3.5.51 \(\int \cot ^3(c+d x) (a+b \tan (c+d x))^4 \, dx\) [451]

Optimal. Leaf size=99 \[ -4 a b \left (a^2-b^2\right ) x-\frac {3 a^3 b \cot (c+d x)}{d}-\frac {b^4 \log (\cos (c+d x))}{d}-\frac {a^2 \left (a^2-6 b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3646, 3716, 3705, 3556} \begin {gather*} -\frac {3 a^3 b \cot (c+d x)}{d}-\frac {a^2 \left (a^2-6 b^2\right ) \log (\sin (c+d x))}{d}-4 a b x \left (a^2-b^2\right )-\frac {a^2 \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {b^4 \log (\cos (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]

[Out]

-4*a*b*(a^2 - b^2)*x - (3*a^3*b*Cot[c + d*x])/d - (b^4*Log[Cos[c + d*x]])/d - (a^2*(a^2 - 6*b^2)*Log[Sin[c + d
*x]])/d - (a^2*Cot[c + d*x]^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 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

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 3716

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 - a*d))*(c^2*C - B*c*d + A*d^2)
*((c + d*Tan[e + f*x])^(n + 1)/(d^2*f*(n + 1)*(c^2 + d^2))), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f
*x])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d
+ a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] &
& NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \cot ^3(c+d x) (a+b \tan (c+d x))^4 \, dx &=-\frac {a^2 \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac {1}{2} \int \cot ^2(c+d x) (a+b \tan (c+d x)) \left (6 a^2 b-2 a \left (a^2-3 b^2\right ) \tan (c+d x)+2 b^3 \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {3 a^3 b \cot (c+d x)}{d}-\frac {a^2 \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac {1}{2} \int \cot (c+d x) \left (-2 a^2 \left (a^2-6 b^2\right )-8 a b \left (a^2-b^2\right ) \tan (c+d x)+2 b^4 \tan ^2(c+d x)\right ) \, dx\\ &=-4 a b \left (a^2-b^2\right ) x-\frac {3 a^3 b \cot (c+d x)}{d}-\frac {a^2 \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}+b^4 \int \tan (c+d x) \, dx-\left (a^2 \left (a^2-6 b^2\right )\right ) \int \cot (c+d x) \, dx\\ &=-4 a b \left (a^2-b^2\right ) x-\frac {3 a^3 b \cot (c+d x)}{d}-\frac {b^4 \log (\cos (c+d x))}{d}-\frac {a^2 \left (a^2-6 b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^2(c+d x) (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 = 0.32, size = 90, normalized size = 0.91 \begin {gather*} -\frac {8 a^3 b \cot (c+d x)+a^4 \cot ^2(c+d x)-(a-i b)^4 \log (i-\cot (c+d x))-(a+i b)^4 \log (i+\cot (c+d x))-2 b^4 \log (\tan (c+d x))}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]

[Out]

-1/2*(8*a^3*b*Cot[c + d*x] + a^4*Cot[c + d*x]^2 - (a - I*b)^4*Log[I - Cot[c + d*x]] - (a + I*b)^4*Log[I + Cot[
c + d*x]] - 2*b^4*Log[Tan[c + d*x]])/d

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Maple [A]
time = 0.21, size = 90, normalized size = 0.91

method	result	size
derivativedivides	\(\frac {a^{4} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+4 a^{3} b \left (-\cot \left (d x +c \right )-d x -c \right )+6 a^{2} b^{2} \ln \left (\sin \left (d x +c \right )\right )+4 a \,b^{3} \left (d x +c \right )-b^{4} \ln \left (\cos \left (d x +c \right )\right )}{d}\)	\(90\)
default	\(\frac {a^{4} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+4 a^{3} b \left (-\cot \left (d x +c \right )-d x -c \right )+6 a^{2} b^{2} \ln \left (\sin \left (d x +c \right )\right )+4 a \,b^{3} \left (d x +c \right )-b^{4} \ln \left (\cos \left (d x +c \right )\right )}{d}\)	\(90\)
norman	\(\frac {\left (-4 a^{3} b +4 a \,b^{3}\right ) x \left (\tan ^{2}\left (d x +c \right )\right )-\frac {a^{4}}{2 d}-\frac {4 a^{3} b \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{2}}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{2} \left (a^{2}-6 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}\)	\(113\)
risch	\(-4 a^{3} b x +4 a \,b^{3} x +i a^{4} x -6 i a^{2} b^{2} x +i b^{4} x +\frac {2 i a^{4} c}{d}-\frac {12 i a^{2} b^{2} c}{d}+\frac {2 i b^{4} c}{d}+\frac {2 a^{3} \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}-4 i b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 i b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {6 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) b^{4}}{d}\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 99, normalized size = 1.00 \begin {gather*} -\frac {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 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {8 \, a^{3} b \tan \left (d x + c\right ) + a^{4}}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.60, size = 126, normalized size = 1.27 \begin {gather*} -\frac {b^{4} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + 8 \, a^{3} b \tan \left (d x + c\right ) + a^{4} + {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + {\left (a^{4} + 8 \, {\left (a^{3} b - a b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2}}{2 \, d \tan \left (d x + c\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 1.36, size = 170, normalized size = 1.72 \begin {gather*} \begin {cases} \tilde {\infty } a^{4} x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan {\left (c \right )}\right )^{4} \cot ^{3}{\left (c \right )} & \text {for}\: d = 0 \\\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} - \frac {a^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} - 4 a^{3} b x - \frac {4 a^{3} b}{d \tan {\left (c + d x \right )}} - \frac {3 a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {6 a^{2} b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 4 a b^{3} x + \frac {b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3*(a+b*tan(d*x+c))**4,x)

[Out]

Piecewise((zoo*a**4*x, (Eq(c, 0) | Eq(c, -d*x)) & (Eq(d, 0) | Eq(c, -d*x))), (x*(a + b*tan(c))**4*cot(c)**3, E
q(d, 0)), (a**4*log(tan(c + d*x)**2 + 1)/(2*d) - a**4*log(tan(c + d*x))/d - a**4/(2*d*tan(c + d*x)**2) - 4*a**
3*b*x - 4*a**3*b/(d*tan(c + d*x)) - 3*a**2*b**2*log(tan(c + d*x)**2 + 1)/d + 6*a**2*b**2*log(tan(c + d*x))/d +
4*a*b**3*x + b**4*log(tan(c + d*x)**2 + 1)/(2*d), True))

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Giac [A]
time = 1.85, size = 132, normalized size = 1.33 \begin {gather*} -\frac {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 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - \frac {3 \, a^{4} \tan \left (d x + c\right )^{2} - 18 \, a^{2} b^{2} \tan \left (d x + c\right )^{2} - 8 \, a^{3} b \tan \left (d x + c\right ) - a^{4}}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/2*(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) + 2*(a^4 - 6*a^2*b^2)*log(
abs(tan(d*x + c))) - (3*a^4*tan(d*x + c)^2 - 18*a^2*b^2*tan(d*x + c)^2 - 8*a^3*b*tan(d*x + c) - a^4)/tan(d*x +
c)^2)/d

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Mupad [B]
time = 4.01, size = 102, normalized size = 1.03 \begin {gather*} -\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^4-6\,a^2\,b^2\right )}{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 {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^4}{2\,d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^2\,\left (\frac {a^4}{2}+4\,b\,\mathrm {tan}\left (c+d\,x\right )\,a^3\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^3*(a + b*tan(c + d*x))^4,x)

[Out]

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

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