3.5.12 \(\int \tan ^5(c+d x) (a+b \tan (c+d x)) \, dx\) [412]

Optimal. Leaf size=93 \[ -b x-\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan (c+d x)}{d}-\frac {a \tan ^2(c+d x)}{2 d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^5(c+d x)}{5 d} \]

[Out]

-b*x-a*ln(cos(d*x+c))/d+b*tan(d*x+c)/d-1/2*a*tan(d*x+c)^2/d-1/3*b*tan(d*x+c)^3/d+1/4*a*tan(d*x+c)^4/d+1/5*b*ta
n(d*x+c)^5/d

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Rubi [A]
time = 0.08, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3609, 3606, 3556} \begin {gather*} \frac {a \tan ^4(c+d x)}{4 d}-\frac {a \tan ^2(c+d x)}{2 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan ^5(c+d x)}{5 d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {b \tan (c+d x)}{d}-b x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^5*(a + b*Tan[c + d*x]),x]

[Out]

-(b*x) - (a*Log[Cos[c + d*x]])/d + (b*Tan[c + d*x])/d - (a*Tan[c + d*x]^2)/(2*d) - (b*Tan[c + d*x]^3)/(3*d) +
(a*Tan[c + d*x]^4)/(4*d) + (b*Tan[c + d*x]^5)/(5*d)

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]

Rubi steps

\begin {align*} \int \tan ^5(c+d x) (a+b \tan (c+d x)) \, dx &=\frac {b \tan ^5(c+d x)}{5 d}+\int \tan ^4(c+d x) (-b+a \tan (c+d x)) \, dx\\ &=\frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^5(c+d x)}{5 d}+\int \tan ^3(c+d x) (-a-b \tan (c+d x)) \, dx\\ &=-\frac {b \tan ^3(c+d x)}{3 d}+\frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^5(c+d x)}{5 d}+\int \tan ^2(c+d x) (b-a \tan (c+d x)) \, dx\\ &=-\frac {a \tan ^2(c+d x)}{2 d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^5(c+d x)}{5 d}+\int \tan (c+d x) (a+b \tan (c+d x)) \, dx\\ &=-b x+\frac {b \tan (c+d x)}{d}-\frac {a \tan ^2(c+d x)}{2 d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^5(c+d x)}{5 d}+a \int \tan (c+d x) \, dx\\ &=-b x-\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan (c+d x)}{d}-\frac {a \tan ^2(c+d x)}{2 d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^5(c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.37, size = 95, normalized size = 1.02 \begin {gather*} -\frac {b \text {ArcTan}(\tan (c+d x))}{d}+\frac {b \tan (c+d x)}{d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {b \tan ^5(c+d x)}{5 d}-\frac {a \left (4 \log (\cos (c+d x))+2 \tan ^2(c+d x)-\tan ^4(c+d x)\right )}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^5*(a + b*Tan[c + d*x]),x]

[Out]

-((b*ArcTan[Tan[c + d*x]])/d) + (b*Tan[c + d*x])/d - (b*Tan[c + d*x]^3)/(3*d) + (b*Tan[c + d*x]^5)/(5*d) - (a*
(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d)

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Maple [A]
time = 0.07, size = 82, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {\frac {b \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {a \left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2}+b \tan \left (d x +c \right )+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(82\)
default	\(\frac {\frac {b \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {a \left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2}+b \tan \left (d x +c \right )+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(82\)
norman	\(\frac {b \tan \left (d x +c \right )}{d}-b x -\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {b \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(90\)
risch	\(-b x +i a x +\frac {2 i a c}{d}+\frac {2 i \left (30 i a \,{\mathrm e}^{8 i \left (d x +c \right )}+45 b \,{\mathrm e}^{8 i \left (d x +c \right )}+60 i a \,{\mathrm e}^{6 i \left (d x +c \right )}+90 b \,{\mathrm e}^{6 i \left (d x +c \right )}+60 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+140 b \,{\mathrm e}^{4 i \left (d x +c \right )}+30 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+70 b \,{\mathrm e}^{2 i \left (d x +c \right )}+23 b \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(160\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^5*(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/5*b*tan(d*x+c)^5+1/4*a*tan(d*x+c)^4-1/3*b*tan(d*x+c)^3-1/2*a*tan(d*x+c)^2+b*tan(d*x+c)+1/2*a*ln(1+tan(d
*x+c)^2)-b*arctan(tan(d*x+c)))

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Maxima [A]
time = 0.54, size = 81, normalized size = 0.87 \begin {gather*} \frac {12 \, b \tan \left (d x + c\right )^{5} + 15 \, a \tan \left (d x + c\right )^{4} - 20 \, b \tan \left (d x + c\right )^{3} - 30 \, a \tan \left (d x + c\right )^{2} - 60 \, {\left (d x + c\right )} b + 30 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \, b \tan \left (d x + c\right )}{60 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5*(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/60*(12*b*tan(d*x + c)^5 + 15*a*tan(d*x + c)^4 - 20*b*tan(d*x + c)^3 - 30*a*tan(d*x + c)^2 - 60*(d*x + c)*b +
30*a*log(tan(d*x + c)^2 + 1) + 60*b*tan(d*x + c))/d

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Fricas [A]
time = 1.64, size = 80, normalized size = 0.86 \begin {gather*} \frac {12 \, b \tan \left (d x + c\right )^{5} + 15 \, a \tan \left (d x + c\right )^{4} - 20 \, b \tan \left (d x + c\right )^{3} - 60 \, b d x - 30 \, a \tan \left (d x + c\right )^{2} - 30 \, a \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 60 \, b \tan \left (d x + c\right )}{60 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5*(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/60*(12*b*tan(d*x + c)^5 + 15*a*tan(d*x + c)^4 - 20*b*tan(d*x + c)^3 - 60*b*d*x - 30*a*tan(d*x + c)^2 - 30*a*
log(1/(tan(d*x + c)^2 + 1)) + 60*b*tan(d*x + c))/d

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Sympy [A]
time = 0.15, size = 97, normalized size = 1.04 \begin {gather*} \begin {cases} \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {a \tan ^{2}{\left (c + d x \right )}}{2 d} - b x + \frac {b \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {b \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {b \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \tan ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**5*(a+b*tan(d*x+c)),x)

[Out]

Piecewise((a*log(tan(c + d*x)**2 + 1)/(2*d) + a*tan(c + d*x)**4/(4*d) - a*tan(c + d*x)**2/(2*d) - b*x + b*tan(
c + d*x)**5/(5*d) - b*tan(c + d*x)**3/(3*d) + b*tan(c + d*x)/d, Ne(d, 0)), (x*(a + b*tan(c))*tan(c)**5, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 947 vs. \(2 (85) = 170\).
time = 2.64, size = 947, normalized size = 10.18 \begin {gather*} -\frac {60 \, b d x \tan \left (d x\right )^{5} \tan \left (c\right )^{5} + 30 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 300 \, b d x \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 45 \, a \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 150 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 60 \, b \tan \left (d x\right )^{5} \tan \left (c\right )^{4} + 60 \, b \tan \left (d x\right )^{4} \tan \left (c\right )^{5} + 600 \, b d x \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 30 \, a \tan \left (d x\right )^{5} \tan \left (c\right )^{3} - 165 \, a \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 30 \, a \tan \left (d x\right )^{3} \tan \left (c\right )^{5} - 20 \, b \tan \left (d x\right )^{5} \tan \left (c\right )^{2} + 300 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 300 \, b \tan \left (d x\right )^{4} \tan \left (c\right )^{3} - 300 \, b \tan \left (d x\right )^{3} \tan \left (c\right )^{4} - 20 \, b \tan \left (d x\right )^{2} \tan \left (c\right )^{5} - 15 \, a \tan \left (d x\right )^{5} \tan \left (c\right ) - 600 \, b d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 150 \, a \tan \left (d x\right )^{4} \tan \left (c\right )^{2} + 180 \, a \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 150 \, a \tan \left (d x\right )^{2} \tan \left (c\right )^{4} - 15 \, a \tan \left (d x\right ) \tan \left (c\right )^{5} + 12 \, b \tan \left (d x\right )^{5} + 100 \, b \tan \left (d x\right )^{4} \tan \left (c\right ) - 300 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 600 \, b \tan \left (d x\right )^{3} \tan \left (c\right )^{2} + 600 \, b \tan \left (d x\right )^{2} \tan \left (c\right )^{3} + 100 \, b \tan \left (d x\right ) \tan \left (c\right )^{4} + 12 \, b \tan \left (c\right )^{5} + 15 \, a \tan \left (d x\right )^{4} + 300 \, b d x \tan \left (d x\right ) \tan \left (c\right ) + 150 \, a \tan \left (d x\right )^{3} \tan \left (c\right ) - 180 \, a \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 150 \, a \tan \left (d x\right ) \tan \left (c\right )^{3} + 15 \, a \tan \left (c\right )^{4} - 20 \, b \tan \left (d x\right )^{3} + 150 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) - 300 \, b \tan \left (d x\right )^{2} \tan \left (c\right ) - 300 \, b \tan \left (d x\right ) \tan \left (c\right )^{2} - 20 \, b \tan \left (c\right )^{3} - 60 \, b d x - 30 \, a \tan \left (d x\right )^{2} + 165 \, a \tan \left (d x\right ) \tan \left (c\right ) - 30 \, a \tan \left (c\right )^{2} - 30 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (c\right )^{2} + 1}\right ) + 60 \, b \tan \left (d x\right ) + 60 \, b \tan \left (c\right ) - 45 \, a}{60 \, {\left (d \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 5 \, d \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 10 \, d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 10 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 5 \, d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5*(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/60*(60*b*d*x*tan(d*x)^5*tan(c)^5 + 30*a*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c
)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^5*tan(c)^5 - 300*b*d*x*tan(d*x)^4*tan(c)^4
+ 45*a*tan(d*x)^5*tan(c)^5 - 150*a*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + ta
n(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^4*tan(c)^4 + 60*b*tan(d*x)^5*tan(c)^4 + 60*b*tan(d*
x)^4*tan(c)^5 + 600*b*d*x*tan(d*x)^3*tan(c)^3 + 30*a*tan(d*x)^5*tan(c)^3 - 165*a*tan(d*x)^4*tan(c)^4 + 30*a*ta
n(d*x)^3*tan(c)^5 - 20*b*tan(d*x)^5*tan(c)^2 + 300*a*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*
x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^3*tan(c)^3 - 300*b*tan(d*x)^4*tan
(c)^3 - 300*b*tan(d*x)^3*tan(c)^4 - 20*b*tan(d*x)^2*tan(c)^5 - 15*a*tan(d*x)^5*tan(c) - 600*b*d*x*tan(d*x)^2*t
an(c)^2 - 150*a*tan(d*x)^4*tan(c)^2 + 180*a*tan(d*x)^3*tan(c)^3 - 150*a*tan(d*x)^2*tan(c)^4 - 15*a*tan(d*x)*ta
n(c)^5 + 12*b*tan(d*x)^5 + 100*b*tan(d*x)^4*tan(c) - 300*a*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) +
tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^2*tan(c)^2 + 600*b*tan(d*x)
^3*tan(c)^2 + 600*b*tan(d*x)^2*tan(c)^3 + 100*b*tan(d*x)*tan(c)^4 + 12*b*tan(c)^5 + 15*a*tan(d*x)^4 + 300*b*d*
x*tan(d*x)*tan(c) + 150*a*tan(d*x)^3*tan(c) - 180*a*tan(d*x)^2*tan(c)^2 + 150*a*tan(d*x)*tan(c)^3 + 15*a*tan(c
)^4 - 20*b*tan(d*x)^3 + 150*a*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x
)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)*tan(c) - 300*b*tan(d*x)^2*tan(c) - 300*b*tan(d*x)*tan(c)
^2 - 20*b*tan(c)^3 - 60*b*d*x - 30*a*tan(d*x)^2 + 165*a*tan(d*x)*tan(c) - 30*a*tan(c)^2 - 30*a*log(4*(tan(d*x)
^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))
+ 60*b*tan(d*x) + 60*b*tan(c) - 45*a)/(d*tan(d*x)^5*tan(c)^5 - 5*d*tan(d*x)^4*tan(c)^4 + 10*d*tan(d*x)^3*tan(c
)^3 - 10*d*tan(d*x)^2*tan(c)^2 + 5*d*tan(d*x)*tan(c) - d)

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Mupad [B]
time = 4.10, size = 76, normalized size = 0.82 \begin {gather*} \frac {b\,\mathrm {tan}\left (c+d\,x\right )+\frac {a\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}-b\,d\,x}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^5*(a + b*tan(c + d*x)),x)

[Out]

(b*tan(c + d*x) + (a*log(tan(c + d*x)^2 + 1))/2 - (a*tan(c + d*x)^2)/2 + (a*tan(c + d*x)^4)/4 - (b*tan(c + d*x
)^3)/3 + (b*tan(c + d*x)^5)/5 - b*d*x)/d

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