2*f^5 + b^18*c^2*f^5 + 8*a^2*b^16*c^2*f^5 + 28*a^4*b^14*c^2*f^5 + 56*a^6*b^12
*c^2*f^5 + 70*a^8*b^10*c^2*f^5 + 56*a^10*b^8*c^2*f^5 + 28*a^12*b^6*c^2*f^5 + 8*a^14*b^4*c^2*f^5 + a^16*b^2*c^2
*f^5 + a^2*b^16*d^2*f^5 + 8*a^4*b^14*d^2*f^5 + 28*a^6*b^12*d^2*f^5 + 56*a^8*b^10*d^2*f^5 + 70*a^10*b^8*d^2*f^5
+ 56*a^12*b^6*d^2*f^5 + 28*a^14*b^4*d^2*f^5 + 8*a^16*b^2*d^2*f^5 - 2*a*b^17*c*d*f^5 - 2*a^17*b*c*d*f^5 - 16*a
^3*b^15*c*d*f^5 - 56*a^5*b^13*c*d*f^5 - 112*a^7*b^11*c*d*f^5 - 140*a^9*b^9*c*d*f^5 - 112*a^11*b^7*c*d*f^5 - 56
*a^13*b^5*c*d*f^5 - 16*a^15*b^3*c*d*f^5)) - ((((c + d*tan(e + f*x))^(1/2)*(8*a*b^20*d^13*f^2 + 4*b^21*c*d^12*f
^2 - 1152*a^3*b^18*d^13*f^2 + 2528*a^5*b^16*d^13*f^2 + 15296*a^7*b^14*d^13*f^2 + 14128*a^9*b^12*d^13*f^2 - 505
6*a^11*b^10*d^13*f^2 - 9248*a^13*b^8*d^13*f^2 + 64*a^15*b^6*d^13*f^2 + 1800*a^17*b^4*d^13*f^2 + 64*a^19*b^2*d^
13*f^2 + 256*b^21*c^3*d^10*f^2 + 576*b^21*c^5*d^8*f^2 + 2624*a^2*b^19*c^3*d^10*f^2 - 3584*a^2*b^19*c^5*d^8*f^2
+ 4800*a^3*b^18*c^2*d^11*f^2 + 3584*a^3*b^18*c^4*d^9*f^2 - 1920*a^4*b^17*c^3*d^10*f^2 - 13056*a^4*b^17*c^5*d^
8*f^2 + 18688*a^5*b^16*c^2*d^11*f^2 + 35072*a^5*b^16*c^4*d^9*f^2 - 19328*a^6*b^15*c^3*d^10*f^2 - 7680*a^6*b^15
*c^5*d^8*f^2 - 6144*a^7*b^14*c^2*d^11*f^2 + 38400*a^7*b^14*c^4*d^9*f^2 - 26496*a^8*b^13*c^3*d^10*f^2 + 8064*a^
8*b^13*c^5*d^8*f^2 - 41472*a^9*b^12*c^2*d^11*f^2 + 6528*a^9*b^12*c^4*d^9*f^2 - 28416*a^10*b^11*c^3*d^10*f^2 +
5632*a^10*b^11*c^5*d^8*f^2 - 6272*a^11*b^10*c^2*d^11*f^2 + 6656*a^11*b^10*c^4*d^9*f^2 - 42624*a^12*b^9*c^3*d^1
0*f^2 - 3840*a^12*b^9*c^5*d^8*f^2 + 42752*a^13*b^8*c^2*d^11*f^2 + 19712*a^13*b^8*c^4*d^9*f^2 - 36992*a^14*b^7*
c^3*d^10*f^2 - 2560*a^14*b^7*c^5*d^8*f^2 + 32256*a^15*b^6*c^2*d^11*f^2 + 8704*a^15*b^6*c^4*d^9*f^2 - 11136*a^1
6*b^5*c^3*d^10*f^2 + 64*a^16*b^5*c^5*d^8*f^2 + 6528*a^17*b^4*c^2*d^11*f^2 - 192*a^17*b^4*c^4*d^9*f^2 + 192*a^1
8*b^3*c^3*d^10*f^2 - 64*a^19*b^2*c^2*d^11*f^2 -...

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