3.447 \(\int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx\)
Optimal. Leaf size=130 \[ \frac{\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac{a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\cos (c+d x))}{d}-4 a b x \left (a^2-b^2\right )+\frac{(a+b \tan (c+d x))^4}{4 d}+\frac{a (a+b \tan (c+d x))^3}{3 d} \]
[Out]
-4*a*b*(a^2 - b^2)*x - ((a^4 - 6*a^2*b^2 + b^4)*Log[Cos[c + d*x]])/d + (a*b*(a^2 - 3*b^2)*Tan[c + d*x])/d + ((
a^2 - b^2)*(a + b*Tan[c + d*x])^2)/(2*d) + (a*(a + b*Tan[c + d*x])^3)/(3*d) + (a + b*Tan[c + d*x])^4/(4*d)
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Rubi [A] time = 0.128687, antiderivative size = 130, normalized size of antiderivative = 1.,
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 =
{3528, 3525, 3475} \[ \frac{\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac{a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\cos (c+d x))}{d}-4 a b x \left (a^2-b^2\right )+\frac{(a+b \tan (c+d x))^4}{4 d}+\frac{a (a+b \tan (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
[In]
Int[Tan[c + d*x]*(a + b*Tan[c + d*x])^4,x]
[Out]
-4*a*b*(a^2 - b^2)*x - ((a^4 - 6*a^2*b^2 + b^4)*Log[Cos[c + d*x]])/d + (a*b*(a^2 - 3*b^2)*Tan[c + d*x])/d + ((
a^2 - b^2)*(a + b*Tan[c + d*x])^2)/(2*d) + (a*(a + b*Tan[c + d*x])^3)/(3*d) + (a + b*Tan[c + d*x])^4/(4*d)
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 3525
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 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx &=\frac{(a+b \tan (c+d x))^4}{4 d}+\int (-b+a \tan (c+d x)) (a+b \tan (c+d x))^3 \, dx\\ &=\frac{a (a+b \tan (c+d x))^3}{3 d}+\frac{(a+b \tan (c+d x))^4}{4 d}+\int (a+b \tan (c+d x))^2 \left (-2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac{a (a+b \tan (c+d x))^3}{3 d}+\frac{(a+b \tan (c+d x))^4}{4 d}+\int (a+b \tan (c+d x)) \left (-b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-4 a b \left (a^2-b^2\right ) x+\frac{a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}+\frac{\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac{a (a+b \tan (c+d x))^3}{3 d}+\frac{(a+b \tan (c+d x))^4}{4 d}+\left (a^4-6 a^2 b^2+b^4\right ) \int \tan (c+d x) \, dx\\ &=-4 a b \left (a^2-b^2\right ) x-\frac{\left (a^4-6 a^2 b^2+b^4\right ) \log (\cos (c+d x))}{d}+\frac{a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}+\frac{\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac{a (a+b \tan (c+d x))^3}{3 d}+\frac{(a+b \tan (c+d x))^4}{4 d}\\ \end{align*}
Mathematica [C] time = 1.77215, size = 123, normalized size = 0.95 \[ \frac{-6 b^2 \left (b^2-6 a^2\right ) \tan ^2(c+d x)+48 a b \left (a^2-b^2\right ) \tan (c+d x)+16 a b^3 \tan ^3(c+d x)+6 \left ((a-i b)^4 \log (\tan (c+d x)+i)+(a+i b)^4 \log (-\tan (c+d x)+i)\right )+3 b^4 \tan ^4(c+d x)}{12 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^4,x]
[Out]
(6*((a + I*b)^4*Log[I - Tan[c + d*x]] + (a - I*b)^4*Log[I + Tan[c + d*x]]) + 48*a*b*(a^2 - b^2)*Tan[c + d*x] -
6*b^2*(-6*a^2 + b^2)*Tan[c + d*x]^2 + 16*a*b^3*Tan[c + d*x]^3 + 3*b^4*Tan[c + d*x]^4)/(12*d)
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Maple [A] time = 0.004, size = 192, normalized size = 1.5 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{4}{b}^{4}}{4\,d}}+{\frac{4\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}a{b}^{3}}{3\,d}}+3\,{\frac{{a}^{2}{b}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}{b}^{4}}{2\,d}}+4\,{\frac{{a}^{3}\tan \left ( dx+c \right ) b}{d}}-4\,{\frac{{b}^{3}a\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{4}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}-3\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{2}{b}^{2}}{d}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{4}}{2\,d}}-4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}b}{d}}+4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{3}}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(d*x+c)*(a+b*tan(d*x+c))^4,x)
[Out]
1/4/d*tan(d*x+c)^4*b^4+4/3/d*tan(d*x+c)^3*a*b^3+3/d*a^2*b^2*tan(d*x+c)^2-1/2/d*tan(d*x+c)^2*b^4+4/d*tan(d*x+c)
*a^3*b-4*a*b^3*tan(d*x+c)/d+1/2/d*a^4*ln(1+tan(d*x+c)^2)-3/d*ln(1+tan(d*x+c)^2)*a^2*b^2+1/2/d*ln(1+tan(d*x+c)^
2)*b^4-4/d*arctan(tan(d*x+c))*a^3*b+4/d*arctan(tan(d*x+c))*a*b^3
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Maxima [A] time = 1.52408, size = 167, normalized size = 1.28 \begin{align*} \frac{3 \, b^{4} \tan \left (d x + c\right )^{4} + 16 \, a b^{3} \tan \left (d x + c\right )^{3} + 6 \,{\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{2} - 48 \,{\left (a^{3} b - a b^{3}\right )}{\left (d x + c\right )} + 6 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 48 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)*(a+b*tan(d*x+c))^4,x, algorithm="maxima")
[Out]
1/12*(3*b^4*tan(d*x + c)^4 + 16*a*b^3*tan(d*x + c)^3 + 6*(6*a^2*b^2 - b^4)*tan(d*x + c)^2 - 48*(a^3*b - a*b^3)
*(d*x + c) + 6*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1) + 48*(a^3*b - a*b^3)*tan(d*x + c))/d
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Fricas [A] time = 2.12221, size = 282, normalized size = 2.17 \begin{align*} \frac{3 \, b^{4} \tan \left (d x + c\right )^{4} + 16 \, a b^{3} \tan \left (d x + c\right )^{3} - 48 \,{\left (a^{3} b - a b^{3}\right )} d x + 6 \,{\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{2} - 6 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 48 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)*(a+b*tan(d*x+c))^4,x, algorithm="fricas")
[Out]
1/12*(3*b^4*tan(d*x + c)^4 + 16*a*b^3*tan(d*x + c)^3 - 48*(a^3*b - a*b^3)*d*x + 6*(6*a^2*b^2 - b^4)*tan(d*x +
c)^2 - 6*(a^4 - 6*a^2*b^2 + b^4)*log(1/(tan(d*x + c)^2 + 1)) + 48*(a^3*b - a*b^3)*tan(d*x + c))/d
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Sympy [A] time = 0.71754, size = 187, normalized size = 1.44 \begin{align*} \begin{cases} \frac{a^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 4 a^{3} b x + \frac{4 a^{3} b \tan{\left (c + d x \right )}}{d} - \frac{3 a^{2} b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{3 a^{2} b^{2} \tan ^{2}{\left (c + d x \right )}}{d} + 4 a b^{3} x + \frac{4 a b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{4 a b^{3} \tan{\left (c + d x \right )}}{d} + \frac{b^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b^{4} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{b^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right )^{4} \tan{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)*(a+b*tan(d*x+c))**4,x)
[Out]
Piecewise((a**4*log(tan(c + d*x)**2 + 1)/(2*d) - 4*a**3*b*x + 4*a**3*b*tan(c + d*x)/d - 3*a**2*b**2*log(tan(c
+ d*x)**2 + 1)/d + 3*a**2*b**2*tan(c + d*x)**2/d + 4*a*b**3*x + 4*a*b**3*tan(c + d*x)**3/(3*d) - 4*a*b**3*tan(
c + d*x)/d + b**4*log(tan(c + d*x)**2 + 1)/(2*d) + b**4*tan(c + d*x)**4/(4*d) - b**4*tan(c + d*x)**2/(2*d), Ne
(d, 0)), (x*(a + b*tan(c))**4*tan(c), True))
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Giac [B] time = 5.69864, size = 2546, normalized size = 19.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)*(a+b*tan(d*x+c))^4,x, algorithm="giac")
[Out]
-1/12*(48*a^3*b*d*x*tan(d*x)^4*tan(c)^4 - 48*a*b^3*d*x*tan(d*x)^4*tan(c)^4 + 6*a^4*log(4*(tan(c)^2 + 1)/(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(d*x)^4*t
an(c)^4 - 36*a^2*b^2*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + t
an(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 + 6*b^4*log(4*(tan(c)^2 + 1)/(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(d*x)^4*tan(c)^4 - 192*a^3*b
*d*x*tan(d*x)^3*tan(c)^3 + 192*a*b^3*d*x*tan(d*x)^3*tan(c)^3 - 36*a^2*b^2*tan(d*x)^4*tan(c)^4 + 9*b^4*tan(d*x)
^4*tan(c)^4 - 24*a^4*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + t
an(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^3 + 144*a^2*b^2*log(4*(tan(c)^2 + 1)/(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(d*x)^3*tan(c)^3 - 24*
b^4*log(4*(tan(c)^2 + 1)/(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(d*x)^3*tan(c)^3 + 48*a^3*b*tan(d*x)^4*tan(c)^3 - 48*a*b^3*tan(d*x)^4*tan(c)^3 + 48*a^3*
b*tan(d*x)^3*tan(c)^4 - 48*a*b^3*tan(d*x)^3*tan(c)^4 + 288*a^3*b*d*x*tan(d*x)^2*tan(c)^2 - 288*a*b^3*d*x*tan(d
*x)^2*tan(c)^2 - 36*a^2*b^2*tan(d*x)^4*tan(c)^2 + 6*b^4*tan(d*x)^4*tan(c)^2 + 72*a^2*b^2*tan(d*x)^3*tan(c)^3 -
24*b^4*tan(d*x)^3*tan(c)^3 - 36*a^2*b^2*tan(d*x)^2*tan(c)^4 + 6*b^4*tan(d*x)^2*tan(c)^4 + 16*a*b^3*tan(d*x)^4
*tan(c) + 36*a^4*log(4*(tan(c)^2 + 1)/(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(d*x)^2*tan(c)^2 - 216*a^2*b^2*log(4*(tan(c)^2 + 1)/(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(d*x)^2*tan(c)^2 + 36*b^4*
log(4*(tan(c)^2 + 1)/(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(d*x)^2*tan(c)^2 - 144*a^3*b*tan(d*x)^3*tan(c)^2 + 192*a*b^3*tan(d*x)^3*tan(c)^2 - 144*a^3*b
*tan(d*x)^2*tan(c)^3 + 192*a*b^3*tan(d*x)^2*tan(c)^3 + 16*a*b^3*tan(d*x)*tan(c)^4 - 3*b^4*tan(d*x)^4 - 192*a^3
*b*d*x*tan(d*x)*tan(c) + 192*a*b^3*d*x*tan(d*x)*tan(c) + 72*a^2*b^2*tan(d*x)^3*tan(c) - 24*b^4*tan(d*x)^3*tan(
c) - 72*a^2*b^2*tan(d*x)^2*tan(c)^2 + 12*b^4*tan(d*x)^2*tan(c)^2 + 72*a^2*b^2*tan(d*x)*tan(c)^3 - 24*b^4*tan(d
*x)*tan(c)^3 - 3*b^4*tan(c)^4 - 16*a*b^3*tan(d*x)^3 - 24*a^4*log(4*(tan(c)^2 + 1)/(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(d*x)*tan(c) + 144*a^2*b^2*log(
4*(tan(c)^2 + 1)/(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)*ta
n(c) + 1))*tan(d*x)*tan(c) - 24*b^4*log(4*(tan(c)^2 + 1)/(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(d*x)*tan(c) + 144*a^3*b*tan(d*x)^2*tan(c) - 192*a*b^3*t
an(d*x)^2*tan(c) + 144*a^3*b*tan(d*x)*tan(c)^2 - 192*a*b^3*tan(d*x)*tan(c)^2 - 16*a*b^3*tan(c)^3 + 48*a^3*b*d*
x - 48*a*b^3*d*x - 36*a^2*b^2*tan(d*x)^2 + 6*b^4*tan(d*x)^2 + 72*a^2*b^2*tan(d*x)*tan(c) - 24*b^4*tan(d*x)*tan
(c) - 36*a^2*b^2*tan(c)^2 + 6*b^4*tan(c)^2 + 6*a^4*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*ta
n(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) - 36*a^2*b^2*log(4*(tan(c)^2 + 1)/(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)) + 6*b^4*log(4*(t
an(c)^2 + 1)/(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)) - 48*a^3*b*tan(d*x) + 48*a*b^3*tan(d*x) - 48*a^3*b*tan(c) + 48*a*b^3*tan(c) - 36*a^2*b^2 + 9*b^4)/(d*ta
n(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)