3.105 \(\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\)

Optimal. Leaf size=353 \[ \frac{2 \sqrt{c+d \tan (e+f x)} \left (A \left (2 a c d+b \left (c^2-d^2\right )\right )+a \left (B c^2-B d^2-2 c C d\right )-b \left (2 B c d+c^2 C-C d^2\right )\right )}{f}+\frac{2 (a B+A b-b C) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 (c+d \tan (e+f x))^{3/2} (a A d+a B c-a C d+A b c-b B d-b c C)}{3 f}-\frac{(b+i a) (c-i d)^{5/2} (A-i B-C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}+\frac{(-b+i a) (c+i d)^{5/2} (A+i B-C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}-\frac{2 (-9 a C d-9 b B d+2 b c C) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f} \]

[Out]

-(((I*a + b)*(A - I*B - C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((I*a - b)*(A
+ I*B - C)*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(a*(B*c^2 - 2*c*C*d - B*d^
2) - b*(c^2*C + 2*B*c*d - C*d^2) + A*(2*a*c*d + b*(c^2 - d^2)))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(A*b*c + a*B*
c - b*c*C + a*A*d - b*B*d - a*C*d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*(A*b + a*B - b*C)*(c + d*Tan[e + f*x
])^(5/2))/(5*f) - (2*(2*b*c*C - 9*b*B*d - 9*a*C*d)*(c + d*Tan[e + f*x])^(7/2))/(63*d^2*f) + (2*b*C*Tan[e + f*x
]*(c + d*Tan[e + f*x])^(7/2))/(9*d*f)

________________________________________________________________________________________

Rubi [A]  time = 1.21232, antiderivative size = 351, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 7, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {3637, 3630, 3528, 3539, 3537, 63, 208} \[ \frac{2 \sqrt{c+d \tan (e+f x)} \left (2 a A c d+a B \left (c^2-d^2\right )-2 a c C d+A b \left (c^2-d^2\right )-b \left (2 B c d+c^2 C-C d^2\right )\right )}{f}+\frac{2 (a B+A b-b C) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 (c+d \tan (e+f x))^{3/2} (a A d+a B c-a C d+A b c-b B d-b c C)}{3 f}-\frac{(b+i a) (c-i d)^{5/2} (A-i B-C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}+\frac{(-b+i a) (c+i d)^{5/2} (A+i B-C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}-\frac{2 (-9 a C d-9 b B d+2 b c C) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]

[Out]

-(((I*a + b)*(A - I*B - C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((I*a - b)*(A
+ I*B - C)*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(2*a*A*c*d - 2*a*c*C*d + A
*b*(c^2 - d^2) + a*B*(c^2 - d^2) - b*(c^2*C + 2*B*c*d - C*d^2))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(A*b*c + a*B*
c - b*c*C + a*A*d - b*B*d - a*C*d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*(A*b + a*B - b*C)*(c + d*Tan[e + f*x
])^(5/2))/(5*f) - (2*(2*b*c*C - 9*b*B*d - 9*a*C*d)*(c + d*Tan[e + f*x])^(7/2))/(63*d^2*f) + (2*b*C*Tan[e + f*x
]*(c + d*Tan[e + f*x])^(7/2))/(9*d*f)

Rule 3637

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]

Rule 3630

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 3528

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 3539

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 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^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 63

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac{2 \int (c+d \tan (e+f x))^{5/2} \left (\frac{1}{2} (2 b c C-9 a A d)-\frac{9}{2} (A b+a B-b C) d \tan (e+f x)+\frac{1}{2} (2 b c C-9 b B d-9 a C d) \tan ^2(e+f x)\right ) \, dx}{9 d}\\ &=-\frac{2 (2 b c C-9 b B d-9 a C d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac{2 \int (c+d \tan (e+f x))^{5/2} \left (\frac{9}{2} (b B-a (A-C)) d-\frac{9}{2} (A b+a B-b C) d \tan (e+f x)\right ) \, dx}{9 d}\\ &=\frac{2 (A b+a B-b C) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac{2 (2 b c C-9 b B d-9 a C d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac{2 \int (c+d \tan (e+f x))^{3/2} \left (\frac{9}{2} d (b B c+b (A-C) d-a (A c-c C-B d))-\frac{9}{2} d (A b c+a B c-b c C+a A d-b B d-a C d) \tan (e+f x)\right ) \, dx}{9 d}\\ &=\frac{2 (A b c+a B c-b c C+a A d-b B d-a C d) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 (A b+a B-b C) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac{2 (2 b c C-9 b B d-9 a C d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac{2 \int \sqrt{c+d \tan (e+f x)} \left (\frac{9}{2} d \left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-\frac{9}{2} d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \tan (e+f x)\right ) \, dx}{9 d}\\ &=\frac{2 \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 (A b c+a B c-b c C+a A d-b B d-a C d) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 (A b+a B-b C) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac{2 (2 b c C-9 b B d-9 a C d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac{2 \int \frac{\frac{9}{2} d \left (a \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )+b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-\frac{9}{2} d \left (A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )\right ) \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{9 d}\\ &=\frac{2 \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 (A b c+a B c-b c C+a A d-b B d-a C d) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 (A b+a B-b C) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac{2 (2 b c C-9 b B d-9 a C d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac{1}{2} \left ((a-i b) (A-i B-C) (c-i d)^3\right ) \int \frac{1+i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx+\frac{1}{2} \left ((a+i b) (A+i B-C) (c+i d)^3\right ) \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx\\ &=\frac{2 \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 (A b c+a B c-b c C+a A d-b B d-a C d) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 (A b+a B-b C) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac{2 (2 b c C-9 b B d-9 a C d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac{\left ((a-i b) (i A+B-i C) (c-i d)^3\right ) \operatorname{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) (A+i B-C) (c+i d)^3\right ) \operatorname{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 (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 (A b c+a B c-b c C+a A d-b B d-a C d) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 (A b+a B-b C) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac{2 (2 b c C-9 b B d-9 a C d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac{\left ((a-i b) (A-i B-C) (c-i d)^3\right ) \operatorname{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) (A+i B-C) (c+i d)^3\right ) \operatorname{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) (A-i B-C) (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}+\frac{(i a-b) (A+i B-C) (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f}+\frac{2 \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 (A b c+a B c-b c C+a A d-b B d-a C d) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 (A b+a B-b C) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac{2 (2 b c C-9 b B d-9 a C d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f}\\ \end{align*}

Mathematica [A]  time = 4.99884, size = 324, normalized size = 0.92 \[ \frac{\frac{63}{2} i d (a-i b) (A-i B-C) \left (\frac{2}{5} (c+d \tan (e+f x))^{5/2}+\frac{2}{3} (c-i d) \left (\sqrt{c+d \tan (e+f x)} (4 c+d \tan (e+f x)-3 i d)-3 (c-i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )\right )\right )-\frac{63}{2} i d (a+i b) (A+i B-C) \left (\frac{2}{5} (c+d \tan (e+f x))^{5/2}+\frac{2}{3} (c+i d) \left (\sqrt{c+d \tan (e+f x)} (4 c+d \tan (e+f x)+3 i d)-3 (c+i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )\right )\right )+\frac{2 (9 a C d+9 b B d-2 b c C) (c+d \tan (e+f x))^{7/2}}{d}+14 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{63 d f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]

[Out]

((2*(-2*b*c*C + 9*b*B*d + 9*a*C*d)*(c + d*Tan[e + f*x])^(7/2))/d + 14*b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(7
/2) + ((63*I)/2)*(a - I*b)*(A - I*B - C)*d*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (2*(c - I*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) - ((63*I)/2)*(a + I*b)*(A + I*B - C)*d*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (2*(c + I*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))/(63*d*f)

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Maple [B]  time = 0.164, size = 7402, normalized size = 21. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x)

[Out]

result too large to display

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (e + f x \right )}\right ) \left (c + d \tan{\left (e + f x \right )}\right )^{\frac{5}{2}} \left (A + B \tan{\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))*(c+d*tan(f*x+e))**(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2),x)

[Out]

Integral((a + b*tan(e + f*x))*(c + d*tan(e + f*x))**(5/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )}{\left (b \tan \left (f x + e\right ) + a\right )}{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="giac")

[Out]

integrate((C*tan(f*x + e)^2 + B*tan(f*x + e) + A)*(b*tan(f*x + e) + a)*(d*tan(f*x + e) + c)^(5/2), x)