3.4.69 \(\int \tan ^2(e+f x) (a+b \tan ^2(e+f x))^p \, dx\) [369]

Optimal. Leaf size=83 \[ \frac {F_1\left (\frac {3}{2};1,-p;\frac {5}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{3 f} \]

[Out]

1/3*AppellF1(3/2,1,-p,5/2,-tan(f*x+e)^2,-b*tan(f*x+e)^2/a)*tan(f*x+e)^3*(a+b*tan(f*x+e)^2)^p/f/((1+b*tan(f*x+e
)^2/a)^p)

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Rubi [A]
time = 0.06, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 525, 524} \begin {gather*} \frac {\tan ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {3}{2};1,-p;\frac {5}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]

[Out]

(AppellF1[3/2, 1, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p)/
(3*f*(1 + (b*Tan[e + f*x]^2)/a)^p)

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

\begin {align*} \int \tan ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\frac {\text {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left (\left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {x^2 \left (1+\frac {b x^2}{a}\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {F_1\left (\frac {3}{2};1,-p;\frac {5}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{3 f}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1992\) vs. \(2(83)=166\).
time = 13.92, size = 1992, normalized size = 24.00 \begin {gather*} \frac {\tan ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{2 p} \left (\, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a}\right ) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}+\frac {3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cos ^2(e+f x)}{-3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (-b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)}\right )}{f \left (2 b p \sec ^2(e+f x) \tan ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{-1+p} \left (\, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a}\right ) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}+\frac {3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cos ^2(e+f x)}{-3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (-b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)}\right )+\sec ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a}\right ) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}+\frac {3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cos ^2(e+f x)}{-3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (-b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)}\right )+\tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (-\frac {2 b p \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a}\right ) \sec ^2(e+f x) \tan (e+f x) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-1-p}}{a}-\frac {6 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cos (e+f x) \sin (e+f x)}{-3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (-b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)}+\frac {3 a \cos ^2(e+f x) \left (\frac {2 b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac {2}{3} F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)\right )}{-3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (-b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)}+\csc (e+f x) \sec (e+f x) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p} \left (-\, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a}\right )+\left (1+\frac {b \tan ^2(e+f x)}{a}\right )^p\right )-\frac {3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cos ^2(e+f x) \left (4 \left (-b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \sec ^2(e+f x) \tan (e+f x)-3 a \left (\frac {2 b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac {2}{3} F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)\right )+2 \tan ^2(e+f x) \left (-b p \left (-\frac {6}{5} F_1\left (\frac {5}{2};1-p,2;\frac {7}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)-\frac {6 b (1-p) F_1\left (\frac {5}{2};2-p,1;\frac {7}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{5 a}\right )+a \left (\frac {6 b p F_1\left (\frac {5}{2};1-p,2;\frac {7}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{5 a}-\frac {12}{5} F_1\left (\frac {5}{2};-p,3;\frac {7}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)\right )\right )\right )}{\left (-3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (-b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)\right ){}^2}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]

[Out]

(Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(2*p)*(Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)]/(1 + (b
*Tan[e + f*x]^2)/a)^p + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^
2)/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1
, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[
e + f*x]^2])*Tan[e + f*x]^2)))/(f*(2*b*p*Sec[e + f*x]^2*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(-1 + p)*(Hyperg
eometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)]/(1 + (b*Tan[e + f*x]^2)/a)^p + (3*a*AppellF1[1/2, -p, 1, 3/
2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]
^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) +
a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)) + Sec[e + f*x]^2*(a +
b*Tan[e + f*x]^2)^p*(Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)]/(1 + (b*Tan[e + f*x]^2)/a)^p + (
3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2)/(-3*a*AppellF1[1/2, -p
, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^
2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]
^2)) + Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p*((-2*b*p*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)]
*Sec[e + f*x]^2*Tan[e + f*x]*(1 + (b*Tan[e + f*x]^2)/a)^(-1 - p))/a - (6*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[
e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x])/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^
2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a
*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*a*Cos[e + f*x]^2*((
2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a
) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3))/(-
3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2
, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f
*x]^2])*Tan[e + f*x]^2) + (Csc[e + f*x]*Sec[e + f*x]*(-Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/a)
] + (1 + (b*Tan[e + f*x]^2)/a)^p))/(1 + (b*Tan[e + f*x]^2)/a)^p - (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e +
f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2*(4*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -T
an[e + f*x]^2]) + a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e
+ f*x] - 3*a*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan
[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e
+ f*x])/3) + 2*Tan[e + f*x]^2*(-(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]
^2]*Sec[e + f*x]^2*Tan[e + f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e
+ f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a))) + a*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/
a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(5*a) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/a
), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/5))))/(-3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a)
, -Tan[e + f*x]^2] + 2*(-(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]) + a*Appe
llF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2)))

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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (\tan ^{2}\left (f x +e \right )\right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x)

[Out]

int(tan(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x)

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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(tan(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^2, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x, algorithm="fricas")

[Out]

integral((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{p} \tan ^{2}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)**2*(a+b*tan(f*x+e)**2)**p,x)

[Out]

Integral((a + b*tan(e + f*x)**2)**p*tan(e + f*x)**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e + f*x)^2*(a + b*tan(e + f*x)^2)^p,x)

[Out]

int(tan(e + f*x)^2*(a + b*tan(e + f*x)^2)^p, x)

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