3.5.49 \(\int \cot (c+d x) (a+b \tan (c+d x))^4 \, dx\) [449]

Optimal. Leaf size=92 \[ 4 a b \left (a^2-b^2\right ) x-\frac {b^2 \left (6 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {a^4 \log (\sin (c+d x))}{d}+\frac {3 a b^3 \tan (c+d x)}{d}+\frac {b^2 (a+b \tan (c+d x))^2}{2 d} \]

[Out]

4*a*b*(a^2-b^2)*x-b^2*(6*a^2-b^2)*ln(cos(d*x+c))/d+a^4*ln(sin(d*x+c))/d+3*a*b^3*tan(d*x+c)/d+1/2*b^2*(a+b*tan(
d*x+c))^2/d

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Rubi [A]
time = 0.13, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3647, 3718, 3705, 3556} \begin {gather*} \frac {a^4 \log (\sin (c+d x))}{d}-\frac {b^2 \left (6 a^2-b^2\right ) \log (\cos (c+d x))}{d}+4 a b x \left (a^2-b^2\right )+\frac {3 a b^3 \tan (c+d x)}{d}+\frac {b^2 (a+b \tan (c+d x))^2}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*a*b*(a^2 - b^2)*x - (b^2*(6*a^2 - b^2)*Log[Cos[c + d*x]])/d + (a^4*Log[Sin[c + d*x]])/d + (3*a*b^3*Tan[c + d
*x])/d + (b^2*(a + b*Tan[c + d*x])^2)/(2*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 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

Rule 3705

Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/tan[(e_.) + (f_.)*(x_)], x_Symbol
] :> Simp[B*x, x] + (Dist[A, Int[1/Tan[e + f*x], x], x] + Dist[C, Int[Tan[e + f*x], x], x]) /; FreeQ[{e, f, A,
B, C}, x] && NeQ[A, C]

Rule 3718

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])
^(n + 1)/(d*f*(n + 2))), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b
+ a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] &&  !LtQ[n, -1]

Rubi steps

\begin {align*} \int \cot (c+d x) (a+b \tan (c+d x))^4 \, dx &=\frac {b^2 (a+b \tan (c+d x))^2}{2 d}+\frac {1}{2} \int \cot (c+d x) (a+b \tan (c+d x)) \left (2 a^3+2 b \left (3 a^2-b^2\right ) \tan (c+d x)+6 a b^2 \tan ^2(c+d x)\right ) \, dx\\ &=\frac {3 a b^3 \tan (c+d x)}{d}+\frac {b^2 (a+b \tan (c+d x))^2}{2 d}-\frac {1}{2} \int \cot (c+d x) \left (-2 a^4-8 a b \left (a^2-b^2\right ) \tan (c+d x)-2 b^2 \left (6 a^2-b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=4 a b \left (a^2-b^2\right ) x+\frac {3 a b^3 \tan (c+d x)}{d}+\frac {b^2 (a+b \tan (c+d x))^2}{2 d}+a^4 \int \cot (c+d x) \, dx+\left (b^2 \left (6 a^2-b^2\right )\right ) \int \tan (c+d x) \, dx\\ &=4 a b \left (a^2-b^2\right ) x-\frac {b^2 \left (6 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {a^4 \log (\sin (c+d x))}{d}+\frac {3 a b^3 \tan (c+d x)}{d}+\frac {b^2 (a+b \tan (c+d x))^2}{2 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.42, size = 94, normalized size = 1.02 \begin {gather*} \frac {-(a+i b)^4 \log (i-\tan (c+d x))+2 a^4 \log (\tan (c+d x))-(a-i b)^4 \log (i+\tan (c+d x))+6 a b^3 \tan (c+d x)+b^2 (a+b \tan (c+d x))^2}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.20, size = 85, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {a^{4} \ln \left (\sin \left (d x +c \right )\right )+4 a^{3} b \left (d x +c \right )-6 a^{2} b^{2} \ln \left (\cos \left (d x +c \right )\right )+4 a \,b^{3} \left (\tan \left (d x +c \right )-d x -c \right )+b^{4} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\)	\(85\)
default	\(\frac {a^{4} \ln \left (\sin \left (d x +c \right )\right )+4 a^{3} b \left (d x +c \right )-6 a^{2} b^{2} \ln \left (\cos \left (d x +c \right )\right )+4 a \,b^{3} \left (\tan \left (d x +c \right )-d x -c \right )+b^{4} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\)	\(85\)
norman	\(\left (4 a^{3} b -4 a \,b^{3}\right ) x +\frac {b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {4 a \,b^{3} \tan \left (d x +c \right )}{d}+\frac {a^{4} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(93\)
risch	\(4 a^{3} b x -4 a \,b^{3} x -i a^{4} x +6 i a^{2} b^{2} x -i b^{4} x +\frac {12 i a^{2} b^{2} c}{d}-\frac {2 i b^{4} c}{d}-\frac {2 i a^{4} c}{d}+\frac {2 i b^{3} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 a \,{\mathrm e}^{2 i \left (d x +c \right )}+4 a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2} b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) b^{4}}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^4*ln(sin(d*x+c))+4*a^3*b*(d*x+c)-6*a^2*b^2*ln(cos(d*x+c))+4*a*b^3*(tan(d*x+c)-d*x-c)+b^4*(1/2*tan(d*x+c
)^2+ln(cos(d*x+c))))

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Maxima [A]
time = 0.52, size = 89, normalized size = 0.97 \begin {gather*} \frac {b^{4} \tan \left (d x + c\right )^{2} + 2 \, a^{4} \log \left (\tan \left (d x + c\right )\right ) + 8 \, a b^{3} \tan \left (d x + c\right ) + 8 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*tan(d*x + c)^2 + 2*a^4*log(tan(d*x + c)) + 8*a*b^3*tan(d*x + c) + 8*(a^3*b - a*b^3)*(d*x + c) - (a^4
- 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1))/d

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Fricas [A]
time = 1.01, size = 101, normalized size = 1.10 \begin {gather*} \frac {b^{4} \tan \left (d x + c\right )^{2} + a^{4} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 8 \, a b^{3} \tan \left (d x + c\right ) + 8 \, {\left (a^{3} b - a b^{3}\right )} d x - {\left (6 \, a^{2} b^{2} - b^{4}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*tan(d*x + c)^2 + a^4*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) + 8*a*b^3*tan(d*x + c) + 8*(a^3*b - a*b
^3)*d*x - (6*a^2*b^2 - b^4)*log(1/(tan(d*x + c)^2 + 1)))/d

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Sympy [A]
time = 0.53, size = 133, normalized size = 1.45 \begin {gather*} \begin {cases} - \frac {a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 4 a^{3} b x + \frac {3 a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - 4 a b^{3} x + \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 ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right )^{4} \cot {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+b*tan(d*x+c))**4,x)

[Out]

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

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Giac [A]
time = 1.43, size = 90, normalized size = 0.98 \begin {gather*} \frac {b^{4} \tan \left (d x + c\right )^{2} + 2 \, a^{4} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 8 \, a b^{3} \tan \left (d x + c\right ) + 8 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*tan(d*x + c)^2 + 2*a^4*log(abs(tan(d*x + c))) + 8*a*b^3*tan(d*x + c) + 8*(a^3*b - a*b^3)*(d*x + c) -
(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1))/d

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Mupad [B]
time = 3.97, size = 92, normalized size = 1.00 \begin {gather*} \frac {b^4\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^4}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^4}{2\,d}+\frac {a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {4\,a\,b^3\,\mathrm {tan}\left (c+d\,x\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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