3.5.20 \(\int \cot ^3(c+d x) (a+b \tan (c+d x)) \, dx\) [420]

Optimal. Leaf size=46 \[ -b x-\frac {b \cot (c+d x)}{d}-\frac {a \cot ^2(c+d x)}{2 d}-\frac {a \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3610, 3612, 3556} \begin {gather*} -\frac {a \cot ^2(c+d x)}{2 d}-\frac {a \log (\sin (c+d x))}{d}-\frac {b \cot (c+d x)}{d}-b x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*x) - (b*Cot[c + d*x])/d - (a*Cot[c + d*x]^2)/(2*d) - (a*Log[Sin[c + d*x]])/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 3610

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

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*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] && NeQ[a*c + b*d, 0]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.29, size = 66, normalized size = 1.43 \begin {gather*} -\frac {b \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d}-\frac {a \left (\cot ^2(c+d x)+2 \log (\cos (c+d x))+2 \log (\tan (c+d x))\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d) - (a*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*
x]] + 2*Log[Tan[c + d*x]]))/(2*d)

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Maple [A]
time = 0.16, size = 46, normalized size = 1.00

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+b \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(46\)
default	\(\frac {a \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+b \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(46\)
norman	\(\frac {-\frac {a}{2 d}-b x \left (\tan ^{2}\left (d x +c \right )\right )-\frac {b \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{2}}-\frac {a \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(72\)
risch	\(-b x +i a x +\frac {2 i a c}{d}-\frac {2 i \left (i a \,{\mathrm e}^{2 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}-b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.52, size = 58, normalized size = 1.26 \begin {gather*} -\frac {2 \, {\left (d x + c\right )} b - a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, a \log \left (\tan \left (d x + c\right )\right ) + \frac {2 \, b \tan \left (d x + c\right ) + a}{\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)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.98, size = 72, normalized size = 1.57 \begin {gather*} -\frac {a \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 (2 \, b d x + a\right )} \tan \left (d x + c\right )^{2} + 2 \, b \tan \left (d x + c\right ) + a}{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)),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (39) = 78\).
time = 0.50, size = 83, normalized size = 1.80 \begin {gather*} \begin {cases} \tilde {\infty } a 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 ) \cot ^{3}{\left (c \right )} & \text {for}\: d = 0 \\\frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {a}{2 d \tan ^{2}{\left (c + d x \right )}} - b x - \frac {b}{d \tan {\left (c + d x \right )}} & \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)),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (44) = 88\).
time = 0.63, size = 113, normalized size = 2.46 \begin {gather*} -\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, {\left (d x + c\right )} b - 8 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 8 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, 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)),x, algorithm="giac")

[Out]

-1/8*(a*tan(1/2*d*x + 1/2*c)^2 + 8*(d*x + c)*b - 8*a*log(tan(1/2*d*x + 1/2*c)^2 + 1) + 8*a*log(abs(tan(1/2*d*x
+ 1/2*c))) - 4*b*tan(1/2*d*x + 1/2*c) - (12*a*tan(1/2*d*x + 1/2*c)^2 - 4*b*tan(1/2*d*x + 1/2*c) - a)/tan(1/2*
d*x + 1/2*c)^2)/d

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Mupad [B]
time = 4.07, size = 83, normalized size = 1.80 \begin {gather*} -\frac {{\mathrm {cot}\left (c+d\,x\right )}^2\,\left (\frac {a}{2}+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (\frac {a}{2}+\frac {b\,1{}\mathrm {i}}{2}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (\frac {a}{2}-\frac {b\,1{}\mathrm {i}}{2}\right )}{d}-\frac {a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\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)),x)

[Out]

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

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