3.448 \(\int \frac{\cot ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx\)

Optimal. Leaf size=121 \[ -\frac{\left (3 a^2+3 a b+b^2\right ) \csc ^2(c+d x)}{2 a^3 d}+\frac{(3 a+b) \csc ^4(c+d x)}{4 a^2 d}+\frac{(a+b)^3 \log \left (a+b \sin ^2(c+d x)\right )}{2 a^4 d}-\frac{(a+b)^3 \log (\sin (c+d x))}{a^4 d}-\frac{\csc ^6(c+d x)}{6 a d} \]

[Out]

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

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Rubi [A]  time = 0.111973, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3194, 88} \[ -\frac{\left (3 a^2+3 a b+b^2\right ) \csc ^2(c+d x)}{2 a^3 d}+\frac{(3 a+b) \csc ^4(c+d x)}{4 a^2 d}+\frac{(a+b)^3 \log \left (a+b \sin ^2(c+d x)\right )}{2 a^4 d}-\frac{(a+b)^3 \log (\sin (c+d x))}{a^4 d}-\frac{\csc ^6(c+d x)}{6 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7/(a + b*Sin[c + d*x]^2),x]

[Out]

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

Rule 3194

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(a + b*ff*x)^p)/(1 - ff*x)^((
m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{\cot ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(1-x)^3}{x^4 (a+b x)} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^4}+\frac{-3 a-b}{a^2 x^3}+\frac{3 a^2+3 a b+b^2}{a^3 x^2}-\frac{(a+b)^3}{a^4 x}+\frac{b (a+b)^3}{a^4 (a+b x)}\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac{\left (3 a^2+3 a b+b^2\right ) \csc ^2(c+d x)}{2 a^3 d}+\frac{(3 a+b) \csc ^4(c+d x)}{4 a^2 d}-\frac{\csc ^6(c+d x)}{6 a d}-\frac{(a+b)^3 \log (\sin (c+d x))}{a^4 d}+\frac{(a+b)^3 \log \left (a+b \sin ^2(c+d x)\right )}{2 a^4 d}\\ \end{align*}

Mathematica [A]  time = 0.268362, size = 100, normalized size = 0.83 \[ -\frac{6 a \left (3 a^2+3 a b+b^2\right ) \csc ^2(c+d x)-3 a^2 (3 a+b) \csc ^4(c+d x)+2 a^3 \csc ^6(c+d x)-6 (a+b)^3 \log \left (a+b \sin ^2(c+d x)\right )+12 (a+b)^3 \log (\sin (c+d x))}{12 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7/(a + b*Sin[c + d*x]^2),x]

[Out]

-(6*a*(3*a^2 + 3*a*b + b^2)*Csc[c + d*x]^2 - 3*a^2*(3*a + b)*Csc[c + d*x]^4 + 2*a^3*Csc[c + d*x]^6 + 12*(a + b
)^3*Log[Sin[c + d*x]] - 6*(a + b)^3*Log[a + b*Sin[c + d*x]^2])/(12*a^4*d)

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Maple [B]  time = 0.108, size = 489, normalized size = 4. \begin{align*}{\frac{1}{48\,da \left ( -1+\cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{5}{32\,da \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{b}{16\,{a}^{2}d \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{19}{32\,da \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{11\,b}{16\,{a}^{2}d \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{{b}^{2}}{4\,d{a}^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{2\,da}}-{\frac{3\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ) b}{2\,{a}^{2}d}}-{\frac{3\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ){b}^{2}}{2\,d{a}^{3}}}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ){b}^{3}}{2\,d{a}^{4}}}-{\frac{1}{48\,da \left ( 1+\cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{5}{32\,da \left ( 1+\cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{b}{16\,{a}^{2}d \left ( 1+\cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{19}{32\,da \left ( 1+\cos \left ( dx+c \right ) \right ) }}-{\frac{11\,b}{16\,{a}^{2}d \left ( 1+\cos \left ( dx+c \right ) \right ) }}-{\frac{{b}^{2}}{4\,d{a}^{3} \left ( 1+\cos \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( 1+\cos \left ( dx+c \right ) \right ) }{2\,da}}-{\frac{3\,\ln \left ( 1+\cos \left ( dx+c \right ) \right ) b}{2\,{a}^{2}d}}-{\frac{3\,\ln \left ( 1+\cos \left ( dx+c \right ) \right ){b}^{2}}{2\,d{a}^{3}}}-{\frac{\ln \left ( 1+\cos \left ( dx+c \right ) \right ){b}^{3}}{2\,d{a}^{4}}}+{\frac{\ln \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b \right ) }{2\,da}}+{\frac{3\,\ln \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b \right ) b}{2\,{a}^{2}d}}+{\frac{3\,\ln \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b \right ){b}^{2}}{2\,d{a}^{3}}}+{\frac{\ln \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b \right ){b}^{3}}{2\,d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^7/(a+sin(d*x+c)^2*b),x)

[Out]

1/48/d/a/(-1+cos(d*x+c))^3+5/32/d/a/(-1+cos(d*x+c))^2+1/16/d/a^2/(-1+cos(d*x+c))^2*b+19/32/d/a/(-1+cos(d*x+c))
+11/16/d/a^2/(-1+cos(d*x+c))*b+1/4/d/a^3/(-1+cos(d*x+c))*b^2-1/2/d/a*ln(-1+cos(d*x+c))-3/2/d/a^2*ln(-1+cos(d*x
+c))*b-3/2/d/a^3*ln(-1+cos(d*x+c))*b^2-1/2/d/a^4*ln(-1+cos(d*x+c))*b^3-1/48/d/a/(1+cos(d*x+c))^3+5/32/a/d/(1+c
os(d*x+c))^2+1/16/d/a^2/(1+cos(d*x+c))^2*b-19/32/a/d/(1+cos(d*x+c))-11/16/d/a^2/(1+cos(d*x+c))*b-1/4/d/a^3/(1+
cos(d*x+c))*b^2-1/2/d/a*ln(1+cos(d*x+c))-3/2/d/a^2*ln(1+cos(d*x+c))*b-3/2/d/a^3*ln(1+cos(d*x+c))*b^2-1/2/d/a^4
*ln(1+cos(d*x+c))*b^3+1/2/d/a*ln(b*cos(d*x+c)^2-a-b)+3/2/d/a^2*ln(b*cos(d*x+c)^2-a-b)*b+3/2/d/a^3*ln(b*cos(d*x
+c)^2-a-b)*b^2+1/2/d/a^4*ln(b*cos(d*x+c)^2-a-b)*b^3

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Maxima [A]  time = 1.01059, size = 185, normalized size = 1.53 \begin{align*} \frac{\frac{6 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (b \sin \left (d x + c\right )^{2} + a\right )}{a^{4}} - \frac{6 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (\sin \left (d x + c\right )^{2}\right )}{a^{4}} - \frac{6 \,{\left (3 \, a^{2} + 3 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{4} - 3 \,{\left (3 \, a^{2} + a b\right )} \sin \left (d x + c\right )^{2} + 2 \, a^{2}}{a^{3} \sin \left (d x + c\right )^{6}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+b*sin(d*x+c)^2),x, algorithm="maxima")

[Out]

1/12*(6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*log(b*sin(d*x + c)^2 + a)/a^4 - 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*log(
sin(d*x + c)^2)/a^4 - (6*(3*a^2 + 3*a*b + b^2)*sin(d*x + c)^4 - 3*(3*a^2 + a*b)*sin(d*x + c)^2 + 2*a^2)/(a^3*s
in(d*x + c)^6))/d

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Fricas [B]  time = 2.83991, size = 861, normalized size = 7.12 \begin{align*} \frac{6 \,{\left (3 \, a^{3} + 3 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )^{4} + 11 \, a^{3} + 15 \, a^{2} b + 6 \, a b^{2} - 3 \,{\left (9 \, a^{3} + 11 \, a^{2} b + 4 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + 6 \,{\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-b \cos \left (d x + c\right )^{2} + a + b\right ) - 12 \,{\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right )}{12 \,{\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+b*sin(d*x+c)^2),x, algorithm="fricas")

[Out]

1/12*(6*(3*a^3 + 3*a^2*b + a*b^2)*cos(d*x + c)^4 + 11*a^3 + 15*a^2*b + 6*a*b^2 - 3*(9*a^3 + 11*a^2*b + 4*a*b^2
)*cos(d*x + c)^2 + 6*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^6 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d
*x + c)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^2)*log(-b*cos(d*x +
c)^2 + a + b) - 12*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^6 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*
x + c)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^2)*log(1/2*sin(d*x +
c)))/(a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**7/(a+b*sin(d*x+c)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.25438, size = 477, normalized size = 3.94 \begin{align*} \frac{\frac{a^{2}{\left (\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )}^{3} + 12 \, a^{2}{\left (\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )}^{2} + 6 \, a b{\left (\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )}^{2} + 84 \, a^{2}{\left (\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )} + 120 \, a b{\left (\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )} + 48 \, b^{2}{\left (\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )}}{a^{3}} + \frac{192 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left ({\left | -a{\left (\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )} + 2 \, a + 4 \, b \right |}\right )}{a^{4}}}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+b*sin(d*x+c)^2),x, algorithm="giac")

[Out]

1/384*((a^2*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1))^3 + 12*a^2*((cos(d
*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1))^2 + 6*a*b*((cos(d*x + c) + 1)/(cos(d*
x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1))^2 + 84*a^2*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(
d*x + c) - 1)/(cos(d*x + c) + 1)) + 120*a*b*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d
*x + c) + 1)) + 48*b^2*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1)))/a^3 +
192*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*log(abs(-a*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(co
s(d*x + c) + 1)) + 2*a + 4*b))/a^4)/d