3.1.5 \(\int \cot ^3(c+d x) (a+b \tan (c+d x)) (B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\) [5]

Optimal. Leaf size=43 \[ -((a B-b C) x)-\frac {a B \cot (c+d x)}{d}+\frac {(b B+a C) \log (\sin (c+d x))}{d} \]

[Out]

-(B*a-C*b)*x-a*B*cot(d*x+c)/d+(B*b+C*a)*ln(sin(d*x+c))/d

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Rubi [A]
time = 0.09, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3713, 3672, 3612, 3556} \begin {gather*} \frac {(a C+b B) \log (\sin (c+d x))}{d}-(x (a B-b C))-\frac {a B \cot (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^3*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

-((a*B - b*C)*x) - (a*B*Cot[c + d*x])/d + ((b*B + a*C)*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 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]

Rule 3672

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

Rule 3713

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

Rubi steps

\begin {align*} \int \cot ^3(c+d x) (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int \cot ^2(c+d x) (a+b \tan (c+d x)) (B+C \tan (c+d x)) \, dx\\ &=-\frac {a B \cot (c+d x)}{d}+\int \cot (c+d x) (b B+a C-(a B-b C) \tan (c+d x)) \, dx\\ &=-(a B-b C) x-\frac {a B \cot (c+d x)}{d}+(b B+a C) \int \cot (c+d x) \, dx\\ &=-(a B-b C) x-\frac {a B \cot (c+d x)}{d}+\frac {(b B+a C) \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 = 0.11, size = 78, normalized size = 1.81 \begin {gather*} b C x-\frac {a B \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d}+\frac {b B (\log (\cos (c+d x))+\log (\tan (c+d x)))}{d}+\frac {a C (\log (\cos (c+d x))+\log (\tan (c+d x)))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

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

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Maple [A]
time = 0.22, size = 53, normalized size = 1.23

method	result	size
derivativedivides	\(\frac {B b \ln \left (\sin \left (d x +c \right )\right )+C b \left (d x +c \right )+a B \left (-\cot \left (d x +c \right )-d x -c \right )+C a \ln \left (\sin \left (d x +c \right )\right )}{d}\)	\(53\)
default	\(\frac {B b \ln \left (\sin \left (d x +c \right )\right )+C b \left (d x +c \right )+a B \left (-\cot \left (d x +c \right )-d x -c \right )+C a \ln \left (\sin \left (d x +c \right )\right )}{d}\)	\(53\)
norman	\(\frac {\left (-a B +C b \right ) x \left (\tan ^{2}\left (d x +c \right )\right )-\frac {a B \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{2}}+\frac {\left (B b +C a \right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (B b +C a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(84\)
risch	\(-i B b x -i C a x -B a x +C b x -\frac {2 i B b c}{d}-\frac {2 i C a c}{d}-\frac {2 i a B}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B b}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C a}{d}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(a+b*tan(d*x+c))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(B*b*ln(sin(d*x+c))+C*b*(d*x+c)+a*B*(-cot(d*x+c)-d*x-c)+C*a*ln(sin(d*x+c)))

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Maxima [A]
time = 0.52, size = 68, normalized size = 1.58 \begin {gather*} -\frac {2 \, {\left (B a - C b\right )} {\left (d x + c\right )} + {\left (C a + B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, {\left (C a + B b\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {2 \, B a}{\tan \left (d x + c\right )}}{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))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 3.56, size = 73, normalized size = 1.70 \begin {gather*} -\frac {2 \, {\left (B a - C b\right )} d x \tan \left (d x + c\right ) - {\left (C a + B b\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 \, B a}{2 \, d \tan \left (d x + c\right )} \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))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (36) = 72\).
time = 0.93, size = 122, normalized size = 2.84 \begin {gather*} \begin {cases} \text {NaN} & \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 ) \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{3}{\left (c \right )} & \text {for}\: d = 0 \\- B a x - \frac {B a}{d \tan {\left (c + d x \right )}} - \frac {B b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {C a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + C b x & \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))*(B*tan(d*x+c)+C*tan(d*x+c)**2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (43) = 86\).
time = 1.13, size = 119, normalized size = 2.77 \begin {gather*} \frac {B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, {\left (B a - C b\right )} {\left (d x + c\right )} - 2 \, {\left (C a + B b\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 2 \, {\left (C a + B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + B a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{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))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 8.87, size = 87, normalized size = 2.02 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,b+C\,a\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,\left (b+a\,1{}\mathrm {i}\right )}{2\,d}-\frac {B\,a\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,\left (a+b\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^3*(B*tan(c + d*x) + C*tan(c + d*x)^2)*(a + b*tan(c + d*x)),x)

[Out]

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

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