3.5.13 \(\int \tan ^4(c+d x) (a+b \tan (c+d x)) \, dx\) [413]

Optimal. Leaf size=77 \[ a x-\frac {b \log (\cos (c+d x))}{d}-\frac {a \tan (c+d x)}{d}-\frac {b \tan ^2(c+d x)}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {b \tan ^4(c+d x)}{4 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, 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 ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+a x+\frac {b \tan ^4(c+d x)}{4 d}-\frac {b \tan ^2(c+d x)}{2 d}-\frac {b \log (\cos (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^4*(a + b*Tan[c + d*x]),x]

[Out]

a*x - (b*Log[Cos[c + d*x]])/d - (a*Tan[c + d*x])/d - (b*Tan[c + d*x]^2)/(2*d) + (a*Tan[c + d*x]^3)/(3*d) + (b*
Tan[c + d*x]^4)/(4*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 ^4(c+d x) (a+b \tan (c+d x)) \, dx &=\frac {b \tan ^4(c+d x)}{4 d}+\int \tan ^3(c+d x) (-b+a \tan (c+d x)) \, dx\\ &=\frac {a \tan ^3(c+d x)}{3 d}+\frac {b \tan ^4(c+d x)}{4 d}+\int \tan ^2(c+d x) (-a-b \tan (c+d x)) \, dx\\ &=-\frac {b \tan ^2(c+d x)}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {b \tan ^4(c+d x)}{4 d}+\int \tan (c+d x) (b-a \tan (c+d x)) \, dx\\ &=a x-\frac {a \tan (c+d x)}{d}-\frac {b \tan ^2(c+d x)}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {b \tan ^4(c+d x)}{4 d}+b \int \tan (c+d x) \, dx\\ &=a x-\frac {b \log (\cos (c+d x))}{d}-\frac {a \tan (c+d x)}{d}-\frac {b \tan ^2(c+d x)}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {b \tan ^4(c+d x)}{4 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.22, size = 79, normalized size = 1.03 \begin {gather*} \frac {a \text {ArcTan}(\tan (c+d x))}{d}-\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}-\frac {b \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]^4*(a + b*Tan[c + d*x]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 71, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {\frac {b \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}-a \tan \left (d x +c \right )+\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+a \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(71\)
default	\(\frac {\frac {b \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}-a \tan \left (d x +c \right )+\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+a \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(71\)
norman	\(a x -\frac {a \tan \left (d x +c \right )}{d}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(76\)
risch	\(i b x +a x +\frac {2 i b c}{d}-\frac {4 \left (3 i a \,{\mathrm e}^{6 i \left (d x +c \right )}+3 b \,{\mathrm e}^{6 i \left (d x +c \right )}+6 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+5 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 i a \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^4*(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 70, normalized size = 0.91 \begin {gather*} \frac {3 \, b \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} - 6 \, b \tan \left (d x + c\right )^{2} + 12 \, {\left (d x + c\right )} a + 6 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 12 \, a \tan \left (d x + c\right )}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^4*(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 1.24, size = 69, normalized size = 0.90 \begin {gather*} \frac {3 \, b \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} + 12 \, a d x - 6 \, b \tan \left (d x + c\right )^{2} - 6 \, b \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 12 \, a \tan \left (d x + c\right )}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^4*(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.11, size = 83, normalized size = 1.08 \begin {gather*} \begin {cases} a x + \frac {a \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {a \tan {\left (c + d x \right )}}{d} + \frac {b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {b \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {b \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \tan ^{4}{\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)**4*(a+b*tan(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (71) = 142\).
time = 1.48, size = 716, normalized size = 9.30 \begin {gather*} \frac {12 \, a d x \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 6 \, b \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} - 48 \, a d x \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 9 \, b \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 24 \, b \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} + 12 \, a \tan \left (d x\right )^{4} \tan \left (c\right )^{3} + 12 \, a \tan \left (d x\right )^{3} \tan \left (c\right )^{4} + 72 \, a d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 6 \, b \tan \left (d x\right )^{4} \tan \left (c\right )^{2} + 24 \, b \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 6 \, b \tan \left (d x\right )^{2} \tan \left (c\right )^{4} - 4 \, a \tan \left (d x\right )^{4} \tan \left (c\right ) - 36 \, b \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} - 48 \, a \tan \left (d x\right )^{3} \tan \left (c\right )^{2} - 48 \, a \tan \left (d x\right )^{2} \tan \left (c\right )^{3} - 4 \, a \tan \left (d x\right ) \tan \left (c\right )^{4} + 3 \, b \tan \left (d x\right )^{4} - 48 \, a d x \tan \left (d x\right ) \tan \left (c\right ) + 24 \, b \tan \left (d x\right )^{3} \tan \left (c\right ) - 12 \, b \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 24 \, b \tan \left (d x\right ) \tan \left (c\right )^{3} + 3 \, b \tan \left (c\right )^{4} + 4 \, a \tan \left (d x\right )^{3} + 24 \, b \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 ) + 48 \, a \tan \left (d x\right )^{2} \tan \left (c\right ) + 48 \, a \tan \left (d x\right ) \tan \left (c\right )^{2} + 4 \, a \tan \left (c\right )^{3} + 12 \, a d x - 6 \, b \tan \left (d x\right )^{2} + 24 \, b \tan \left (d x\right ) \tan \left (c\right ) - 6 \, b \tan \left (c\right )^{2} - 6 \, b \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 ) - 12 \, a \tan \left (d x\right ) - 12 \, a \tan \left (c\right ) - 9 \, b}{12 \, {\left (d \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 4 \, d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 6 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 4 \, 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)^4*(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

1/12*(12*a*d*x*tan(d*x)^4*tan(c)^4 - 6*b*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)^4*tan(c)^4 - 48*a*d*x*tan(d*x)^3*tan(c)^3 - 9
*b*tan(d*x)^4*tan(c)^4 + 24*b*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 + 12*a*tan(d*x)^4*tan(c)^3 + 12*a*tan(d*x)^3*
tan(c)^4 + 72*a*d*x*tan(d*x)^2*tan(c)^2 - 6*b*tan(d*x)^4*tan(c)^2 + 24*b*tan(d*x)^3*tan(c)^3 - 6*b*tan(d*x)^2*
tan(c)^4 - 4*a*tan(d*x)^4*tan(c) - 36*b*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 - 48*a*tan(d*x)^3*tan(c)^2 - 48*a*t
an(d*x)^2*tan(c)^3 - 4*a*tan(d*x)*tan(c)^4 + 3*b*tan(d*x)^4 - 48*a*d*x*tan(d*x)*tan(c) + 24*b*tan(d*x)^3*tan(c
) - 12*b*tan(d*x)^2*tan(c)^2 + 24*b*tan(d*x)*tan(c)^3 + 3*b*tan(c)^4 + 4*a*tan(d*x)^3 + 24*b*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))*ta
n(d*x)*tan(c) + 48*a*tan(d*x)^2*tan(c) + 48*a*tan(d*x)*tan(c)^2 + 4*a*tan(c)^3 + 12*a*d*x - 6*b*tan(d*x)^2 + 2
4*b*tan(d*x)*tan(c) - 6*b*tan(c)^2 - 6*b*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)) - 12*a*tan(d*x) - 12*a*tan(c) - 9*b)/(d*tan(d*x)^4*tan
(c)^4 - 4*d*tan(d*x)^3*tan(c)^3 + 6*d*tan(d*x)^2*tan(c)^2 - 4*d*tan(d*x)*tan(c) + d)

________________________________________________________________________________________

Mupad [B]
time = 4.01, size = 65, normalized size = 0.84 \begin {gather*} \frac {\frac {b\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-a\,\mathrm {tan}\left (c+d\,x\right )+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}+a\,d\,x}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^4*(a + b*tan(c + d*x)),x)

[Out]

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

________________________________________________________________________________________