3.13.4 \(\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx\) [1204]

Optimal. Leaf size=219 \[ -\left (\left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right ) x\right )-\frac {\left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) \log (\cos (e+f x))}{f}+\frac {2 d (b c+a d) (a c-b d) \tan (e+f x)}{f}+\frac {\left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3624, 3609, 3606, 3556} \begin {gather*} -\frac {\left (a^2 \left (3 c^2 d-d^3\right )+2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}-x \left (-\left (a^2 \left (c^3-3 c d^2\right )\right )+2 a b d \left (3 c^2-d^2\right )+b^2 c \left (c^2-3 d^2\right )\right )+\frac {\left (a^2 d+2 a b c-b^2 d\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {2 d (a d+b c) (a c-b d) \tan (e+f x)}{f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*c*(c^2 - 3*d^2) + 2*a*b*d*(3*c^2 - d^2) - a^2*(c^3 - 3*c*d^2))*x) - ((2*a*b*c*(c^2 - 3*d^2) - b^2*d*(3*
c^2 - d^2) + a^2*(3*c^2*d - d^3))*Log[Cos[e + f*x]])/f + (2*d*(b*c + a*d)*(a*c - b*d)*Tan[e + f*x])/f + ((2*a*
b*c + a^2*d - b^2*d)*(c + d*Tan[e + f*x])^2)/(2*f) + (2*a*b*(c + d*Tan[e + f*x])^3)/(3*f) + (b^2*(c + d*Tan[e
+ f*x])^4)/(4*d*f)

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3606

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[b*d*(Tan[e + f*x]/f), x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

Rule 3609

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 3624

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[
d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e
+ f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] &&  !LeQ[m, -1] &&  !(EqQ[m, 2] && EqQ
[a, 0])

Rubi steps

\begin {align*} \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx &=\frac {b^2 (c+d \tan (e+f x))^4}{4 d f}+\int \left (a^2-b^2+2 a b \tan (e+f x)\right ) (c+d \tan (e+f x))^3 \, dx\\ &=\frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f}+\int (c+d \tan (e+f x))^2 \left (a^2 c-b^2 c-2 a b d+\left (2 a b c+a^2 d-b^2 d\right ) \tan (e+f x)\right ) \, dx\\ &=\frac {\left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f}+\int (c+d \tan (e+f x)) ((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)) \, dx\\ &=-\left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right ) x+\frac {2 d (b c+a d) (a c-b d) \tan (e+f x)}{f}+\frac {\left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f}+\left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) \int \tan (e+f x) \, dx\\ &=-\left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right ) x-\frac {\left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) \log (\cos (e+f x))}{f}+\frac {2 d (b c+a d) (a c-b d) \tan (e+f x)}{f}+\frac {\left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.30, size = 221, normalized size = 1.01 \begin {gather*} \frac {3 b^2 (c+d \tan (e+f x))^4-6 \left (2 a b c-a^2 d+b^2 d\right ) \left ((i c-d)^3 \log (i-\tan (e+f x))-(i c+d)^3 \log (i+\tan (e+f x))+6 c d^2 \tan (e+f x)+d^3 \tan ^2(e+f x)\right )-4 a b \left (3 i (c+i d)^4 \log (i-\tan (e+f x))-3 i (c-i d)^4 \log (i+\tan (e+f x))+6 d^2 \left (-6 c^2+d^2\right ) \tan (e+f x)-12 c d^3 \tan ^2(e+f x)-2 d^4 \tan ^3(e+f x)\right )}{12 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^2*(c + d*Tan[e + f*x])^4 - 6*(2*a*b*c - a^2*d + b^2*d)*((I*c - d)^3*Log[I - Tan[e + f*x]] - (I*c + d)^3*L
og[I + Tan[e + f*x]] + 6*c*d^2*Tan[e + f*x] + d^3*Tan[e + f*x]^2) - 4*a*b*((3*I)*(c + I*d)^4*Log[I - Tan[e + f
*x]] - (3*I)*(c - I*d)^4*Log[I + Tan[e + f*x]] + 6*d^2*(-6*c^2 + d^2)*Tan[e + f*x] - 12*c*d^3*Tan[e + f*x]^2 -
2*d^4*Tan[e + f*x]^3))/(12*d*f)

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Maple [A]
time = 0.13, size = 307, normalized size = 1.40 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(1/4*b^2*d^3*tan(f*x+e)^4+2/3*a*b*d^3*tan(f*x+e)^3+b^2*c*d^2*tan(f*x+e)^3+1/2*a^2*d^3*tan(f*x+e)^2+3*a*b*c
*d^2*tan(f*x+e)^2+3/2*b^2*c^2*d*tan(f*x+e)^2-1/2*b^2*d^3*tan(f*x+e)^2+3*a^2*c*d^2*tan(f*x+e)+6*a*b*c^2*d*tan(f
*x+e)-2*a*b*d^3*tan(f*x+e)+b^2*c^3*tan(f*x+e)-3*b^2*c*d^2*tan(f*x+e)+1/2*(3*a^2*c^2*d-a^2*d^3+2*a*b*c^3-6*a*b*
c*d^2-3*b^2*c^2*d+b^2*d^3)*ln(1+tan(f*x+e)^2)+(a^2*c^3-3*a^2*c*d^2-6*a*b*c^2*d+2*a*b*d^3-b^2*c^3+3*b^2*c*d^2)*
arctan(tan(f*x+e)))

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Maxima [A]
time = 0.54, size = 252, normalized size = 1.15 \begin {gather*} \frac {3 \, b^{2} d^{3} \tan \left (f x + e\right )^{4} + 4 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} \tan \left (f x + e\right )^{3} + 6 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + {\left (a^{2} - b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} - 12 \, {\left (6 \, a b c^{2} d - 2 \, a b d^{3} - {\left (a^{2} - b^{2}\right )} c^{3} + 3 \, {\left (a^{2} - b^{2}\right )} c d^{2}\right )} {\left (f x + e\right )} + 6 \, {\left (2 \, a b c^{3} - 6 \, a b c d^{2} + 3 \, {\left (a^{2} - b^{2}\right )} c^{2} d - {\left (a^{2} - b^{2}\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 12 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d - 2 \, a b d^{3} + 3 \, {\left (a^{2} - b^{2}\right )} c d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

1/12*(3*b^2*d^3*tan(f*x + e)^4 + 4*(3*b^2*c*d^2 + 2*a*b*d^3)*tan(f*x + e)^3 + 6*(3*b^2*c^2*d + 6*a*b*c*d^2 + (
a^2 - b^2)*d^3)*tan(f*x + e)^2 - 12*(6*a*b*c^2*d - 2*a*b*d^3 - (a^2 - b^2)*c^3 + 3*(a^2 - b^2)*c*d^2)*(f*x + e
) + 6*(2*a*b*c^3 - 6*a*b*c*d^2 + 3*(a^2 - b^2)*c^2*d - (a^2 - b^2)*d^3)*log(tan(f*x + e)^2 + 1) + 12*(b^2*c^3
+ 6*a*b*c^2*d - 2*a*b*d^3 + 3*(a^2 - b^2)*c*d^2)*tan(f*x + e))/f

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Fricas [A]
time = 0.95, size = 250, normalized size = 1.14 \begin {gather*} \frac {3 \, b^{2} d^{3} \tan \left (f x + e\right )^{4} + 4 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} \tan \left (f x + e\right )^{3} - 12 \, {\left (6 \, a b c^{2} d - 2 \, a b d^{3} - {\left (a^{2} - b^{2}\right )} c^{3} + 3 \, {\left (a^{2} - b^{2}\right )} c d^{2}\right )} f x + 6 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + {\left (a^{2} - b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} - 6 \, {\left (2 \, a b c^{3} - 6 \, a b c d^{2} + 3 \, {\left (a^{2} - b^{2}\right )} c^{2} d - {\left (a^{2} - b^{2}\right )} d^{3}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 12 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d - 2 \, a b d^{3} + 3 \, {\left (a^{2} - b^{2}\right )} c d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/12*(3*b^2*d^3*tan(f*x + e)^4 + 4*(3*b^2*c*d^2 + 2*a*b*d^3)*tan(f*x + e)^3 - 12*(6*a*b*c^2*d - 2*a*b*d^3 - (a
^2 - b^2)*c^3 + 3*(a^2 - b^2)*c*d^2)*f*x + 6*(3*b^2*c^2*d + 6*a*b*c*d^2 + (a^2 - b^2)*d^3)*tan(f*x + e)^2 - 6*
(2*a*b*c^3 - 6*a*b*c*d^2 + 3*(a^2 - b^2)*c^2*d - (a^2 - b^2)*d^3)*log(1/(tan(f*x + e)^2 + 1)) + 12*(b^2*c^3 +
6*a*b*c^2*d - 2*a*b*d^3 + 3*(a^2 - b^2)*c*d^2)*tan(f*x + e))/f

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (194) = 388\).
time = 0.18, size = 445, normalized size = 2.03 \begin {gather*} \begin {cases} a^{2} c^{3} x + \frac {3 a^{2} c^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - 3 a^{2} c d^{2} x + \frac {3 a^{2} c d^{2} \tan {\left (e + f x \right )}}{f} - \frac {a^{2} d^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {a^{2} d^{3} \tan ^{2}{\left (e + f x \right )}}{2 f} + \frac {a b c^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} - 6 a b c^{2} d x + \frac {6 a b c^{2} d \tan {\left (e + f x \right )}}{f} - \frac {3 a b c d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {3 a b c d^{2} \tan ^{2}{\left (e + f x \right )}}{f} + 2 a b d^{3} x + \frac {2 a b d^{3} \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 a b d^{3} \tan {\left (e + f x \right )}}{f} - b^{2} c^{3} x + \frac {b^{2} c^{3} \tan {\left (e + f x \right )}}{f} - \frac {3 b^{2} c^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 b^{2} c^{2} d \tan ^{2}{\left (e + f x \right )}}{2 f} + 3 b^{2} c d^{2} x + \frac {b^{2} c d^{2} \tan ^{3}{\left (e + f x \right )}}{f} - \frac {3 b^{2} c d^{2} \tan {\left (e + f x \right )}}{f} + \frac {b^{2} d^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {b^{2} d^{3} \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {b^{2} d^{3} \tan ^{2}{\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \left (a + b \tan {\left (e \right )}\right )^{2} \left (c + d \tan {\left (e \right )}\right )^{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**3*x + 3*a**2*c**2*d*log(tan(e + f*x)**2 + 1)/(2*f) - 3*a**2*c*d**2*x + 3*a**2*c*d**2*tan(e
+ f*x)/f - a**2*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + a**2*d**3*tan(e + f*x)**2/(2*f) + a*b*c**3*log(tan(e + f
*x)**2 + 1)/f - 6*a*b*c**2*d*x + 6*a*b*c**2*d*tan(e + f*x)/f - 3*a*b*c*d**2*log(tan(e + f*x)**2 + 1)/f + 3*a*b
*c*d**2*tan(e + f*x)**2/f + 2*a*b*d**3*x + 2*a*b*d**3*tan(e + f*x)**3/(3*f) - 2*a*b*d**3*tan(e + f*x)/f - b**2
*c**3*x + b**2*c**3*tan(e + f*x)/f - 3*b**2*c**2*d*log(tan(e + f*x)**2 + 1)/(2*f) + 3*b**2*c**2*d*tan(e + f*x)
**2/(2*f) + 3*b**2*c*d**2*x + b**2*c*d**2*tan(e + f*x)**3/f - 3*b**2*c*d**2*tan(e + f*x)/f + b**2*d**3*log(tan
(e + f*x)**2 + 1)/(2*f) + b**2*d**3*tan(e + f*x)**4/(4*f) - b**2*d**3*tan(e + f*x)**2/(2*f), Ne(f, 0)), (x*(a
+ b*tan(e))**2*(c + d*tan(e))**3, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4557 vs. \(2 (218) = 436\).
time = 2.67, size = 4557, normalized size = 20.81 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3,x, algorithm="giac")

[Out]

1/12*(12*a^2*c^3*f*x*tan(f*x)^4*tan(e)^4 - 12*b^2*c^3*f*x*tan(f*x)^4*tan(e)^4 - 72*a*b*c^2*d*f*x*tan(f*x)^4*ta
n(e)^4 - 36*a^2*c*d^2*f*x*tan(f*x)^4*tan(e)^4 + 36*b^2*c*d^2*f*x*tan(f*x)^4*tan(e)^4 + 24*a*b*d^3*f*x*tan(f*x)
^4*tan(e)^4 - 12*a*b*c^3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 -
2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 18*a^2*c^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(
f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4
+ 18*b^2*c^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x
)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 36*a*b*c*d^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan
(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 6*a^2*d^
3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)
/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 6*b^2*d^3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2
*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 48*a^2*c^3*f*x*tan(f*x)^
3*tan(e)^3 + 48*b^2*c^3*f*x*tan(f*x)^3*tan(e)^3 + 288*a*b*c^2*d*f*x*tan(f*x)^3*tan(e)^3 + 144*a^2*c*d^2*f*x*ta
n(f*x)^3*tan(e)^3 - 144*b^2*c*d^2*f*x*tan(f*x)^3*tan(e)^3 - 96*a*b*d^3*f*x*tan(f*x)^3*tan(e)^3 + 18*b^2*c^2*d*
tan(f*x)^4*tan(e)^4 + 36*a*b*c*d^2*tan(f*x)^4*tan(e)^4 + 6*a^2*d^3*tan(f*x)^4*tan(e)^4 - 9*b^2*d^3*tan(f*x)^4*
tan(e)^4 + 48*a*b*c^3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*
tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 + 72*a^2*c^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x
)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 7
2*b^2*c^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*t
an(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 144*a*b*c*d^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e
) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 24*a^2*d^3
*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/
(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 + 24*b^2*d^3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2
*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 12*b^2*c^3*tan(f*x)^4*ta
n(e)^3 - 72*a*b*c^2*d*tan(f*x)^4*tan(e)^3 - 36*a^2*c*d^2*tan(f*x)^4*tan(e)^3 + 36*b^2*c*d^2*tan(f*x)^4*tan(e)^
3 + 24*a*b*d^3*tan(f*x)^4*tan(e)^3 - 12*b^2*c^3*tan(f*x)^3*tan(e)^4 - 72*a*b*c^2*d*tan(f*x)^3*tan(e)^4 - 36*a^
2*c*d^2*tan(f*x)^3*tan(e)^4 + 36*b^2*c*d^2*tan(f*x)^3*tan(e)^4 + 24*a*b*d^3*tan(f*x)^3*tan(e)^4 + 72*a^2*c^3*f
*x*tan(f*x)^2*tan(e)^2 - 72*b^2*c^3*f*x*tan(f*x)^2*tan(e)^2 - 432*a*b*c^2*d*f*x*tan(f*x)^2*tan(e)^2 - 216*a^2*
c*d^2*f*x*tan(f*x)^2*tan(e)^2 + 216*b^2*c*d^2*f*x*tan(f*x)^2*tan(e)^2 + 144*a*b*d^3*f*x*tan(f*x)^2*tan(e)^2 +
18*b^2*c^2*d*tan(f*x)^4*tan(e)^2 + 36*a*b*c*d^2*tan(f*x)^4*tan(e)^2 + 6*a^2*d^3*tan(f*x)^4*tan(e)^2 - 6*b^2*d^
3*tan(f*x)^4*tan(e)^2 - 36*b^2*c^2*d*tan(f*x)^3*tan(e)^3 - 72*a*b*c*d^2*tan(f*x)^3*tan(e)^3 - 12*a^2*d^3*tan(f
*x)^3*tan(e)^3 + 24*b^2*d^3*tan(f*x)^3*tan(e)^3 + 18*b^2*c^2*d*tan(f*x)^2*tan(e)^4 + 36*a*b*c*d^2*tan(f*x)^2*t
an(e)^4 + 6*a^2*d^3*tan(f*x)^2*tan(e)^4 - 6*b^2*d^3*tan(f*x)^2*tan(e)^4 - 12*b^2*c*d^2*tan(f*x)^4*tan(e) - 8*a
*b*d^3*tan(f*x)^4*tan(e) - 72*a*b*c^3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 +
tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 - 108*a^2*c^2*d*log(4*(tan(f*x)^4*tan
(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*
x)^2*tan(e)^2 + 108*b^2*c^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x
)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 + 216*a*b*c*d^2*log(4*(tan(f*x)^4*tan(e)^2 -
2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2*tan
(e)^2 + 36*a^2*d^3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan
(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 - 36*b^2*d^3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*t
an(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 + 36*b^2
*c^3*tan(f*x)^3*tan(e)^2 + 216*a*b*c^2*d*tan(f*x)^3*tan(e)^2 + 108*a^2*c*d^2*tan(f*x)^3*tan(e)^2 - 144*b^2*c*d
^2*tan(f*x)^3*tan(e)^2 - 96*a*b*d^3*tan(f*x)^3*tan(e)^2 + 36*b^2*c^3*tan(f*x)^2*tan(e)^3 + 216*a*b*c^2*d*tan(f
*x)^2*tan(e)^3 + 108*a^2*c*d^2*tan(f*x)^2*tan(e)^3 - 144*b^2*c*d^2*tan(f*x)^2*tan(e)^3 - 96*a*b*d^3*tan(f*x)^2
*tan(e)^3 - 12*b^2*c*d^2*tan(f*x)*tan(e)^4 - 8*a*b*d^3*tan(f*x)*tan(e)^4 + 3*b^2*d^3*tan(f*x)^4 - 48*a^2*c^3*f
*x*tan(f*x)*tan(e) + 48*b^2*c^3*f*x*tan(f*x)*ta...

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Mupad [B]
time = 5.27, size = 259, normalized size = 1.18 \begin {gather*} x\,\left (a^2\,c^3-3\,a^2\,c\,d^2-6\,a\,b\,c^2\,d+2\,a\,b\,d^3-b^2\,c^3+3\,b^2\,c\,d^2\right )+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (b^2\,c^3-b\,d^2\,\left (2\,a\,d+3\,b\,c\right )+3\,a^2\,c\,d^2+6\,a\,b\,c^2\,d\right )}{f}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (-\frac {3\,a^2\,c^2\,d}{2}+\frac {a^2\,d^3}{2}-a\,b\,c^3+3\,a\,b\,c\,d^2+\frac {3\,b^2\,c^2\,d}{2}-\frac {b^2\,d^3}{2}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {a^2\,d^3}{2}+3\,a\,b\,c\,d^2+\frac {3\,b^2\,c^2\,d}{2}-\frac {b^2\,d^3}{2}\right )}{f}+\frac {b^2\,d^3\,{\mathrm {tan}\left (e+f\,x\right )}^4}{4\,f}+\frac {b\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (2\,a\,d+3\,b\,c\right )}{3\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(a^2*c^3 - b^2*c^3 - 3*a^2*c*d^2 + 3*b^2*c*d^2 + 2*a*b*d^3 - 6*a*b*c^2*d) + (tan(e + f*x)*(b^2*c^3 - b*d^2*(
2*a*d + 3*b*c) + 3*a^2*c*d^2 + 6*a*b*c^2*d))/f - (log(tan(e + f*x)^2 + 1)*((a^2*d^3)/2 - (b^2*d^3)/2 - (3*a^2*
c^2*d)/2 + (3*b^2*c^2*d)/2 - a*b*c^3 + 3*a*b*c*d^2))/f + (tan(e + f*x)^2*((a^2*d^3)/2 - (b^2*d^3)/2 + (3*b^2*c
^2*d)/2 + 3*a*b*c*d^2))/f + (b^2*d^3*tan(e + f*x)^4)/(4*f) + (b*d^2*tan(e + f*x)^3*(2*a*d + 3*b*c))/(3*f)

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