3.2.46 \(\int \cos ^3(a+b x) \cot ^3(a+b x) \, dx\) [146]

Optimal. Leaf size=66 \[ \frac {5 \tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {5 \cos (a+b x)}{2 b}-\frac {5 \cos ^3(a+b x)}{6 b}-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b} \]

[Out]

5/2*arctanh(cos(b*x+a))/b-5/2*cos(b*x+a)/b-5/6*cos(b*x+a)^3/b-1/2*cos(b*x+a)^3*cot(b*x+a)^2/b

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Rubi [A]
time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2672, 294, 308, 212} \begin {gather*} -\frac {5 \cos ^3(a+b x)}{6 b}-\frac {5 \cos (a+b x)}{2 b}-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}+\frac {5 \tanh ^{-1}(\cos (a+b x))}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^3*Cot[a + b*x]^3,x]

[Out]

(5*ArcTanh[Cos[a + b*x]])/(2*b) - (5*Cos[a + b*x])/(2*b) - (5*Cos[a + b*x]^3)/(6*b) - (Cos[a + b*x]^3*Cot[a +
b*x]^2)/(2*b)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2672

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rubi steps

\begin {align*} \int \cos ^3(a+b x) \cot ^3(a+b x) \, dx &=-\frac {\text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}+\frac {5 \text {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cos (a+b x)\right )}{2 b}\\ &=-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}+\frac {5 \text {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cos (a+b x)\right )}{2 b}\\ &=-\frac {5 \cos (a+b x)}{2 b}-\frac {5 \cos ^3(a+b x)}{6 b}-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}+\frac {5 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (a+b x)\right )}{2 b}\\ &=\frac {5 \tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {5 \cos (a+b x)}{2 b}-\frac {5 \cos ^3(a+b x)}{6 b}-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 103, normalized size = 1.56 \begin {gather*} -\frac {9 \cos (a+b x)}{4 b}-\frac {\cos (3 (a+b x))}{12 b}-\frac {\csc ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {5 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {5 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {\sec ^2\left (\frac {1}{2} (a+b x)\right )}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^3*Cot[a + b*x]^3,x]

[Out]

(-9*Cos[a + b*x])/(4*b) - Cos[3*(a + b*x)]/(12*b) - Csc[(a + b*x)/2]^2/(8*b) + (5*Log[Cos[(a + b*x)/2]])/(2*b)
- (5*Log[Sin[(a + b*x)/2]])/(2*b) + Sec[(a + b*x)/2]^2/(8*b)

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Maple [A]
time = 0.06, size = 70, normalized size = 1.06

method	result	size
derivativedivides	\(\frac {-\frac {\cos ^{7}\left (b x +a \right )}{2 \sin \left (b x +a \right )^{2}}-\frac {\left (\cos ^{5}\left (b x +a \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (b x +a \right )\right )}{6}-\frac {5 \cos \left (b x +a \right )}{2}-\frac {5 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{2}}{b}\)	\(70\)
default	\(\frac {-\frac {\cos ^{7}\left (b x +a \right )}{2 \sin \left (b x +a \right )^{2}}-\frac {\left (\cos ^{5}\left (b x +a \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (b x +a \right )\right )}{6}-\frac {5 \cos \left (b x +a \right )}{2}-\frac {5 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{2}}{b}\)	\(70\)
norman	\(\frac {-\frac {1}{8 b}+\frac {\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}-\frac {75 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {65 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{12 b}-\frac {55 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}-\frac {5 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b}\)	\(114\)
risch	\(-\frac {{\mathrm e}^{3 i \left (b x +a \right )}}{24 b}-\frac {9 \,{\mathrm e}^{i \left (b x +a \right )}}{8 b}-\frac {9 \,{\mathrm e}^{-i \left (b x +a \right )}}{8 b}-\frac {{\mathrm e}^{-3 i \left (b x +a \right )}}{24 b}+\frac {{\mathrm e}^{3 i \left (b x +a \right )}+{\mathrm e}^{i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {5 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{2 b}+\frac {5 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{2 b}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^6/sin(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/b*(-1/2*cos(b*x+a)^7/sin(b*x+a)^2-1/2*cos(b*x+a)^5-5/6*cos(b*x+a)^3-5/2*cos(b*x+a)-5/2*ln(csc(b*x+a)-cot(b*x
+a)))

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Maxima [A]
time = 0.28, size = 66, normalized size = 1.00 \begin {gather*} -\frac {4 \, \cos \left (b x + a\right )^{3} - \frac {6 \, \cos \left (b x + a\right )}{\cos \left (b x + a\right )^{2} - 1} + 24 \, \cos \left (b x + a\right ) - 15 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 15 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{12 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^6/sin(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/12*(4*cos(b*x + a)^3 - 6*cos(b*x + a)/(cos(b*x + a)^2 - 1) + 24*cos(b*x + a) - 15*log(cos(b*x + a) + 1) + 1
5*log(cos(b*x + a) - 1))/b

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Fricas [A]
time = 0.38, size = 93, normalized size = 1.41 \begin {gather*} -\frac {4 \, \cos \left (b x + a\right )^{5} + 20 \, \cos \left (b x + a\right )^{3} - 15 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 30 \, \cos \left (b x + a\right )}{12 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^6/sin(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/12*(4*cos(b*x + a)^5 + 20*cos(b*x + a)^3 - 15*(cos(b*x + a)^2 - 1)*log(1/2*cos(b*x + a) + 1/2) + 15*(cos(b*
x + a)^2 - 1)*log(-1/2*cos(b*x + a) + 1/2) - 30*cos(b*x + a))/(b*cos(b*x + a)^2 - b)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 719 vs. \(2 (58) = 116\).
time = 2.53, size = 719, normalized size = 10.89 \begin {gather*} \begin {cases} - \frac {60 \log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \tan ^{8}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{24 b \tan ^{8}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{6}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 24 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} - \frac {180 \log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \tan ^{6}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{24 b \tan ^{8}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{6}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 24 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} - \frac {180 \log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{24 b \tan ^{8}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{6}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 24 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} - \frac {60 \log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{24 b \tan ^{8}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{6}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 24 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} + \frac {3 \tan ^{10}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{24 b \tan ^{8}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{6}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 24 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} - \frac {165 \tan ^{6}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{24 b \tan ^{8}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{6}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 24 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} - \frac {225 \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{24 b \tan ^{8}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{6}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 24 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} - \frac {130 \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{24 b \tan ^{8}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{6}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 24 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} - \frac {3}{24 b \tan ^{8}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{6}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 72 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 24 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{6}{\left (a \right )}}{\sin ^{3}{\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**6/sin(b*x+a)**3,x)

[Out]

Piecewise((-60*log(tan(a/2 + b*x/2))*tan(a/2 + b*x/2)**8/(24*b*tan(a/2 + b*x/2)**8 + 72*b*tan(a/2 + b*x/2)**6
+ 72*b*tan(a/2 + b*x/2)**4 + 24*b*tan(a/2 + b*x/2)**2) - 180*log(tan(a/2 + b*x/2))*tan(a/2 + b*x/2)**6/(24*b*t
an(a/2 + b*x/2)**8 + 72*b*tan(a/2 + b*x/2)**6 + 72*b*tan(a/2 + b*x/2)**4 + 24*b*tan(a/2 + b*x/2)**2) - 180*log
(tan(a/2 + b*x/2))*tan(a/2 + b*x/2)**4/(24*b*tan(a/2 + b*x/2)**8 + 72*b*tan(a/2 + b*x/2)**6 + 72*b*tan(a/2 + b
*x/2)**4 + 24*b*tan(a/2 + b*x/2)**2) - 60*log(tan(a/2 + b*x/2))*tan(a/2 + b*x/2)**2/(24*b*tan(a/2 + b*x/2)**8
+ 72*b*tan(a/2 + b*x/2)**6 + 72*b*tan(a/2 + b*x/2)**4 + 24*b*tan(a/2 + b*x/2)**2) + 3*tan(a/2 + b*x/2)**10/(24
*b*tan(a/2 + b*x/2)**8 + 72*b*tan(a/2 + b*x/2)**6 + 72*b*tan(a/2 + b*x/2)**4 + 24*b*tan(a/2 + b*x/2)**2) - 165
*tan(a/2 + b*x/2)**6/(24*b*tan(a/2 + b*x/2)**8 + 72*b*tan(a/2 + b*x/2)**6 + 72*b*tan(a/2 + b*x/2)**4 + 24*b*ta
n(a/2 + b*x/2)**2) - 225*tan(a/2 + b*x/2)**4/(24*b*tan(a/2 + b*x/2)**8 + 72*b*tan(a/2 + b*x/2)**6 + 72*b*tan(a
/2 + b*x/2)**4 + 24*b*tan(a/2 + b*x/2)**2) - 130*tan(a/2 + b*x/2)**2/(24*b*tan(a/2 + b*x/2)**8 + 72*b*tan(a/2
+ b*x/2)**6 + 72*b*tan(a/2 + b*x/2)**4 + 24*b*tan(a/2 + b*x/2)**2) - 3/(24*b*tan(a/2 + b*x/2)**8 + 72*b*tan(a/
2 + b*x/2)**6 + 72*b*tan(a/2 + b*x/2)**4 + 24*b*tan(a/2 + b*x/2)**2), Ne(b, 0)), (x*cos(a)**6/sin(a)**3, True)
)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (58) = 116\).
time = 3.96, size = 163, normalized size = 2.47 \begin {gather*} \frac {\frac {3 \, {\left (\frac {10 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} - \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {16 \, {\left (\frac {12 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 7\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{3}} - 30 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{24 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^6/sin(b*x+a)^3,x, algorithm="giac")

[Out]

1/24*(3*(10*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)*(cos(b*x + a) + 1)/(cos(b*x + a) - 1) - 3*(cos(b*x + a)
- 1)/(cos(b*x + a) + 1) - 16*(12*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 9*(cos(b*x + a) - 1)^2/(cos(b*x + a)
+ 1)^2 - 7)/((cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)^3 - 30*log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) +
1)))/b

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Mupad [B]
time = 1.35, size = 129, normalized size = 1.95 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2}{8\,b}-\frac {5\,\ln \left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{2\,b}-\frac {\frac {49\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^6}{8}+\frac {67\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4}{8}+\frac {121\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2}{24}+\frac {1}{8}}{b\,\left ({\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^8+3\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^6/sin(a + b*x)^3,x)

[Out]

tan(a/2 + (b*x)/2)^2/(8*b) - (5*log(tan(a/2 + (b*x)/2)))/(2*b) - ((121*tan(a/2 + (b*x)/2)^2)/24 + (67*tan(a/2
+ (b*x)/2)^4)/8 + (49*tan(a/2 + (b*x)/2)^6)/8 + 1/8)/(b*(tan(a/2 + (b*x)/2)^2 + 3*tan(a/2 + (b*x)/2)^4 + 3*tan
(a/2 + (b*x)/2)^6 + tan(a/2 + (b*x)/2)^8))

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