3.13.39 \(\int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^2} \, dx\) [1239]

Optimal. Leaf size=239 \[ -\frac {i (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 f}+\frac {i (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}-\frac {\sqrt {b c-a d} \left (4 a b c-a^2 d+3 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} \left (a^2+b^2\right )^2 f}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))} \]

[Out]

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

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Rubi [A]
time = 0.62, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3648, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac {\sqrt {b c-a d} \left (a^2 (-d)+4 a b c+3 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} f \left (a^2+b^2\right )^2}-\frac {i (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2}+\frac {i (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f) + (I*(c + I*d)^(3/2)*Ar
cTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) - (Sqrt[b*c - a*d]*(4*a*b*c - a^2*d + 3*b^2*d)*
ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)^2*f) - ((b*c - a*d)*Sqrt[c +
d*Tan[e + f*x]])/((a^2 + b^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 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))^2} \, dx &=-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {\int \frac {\frac {1}{2} \left (-3 b c d-a \left (2 c^2-d^2\right )\right )-\left (2 a c d-b \left (c^2-d^2\right )\right ) \tan (e+f x)+\frac {1}{2} d (b c-a d) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{a^2+b^2}\\ &=-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {\int \frac {-(a c+b c-a d+b d) (a c-b c+a d+b d)+2 (b c-a d) (a c+b d) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right )^2}+\frac {\left ((b c-a d) \left (4 a b c-a^2 d+3 b^2 d\right )\right ) \int \frac {1+\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}\\ &=-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) 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)^2}+\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)^2}+\frac {\left ((b c-a d) \left (4 a b c-a^2 d+3 b^2 d\right )\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f}\\ &=-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {\left (i (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 (a-i b)^2 f}-\frac {\left (i (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 (a+i b)^2 f}+\frac {\left ((b c-a d) \left (4 a b c-a^2 d+3 b^2 d\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 )}{\left (a^2+b^2\right )^2 d f}\\ &=-\frac {\sqrt {b c-a d} \left (4 a b c-a^2 d+3 b^2 d\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 )^2 f}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) 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)^2 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)^2 d f}\\ &=-\frac {i (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 f}+\frac {i (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}-\frac {\sqrt {b c-a d} \left (4 a b c-a^2 d+3 b^2 d\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 )^2 f}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 3.29, size = 316, normalized size = 1.32 \begin {gather*} \frac {-\frac {4 \left (\frac {3}{4} i (a+i b)^2 b^2 (c-i d)^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )-\frac {3}{4} i (a-i b)^2 b^2 (c+i d)^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+\frac {3}{4} b^{3/2} (b c-a d)^{3/2} \left (4 a b c-a^2 d+3 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )\right )}{b^2 \left (a^2+b^2\right )}+3 d (b c-a d) \sqrt {c+d \tan (e+f x)}+3 b d (c+d \tan (e+f x))^{3/2}-\frac {3 b^2 (c+d \tan (e+f x))^{5/2}}{a+b \tan (e+f x)}}{3 \left (a^2+b^2\right ) (b c-a d) f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1412\) vs. \(2(207)=414\).
time = 0.54, size = 1413, normalized size = 5.91 Too large to display

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))^2,x,method=_RETURNVERBOSE)

[Out]

2/f*d^3*((a*d-b*c)/d^3/(a^2+b^2)^2*((1/2*a^2*d+1/2*b^2*d)*(c+d*tan(f*x+e))^(1/2)/((c+d*tan(f*x+e))*b+a*d-b*c)+
1/2*(a^2*d-4*a*b*c-3*b^2*d)/((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^3/(a
^2+b^2)^2*(1/4/d*(1/2*((c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)*a*b*d-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2+(
2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^2-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2
*c^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^2)*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^2*d^2+4*(c^2+d^2)^(1/2)*a*b*c*d+2*(c^2+d^2)^(1/2)*b^2*d^2+1/2*((
c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d-(c^2+
d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2+(2*(c^2+d^2)^(1/2)+2*c)^(
1/2)*a^2*d^2-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2-(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*b^2*d^2)*(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^2*c-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)
+2*c)^(1/2)*b^2*c+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^2+4*(2*(c^2+d^2)^(
1/2)+2*c)^(1/2)*a*b*c*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^2)*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^2*d^2+4
*(c^2+d^2)^(1/2)*a*b*c*d+2*(c^2+d^2)^(1/2)*b^2*d^2-1/2*(-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c-2
*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c+(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^2+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c*d
-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^2)*(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))^(3/2)/(a+b*tan(f*x+e))^2,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))^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 (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, 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))**2,x)

[Out]

Integral((c + d*tan(e + f*x))**(3/2)/(a + b*tan(e + f*x))**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((c+d*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^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 = 12.95, size = 2500, normalized size = 10.46 \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))^2,x)

[Out]

(atan(((((16*(c + d*tan(e + f*x))^(1/2)*(2*b^9*d^16 + a^8*b*d^16 - 5*a^2*b^7*d^16 + 17*a^4*b^5*d^16 - 7*a^6*b^
3*d^16 - b^9*c^2*d^14 + 66*b^9*c^4*d^12 - b^9*c^6*d^10 + 2*b^9*c^8*d^8 - 204*a*b^8*c^3*d^13 + 234*a*b^8*c^5*d^
11 - 24*a*b^8*c^7*d^9 - 126*a^3*b^6*c*d^15 + 94*a^5*b^4*c*d^15 - 18*a^7*b^2*c*d^15 - 6*a^8*b*c^2*d^14 + a^8*b*
c^4*d^12 + 277*a^2*b^7*c^2*d^14 - 715*a^2*b^7*c^4*d^12 + 367*a^2*b^7*c^6*d^10 - 12*a^2*b^7*c^8*d^8 + 892*a^3*b
^6*c^3*d^13 - 998*a^3*b^6*c^5*d^11 + 192*a^3*b^6*c^7*d^9 - 457*a^4*b^5*c^2*d^14 + 1173*a^4*b^5*c^4*d^12 - 495*
a^4*b^5*c^6*d^10 + 18*a^4*b^5*c^8*d^8 - 628*a^5*b^4*c^3*d^13 + 550*a^5*b^4*c^5*d^11 - 40*a^5*b^4*c^7*d^9 + 155
*a^6*b^3*c^2*d^14 - 285*a^6*b^3*c^4*d^12 + 33*a^6*b^3*c^6*d^10 + 68*a^7*b^2*c^3*d^13 - 10*a^7*b^2*c^5*d^11 + 1
8*a*b^8*c*d^15))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4) + (((16*(78*a^2*b^9*d^15*
f^2 - 2*a^10*b*d^15*f^2 - 8*a^4*b^7*d^15*f^2 - 60*a^6*b^5*d^15*f^2 + 24*a^8*b^3*d^15*f^2 + 50*b^11*c^2*d^13*f^
2 + 22*b^11*c^4*d^11*f^2 - 28*b^11*c^6*d^9*f^2 - 546*a^2*b^9*c^2*d^13*f^2 - 296*a^2*b^9*c^4*d^11*f^2 + 328*a^2
*b^9*c^6*d^9*f^2 - 240*a^3*b^8*c^3*d^12*f^2 - 544*a^3*b^8*c^5*d^10*f^2 + 64*a^3*b^8*c^7*d^8*f^2 + 108*a^4*b^7*
c^2*d^13*f^2 + 148*a^4*b^7*c^4*d^11*f^2 + 32*a^4*b^7*c^6*d^9*f^2 - 296*a^5*b^6*c^3*d^12*f^2 - 456*a^5*b^6*c^5*
d^10*f^2 + 96*a^5*b^6*c^7*d^8*f^2 + 580*a^6*b^5*c^2*d^13*f^2 + 312*a^6*b^5*c^4*d^11*f^2 - 328*a^6*b^5*c^6*d^9*
f^2 + 144*a^7*b^4*c^3*d^12*f^2 + 352*a^7*b^4*c^5*d^10*f^2 - 126*a^8*b^3*c^2*d^13*f^2 - 154*a^8*b^3*c^4*d^11*f^
2 - 4*a^8*b^3*c^6*d^9*f^2 + 36*a^9*b^2*c^3*d^12*f^2 + 4*a^9*b^2*c^5*d^10*f^2 - 128*a*b^10*c*d^14*f^2 + 164*a*b
^10*c^3*d^12*f^2 + 260*a*b^10*c^5*d^10*f^2 - 32*a*b^10*c^7*d^8*f^2 + 368*a^3*b^8*c*d^14*f^2 + 256*a^5*b^6*c*d^
14*f^2 - 208*a^7*b^4*c*d^14*f^2 + 32*a^9*b^2*c*d^14*f^2 - 2*a^10*b*c^2*d^13*f^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b
^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - ((-b*(a*d - b*c))^(1/2)*((16*(c + d*tan(e + f*x))^(1/2)*(44*b^13*c*d
^12*f^2 - 60*a*b^12*d^13*f^2 + 20*a^3*b^10*d^13*f^2 + 168*a^5*b^8*d^13*f^2 + 40*a^7*b^6*d^13*f^2 - 44*a^9*b^4*
d^13*f^2 + 4*a^11*b^2*d^13*f^2 + 92*b^13*c^3*d^10*f^2 - 20*b^13*c^5*d^8*f^2 - 76*a^2*b^11*c^3*d^10*f^2 + 116*a
^2*b^11*c^5*d^8*f^2 - 288*a^3*b^10*c^2*d^11*f^2 + 396*a^3*b^10*c^4*d^9*f^2 - 456*a^4*b^9*c^3*d^10*f^2 + 216*a^
4*b^9*c^5*d^8*f^2 - 720*a^5*b^8*c^2*d^11*f^2 + 8*a^5*b^8*c^4*d^9*f^2 - 504*a^6*b^7*c^3*d^10*f^2 + 8*a^6*b^7*c^
5*d^8*f^2 - 224*a^7*b^6*c^2*d^11*f^2 + 120*a^7*b^6*c^4*d^9*f^2 - 404*a^8*b^5*c^3*d^10*f^2 - 68*a^8*b^5*c^5*d^8
*f^2 + 200*a^9*b^4*c^2*d^11*f^2 + 180*a^9*b^4*c^4*d^9*f^2 - 188*a^10*b^3*c^3*d^10*f^2 + 4*a^10*b^3*c^5*d^8*f^2
+ 64*a^11*b^2*c^2*d^11*f^2 - 4*a^11*b^2*c^4*d^9*f^2 - 4*a^12*b*c*d^12*f^2 + 72*a*b^12*c^2*d^11*f^2 + 324*a*b^
12*c^4*d^9*f^2 - 280*a^2*b^11*c*d^12*f^2 - 348*a^4*b^9*c*d^12*f^2 + 304*a^6*b^7*c*d^12*f^2 + 308*a^8*b^5*c*d^1
2*f^2 - 24*a^10*b^3*c*d^12*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4) + ((-b*(a
*d - b*c))^(1/2)*((16*(40*b^15*c*d^11*f^4 - 40*a*b^14*d^12*f^4 - 192*a^3*b^12*d^12*f^4 - 360*a^5*b^10*d^12*f^4
- 320*a^7*b^8*d^12*f^4 - 120*a^9*b^6*d^12*f^4 + 8*a^13*b^2*d^12*f^4 + 40*b^15*c^3*d^9*f^4 + 160*a^2*b^13*c^3*
d^9*f^4 - 32*a^3*b^12*c^2*d^10*f^4 + 160*a^3*b^12*c^4*d^8*f^4 + 200*a^4*b^11*c^3*d^9*f^4 - 40*a^5*b^10*c^2*d^1
0*f^4 + 320*a^5*b^10*c^4*d^8*f^4 + 320*a^7*b^8*c^4*d^8*f^4 - 200*a^8*b^7*c^3*d^9*f^4 + 40*a^9*b^6*c^2*d^10*f^4
+ 160*a^9*b^6*c^4*d^8*f^4 - 160*a^10*b^5*c^3*d^9*f^4 + 32*a^11*b^4*c^2*d^10*f^4 + 32*a^11*b^4*c^4*d^8*f^4 - 4
0*a^12*b^3*c^3*d^9*f^4 + 8*a^13*b^2*c^2*d^10*f^4 - 8*a*b^14*c^2*d^10*f^4 + 32*a*b^14*c^4*d^8*f^4 + 160*a^2*b^1
3*c*d^11*f^4 + 200*a^4*b^11*c*d^11*f^4 - 200*a^8*b^7*c*d^11*f^4 - 160*a^10*b^5*c*d^11*f^4 - 40*a^12*b^3*c*d^11
*f^4))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - (8*(-b*(a*d - b*c))^(1/2)*(c + d*
tan(e + f*x))^(1/2)*(3*b^2*d - a^2*d + 4*a*b*c)*(32*b^17*d^10*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*
f^4 + 160*a^6*b^11*d^10*f^4 - 160*a^8*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b
^3*d^10*f^4 + 48*b^17*c^2*d^8*f^4 + 272*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8
*f^4 + 400*a^8*b^9*c^2*d^8*f^4 + 48*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 +
16*a*b^16*c*d^9*f^4 + 112*a^3*b^14*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*
c*d^9*f^4 + 336*a^11*b^6*c*d^9*f^4 + 112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^9*f^4))/((b^5*f + 2*a^2*b^3*f +
a^4*b*f)*(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4)))*(3*b^2*d - a^2*d + 4*a*b*c))/(2
*(b^5*f + 2*a^2*b^3*f + a^4*b*f)))*(3*b^2*d - a^2*d + 4*a*b*c))/(2*(b^5*f + 2*a^2*b^3*f + a^4*b*f)))*(-b*(a*d
- b*c))^(1/2)*(3*b^2*d - a^2*d + 4*a*b*c))/(2*(b^5*f + 2*a^2*b^3*f + a^4*b*f)))*(-b*(a*d - b*c))^(1/2)*(3*b^2*
d - a^2*d + 4*a*b*c)*1i)/(2*(b^5*f + 2*a^2*b^3*f + a^4*b*f)) + (((16*(c + d*tan(e + f*x))^(1/2)*(2*b^9*d^16 +
a^8*b*d^16 - 5*a^2*b^7*d^16 + 17*a^4*b^5*d^16 -...

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