3.13.40 \(\int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx\) [1240]

Optimal. Leaf size=341 \[ -\frac {(c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}+\frac {\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} \left (a^2+b^2\right )^3 \sqrt {b c-a d} f}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))} \]

[Out]

-(c-I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(I*a+b)^3/f+(c+I*d)^(3/2)*arctanh((c+d*tan(f*x+e)
)^(1/2)/(c+I*d)^(1/2))/(I*a-b)^3/f+1/4*(24*a^3*b*c*d-40*a*b^3*c*d-3*a^4*d^2-2*a^2*b^2*(12*c^2-13*d^2)+b^4*(8*c
^2-3*d^2))*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2)/(-a*d+b*c)^(1/2))/(a^2+b^2)^3/f/b^(1/2)/(-a*d+b*c)^(1/2)-1/2
*(-a*d+b*c)*(c+d*tan(f*x+e))^(1/2)/(a^2+b^2)/f/(a+b*tan(f*x+e))^2-1/4*(-3*a^2*d+8*a*b*c+5*b^2*d)*(c+d*tan(f*x+
e))^(1/2)/(a^2+b^2)^2/f/(a+b*tan(f*x+e))

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Rubi [A]
time = 1.30, antiderivative size = 341, 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 = {3648, 3730, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {\left (-3 a^4 d^2+24 a^3 b c d-2 a^2 b^2 \left (12 c^2-13 d^2\right )-40 a b^3 c d+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} f \left (a^2+b^2\right )^3 \sqrt {b c-a d}}-\frac {(c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)^3}+\frac {(c+i d)^{3/2} \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[(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^3,x]

[Out]

-(((c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + ((c + I*d)^(3/2)*ArcTan
h[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + ((24*a^3*b*c*d - 40*a*b^3*c*d - 3*a^4*d^2 - 2*a^2
*b^2*(12*c^2 - 13*d^2) + b^4*(8*c^2 - 3*d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4*
Sqrt[b]*(a^2 + b^2)^3*Sqrt[b*c - a*d]*f) - ((b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(2*(a^2 + b^2)*f*(a + b*Tan[
e + f*x])^2) - ((8*a*b*c - 3*a^2*d + 5*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(4*(a^2 + b^2)^2*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 3648

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n - 1)/(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 - 2)*Simp[a*c^2*(m + 1) + a*
d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2*a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*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] && LtQ[1, n, 2] && 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 {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx &=-\frac {(b c-a d) \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} \left (-5 b c d-a \left (4 c^2-d^2\right )\right )-2 \left (2 a c d-b \left (c^2-d^2\right )\right ) \tan (e+f x)+\frac {3}{2} d (b c-a 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 c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\int \frac {\frac {1}{4} (b c-a d) \left (8 a^2 c^2-8 b^2 c^2+24 a b c d-5 a^2 d^2+3 b^2 d^2\right )-4 (b c-a d)^2 (a c+b d) \tan (e+f x)-\frac {1}{4} d (b c-a d) \left (8 a b c-3 a^2 d+5 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 c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\int \frac {2 (b c-a d) \left (a^3 c^2-3 a b^2 c^2+6 a^2 b c d-2 b^3 c d-a^3 d^2+3 a b^2 d^2\right )-2 (b c-a d) \left (3 a^2 b c^2-b^3 c^2-2 a^3 c d+6 a b^2 c d-3 a^2 b d^2+b^3 d^2\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 (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\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}\\ &=-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {(c-i d)^2 \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)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^3}-\frac {\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\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 f}\\ &=-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {(c-i d)^2 \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 {(c+i d)^2 \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 {\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\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 f}\\ &=\frac {\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} \left (a^2+b^2\right )^3 \sqrt {b c-a d} f}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {(c-i d)^2 \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 {(c+i d)^2 \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 {(c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}+\frac {\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} \left (a^2+b^2\right )^3 \sqrt {b c-a d} f}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 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. \(2093\) vs. \(2(341)=682\).
time = 6.34, size = 2093, normalized size = 6.14 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b^2*(c + d*Tan[e + f*x])^(5/2))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((b*d*(c + d*Tan[
e + f*x])^(3/2))/(f*(a + b*Tan[e + f*x]))) + (2*(-1/2*(b*d*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(f*(a + b*Tan
[e + f*x])) - (2*(-((((I*Sqrt[c - I*d]*(b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 +
5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) + a*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c
*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d
)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a
*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2) - I*(a*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 +
(b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - b*((b^2*
(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c -
a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)
)/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2)))*ArcTanh[Sqrt[c +
d*Tan[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*(b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 +
(b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) + a*((b^2*(
b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c -
a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))
/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2) + I*(a*(b*c - a*d)*(
(3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*
c*d - b*d^2))/2) - b*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) +
(a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*
(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/
2)))/2)))*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*b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c -
a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)) + (a^2*d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*
(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2 + b^2*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c
*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d
)*(b*c^2 - 2*a*c*d - b*d^2))/2)))*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^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((
3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 +
b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))))/b))/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. \(1871\) vs. \(2(303)=606\).
time = 0.61, size = 1872, normalized size = 5.49

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1872\)
default	\(\text {Expression too large to display}\)	\(1872\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f*d^4*(1/d^4/(a^2+b^2)^3*(((3/8*a^4*b*d^2-a^3*c*d*b^2-1/4*a^2*b^3*d^2-c*d*a*b^4-5/8*b^5*d^2)*(c+d*tan(f*x+e)
)^(3/2)+1/8*d*(5*a^5*d^2-13*a^4*b*c*d+8*a^3*b^2*c^2+2*a^3*b^2*d^2-10*a^2*b^3*c*d+8*a*b^4*c^2-3*a*b^4*d^2+3*b^5
*c*d)*(c+d*tan(f*x+e))^(1/2))/((c+d*tan(f*x+e))*b+a*d-b*c)^2+1/8*(3*a^4*d^2-24*a^3*b*c*d+24*a^2*b^2*c^2-26*a^2
*b^2*d^2+40*a*b^3*c*d-8*b^4*c^2+3*b^4*d^2)/((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*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((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*(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^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)))))

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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))^(3/2)/(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))^(3/2)/(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 {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\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))**(3/2)/(a+b*tan(f*x+e))**3,x)

[Out]

Integral((c + d*tan(e + f*x))**(3/2)/(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))^(3/2)/(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 = 36.70, size = 2500, normalized size = 7.33 \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))^(3/2)/(a + b*tan(e + f*x))^3,x)

[Out]

(((c + d*tan(e + f*x))^(1/2)*(5*a^3*d^3 - 3*a*b^2*d^3 + 3*b^3*c*d^2 + 8*a*b^2*c^2*d - 13*a^2*b*c*d^2))/(4*(a^4
+ b^4 + 2*a^2*b^2)) - (b*(c + d*tan(e + f*x))^(3/2)*(5*b^2*d^2 - 3*a^2*d^2 + 8*a*b*c*d))/(4*(a^4 + b^4 + 2*a^
2*b^2)))/(a^2*d^2*f - (2*b^2*c*f - 2*a*b*d*f)*(c + d*tan(e + f*x)) + b^2*c^2*f + b^2*f*(c + d*tan(e + f*x))^2
- 2*a*b*c*d*f) - atan(((((1036*a*b^15*d^15*f^2 - 36*a^15*b*d^15*f^2 - 604*b^16*c*d^14*f^2 - 8988*a^3*b^13*d^15
*f^2 + 6044*a^5*b^11*d^15*f^2 + 34388*a^7*b^9*d^15*f^2 + 10596*a^9*b^7*d^15*f^2 - 6676*a^11*b^5*d^15*f^2 + 101
2*a^13*b^3*d^15*f^2 + 932*b^16*c^3*d^12*f^2 + 1344*b^16*c^5*d^10*f^2 - 192*b^16*c^7*d^8*f^2 - 30836*a^2*b^14*c
^3*d^12*f^2 - 48000*a^2*b^14*c^5*d^10*f^2 + 5248*a^2*b^14*c^7*d^8*f^2 + 95076*a^3*b^13*c^2*d^13*f^2 + 57600*a^
3*b^13*c^4*d^11*f^2 - 46464*a^3*b^13*c^6*d^9*f^2 + 17172*a^4*b^12*c^3*d^12*f^2 + 69696*a^4*b^12*c^5*d^10*f^2 -
5696*a^4*b^12*c^7*d^8*f^2 + 47004*a^5*b^11*c^2*d^13*f^2 + 10944*a^5*b^11*c^4*d^11*f^2 - 30016*a^5*b^11*c^6*d^
9*f^2 + 85404*a^6*b^10*c^3*d^12*f^2 + 169344*a^6*b^10*c^5*d^10*f^2 - 17664*a^6*b^10*c^7*d^8*f^2 - 171180*a^7*b
^9*c^2*d^13*f^2 - 119808*a^7*b^9*c^4*d^11*f^2 + 85760*a^7*b^9*c^6*d^9*f^2 - 4308*a^8*b^8*c^3*d^12*f^2 - 49728*
a^8*b^8*c^5*d^10*f^2 + 3776*a^8*b^8*c^7*d^8*f^2 - 50972*a^9*b^7*c^2*d^13*f^2 - 24768*a^9*b^7*c^4*d^11*f^2 + 36
800*a^9*b^7*c^6*d^9*f^2 - 35356*a^10*b^6*c^3*d^12*f^2 - 80512*a^10*b^6*c^5*d^10*f^2 + 10368*a^10*b^6*c^7*d^8*f
^2 + 55916*a^11*b^5*c^2*d^13*f^2 + 37632*a^11*b^5*c^4*d^11*f^2 - 24960*a^11*b^5*c^6*d^9*f^2 + 6428*a^12*b^4*c^
3*d^12*f^2 + 19648*a^12*b^4*c^5*d^10*f^2 + 64*a^12*b^4*c^7*d^8*f^2 - 4876*a^13*b^3*c^2*d^13*f^2 - 6080*a^13*b^
3*c^4*d^11*f^2 - 192*a^13*b^3*c^6*d^9*f^2 + 1012*a^14*b^2*c^3*d^12*f^2 + 128*a^14*b^2*c^5*d^10*f^2 - 11380*a*b
^15*c^2*d^13*f^2 - 4672*a*b^15*c^4*d^11*f^2 + 7744*a*b^15*c^6*d^9*f^2 + 22412*a^2*b^14*c*d^14*f^2 - 58220*a^4*
b^12*c*d^14*f^2 - 101604*a^6*b^10*c*d^14*f^2 + 49196*a^8*b^8*c*d^14*f^2 + 55524*a^10*b^6*c*d^14*f^2 - 13156*a^
12*b^4*c*d^14*f^2 + 884*a^14*b^2*c*d^14*f^2 - 36*a^15*b*c^2*d^13*f^2)/(2*(a^16*f^5 + b^16*f^5 + 8*a^2*b^14*f^5
+ 28*a^4*b^12*f^5 + 56*a^6*b^10*f^5 + 70*a^8*b^8*f^5 + 56*a^10*b^6*f^5 + 28*a^12*b^4*f^5 + 8*a^14*b^2*f^5)) +
(((8448*a^4*b^18*d^12*f^4 - 640*a^2*b^20*d^12*f^4 - 384*b^22*d^12*f^4 + 44544*a^6*b^16*d^12*f^4 + 102144*a^8*
b^14*d^12*f^4 + 134400*a^10*b^12*d^12*f^4 + 107520*a^12*b^10*d^12*f^4 + 50688*a^14*b^8*d^12*f^4 + 11904*a^16*b
^6*d^12*f^4 + 384*a^18*b^4*d^12*f^4 - 256*a^20*b^2*d^12*f^4 + 384*b^22*c^2*d^10*f^4 + 768*b^22*c^4*d^8*f^4 + 4
224*a^2*b^20*c^2*d^10*f^4 + 4864*a^2*b^20*c^4*d^8*f^4 - 27136*a^3*b^19*c^3*d^9*f^4 + 19712*a^4*b^18*c^2*d^10*f
^4 + 11264*a^4*b^18*c^4*d^8*f^4 - 88064*a^5*b^17*c^3*d^9*f^4 + 51712*a^6*b^16*c^2*d^10*f^4 + 7168*a^6*b^16*c^4
*d^8*f^4 - 157696*a^7*b^15*c^3*d^9*f^4 + 84224*a^8*b^14*c^2*d^10*f^4 - 17920*a^8*b^14*c^4*d^8*f^4 - 164864*a^9
*b^13*c^3*d^9*f^4 + 87808*a^10*b^12*c^2*d^10*f^4 - 46592*a^10*b^12*c^4*d^8*f^4 - 93184*a^11*b^11*c^3*d^9*f^4 +
57344*a^12*b^10*c^2*d^10*f^4 - 50176*a^12*b^10*c^4*d^8*f^4 - 14336*a^13*b^9*c^3*d^9*f^4 + 20992*a^14*b^8*c^2*
d^10*f^4 - 29696*a^14*b^8*c^4*d^8*f^4 + 14336*a^15*b^7*c^3*d^9*f^4 + 2432*a^16*b^6*c^2*d^10*f^4 - 9472*a^16*b^
6*c^4*d^8*f^4 + 8704*a^17*b^5*c^3*d^9*f^4 - 896*a^18*b^4*c^2*d^10*f^4 - 1280*a^18*b^4*c^4*d^8*f^4 + 1536*a^19*
b^3*c^3*d^9*f^4 - 256*a^20*b^2*c^2*d^10*f^4 - 3584*a*b^21*c*d^11*f^4 - 3584*a*b^21*c^3*d^9*f^4 - 27136*a^3*b^1
9*c*d^11*f^4 - 88064*a^5*b^17*c*d^11*f^4 - 157696*a^7*b^15*c*d^11*f^4 - 164864*a^9*b^13*c*d^11*f^4 - 93184*a^1
1*b^11*c*d^11*f^4 - 14336*a^13*b^9*c*d^11*f^4 + 14336*a^15*b^7*c*d^11*f^4 + 8704*a^17*b^5*c*d^11*f^4 + 1536*a^
19*b^3*c*d^11*f^4)/(2*(a^16*f^5 + b^16*f^5 + 8*a^2*b^14*f^5 + 28*a^4*b^12*f^5 + 56*a^6*b^10*f^5 + 70*a^8*b^8*f
^5 + 56*a^10*b^6*f^5 + 28*a^12*b^4*f^5 + 8*a^14*b^2*f^5)) + ((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)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)*(16*a^12*f^4 + 1
6*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2) +
4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 6
0*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^
2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 1
5*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2)*(512*b^25*d^10*f^4 + 4608*a^2*b^23*d
^10*f^4 + 17920*a^4*b^21*d^10*f^4 + 38400*a^6*b^19*d^10*f^4 + 46080*a^8*b^17*d^10*f^4 + 21504*a^10*b^15*d^10*f
^4 - 21504*a^12*b^13*d^10*f^4 - 46080*a^14*b^11*d^10*f^4 - 38400*a^16*b^9*d^10*f^4 - 17920*a^18*b^7*d^10*f^4 -
4608*a^20*b^5*d^10*f^4 - 512*a^22*b^3*d^10*f^4...

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