3.404 \(\int \frac{\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx\)

Optimal. Leaf size=131 \[ \frac{\left (\sqrt{a}+\sqrt{b}\right )^3 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac{\left (\sqrt{a}-\sqrt{b}\right )^3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}+\frac{\sin ^3(c+d x)}{3 b d}-\frac{3 \sin (c+d x)}{b d} \]

[Out]

((Sqrt[a] + Sqrt[b])^3*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(7/4)*d) - ((Sqrt[a] - Sqrt[b])^3*
ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(7/4)*d) - (3*Sin[c + d*x])/(b*d) + Sin[c + d*x]^3/(3*b*
d)

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Rubi [A]  time = 0.189984, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3223, 1171, 1167, 205, 208} \[ \frac{\left (\sqrt{a}+\sqrt{b}\right )^3 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac{\left (\sqrt{a}-\sqrt{b}\right )^3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}+\frac{\sin ^3(c+d x)}{3 b d}-\frac{3 \sin (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7/(a - b*Sin[c + d*x]^4),x]

[Out]

((Sqrt[a] + Sqrt[b])^3*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(7/4)*d) - ((Sqrt[a] - Sqrt[b])^3*
ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(7/4)*d) - (3*Sin[c + d*x])/(b*d) + Sin[c + d*x]^3/(3*b*
d)

Rule 3223

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

Rule 1167

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q)
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[-(a*c)]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{3}{b}+\frac{x^2}{b}+\frac{3 a+b-(a+3 b) x^2}{b \left (a-b x^4\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{3 \sin (c+d x)}{b d}+\frac{\sin ^3(c+d x)}{3 b d}+\frac{\operatorname{Subst}\left (\int \frac{3 a+b+(-a-3 b) x^2}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{b d}\\ &=-\frac{3 \sin (c+d x)}{b d}+\frac{\sin ^3(c+d x)}{3 b d}-\frac{\left (\sqrt{a}-\sqrt{b}\right )^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt{a} b d}-\frac{\left (\sqrt{a}+\sqrt{b}\right )^3 \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt{a} b d}\\ &=\frac{\left (\sqrt{a}+\sqrt{b}\right )^3 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac{\left (\sqrt{a}-\sqrt{b}\right )^3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac{3 \sin (c+d x)}{b d}+\frac{\sin ^3(c+d x)}{3 b d}\\ \end{align*}

Mathematica [C]  time = 0.26784, size = 207, normalized size = 1.58 \[ \frac{4 a^{3/4} b^{3/4} \sin ^3(c+d x)-36 a^{3/4} b^{3/4} \sin (c+d x)+3 \left (\sqrt{a}-\sqrt{b}\right )^3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} \sin (c+d x)\right )-3 \left (\sqrt{a}-\sqrt{b}\right )^3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} \sin (c+d x)\right )+3 i \left (\sqrt{a}+\sqrt{b}\right )^3 \log \left (\sqrt [4]{a}-i \sqrt [4]{b} \sin (c+d x)\right )-3 i \left (\sqrt{a}+\sqrt{b}\right )^3 \log \left (\sqrt [4]{a}+i \sqrt [4]{b} \sin (c+d x)\right )}{12 a^{3/4} b^{7/4} d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7/(a - b*Sin[c + d*x]^4),x]

[Out]

(3*(Sqrt[a] - Sqrt[b])^3*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]] + (3*I)*(Sqrt[a] + Sqrt[b])^3*Log[a^(1/4) - I*b^(
1/4)*Sin[c + d*x]] - (3*I)*(Sqrt[a] + Sqrt[b])^3*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x]] - 3*(Sqrt[a] - Sqrt[b])
^3*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]] - 36*a^(3/4)*b^(3/4)*Sin[c + d*x] + 4*a^(3/4)*b^(3/4)*Sin[c + d*x]^3)/(
12*a^(3/4)*b^(7/4)*d)

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Maple [B]  time = 0.084, size = 350, normalized size = 2.7 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,bd}}-3\,{\frac{\sin \left ( dx+c \right ) }{bd}}+{\frac{3}{2\,bd}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sin \left ( dx+c \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{1}{2\,da}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sin \left ( dx+c \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{3}{4\,bd}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( \sin \left ( dx+c \right ) +\sqrt [4]{{\frac{a}{b}}} \right ) \left ( \sin \left ( dx+c \right ) -\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{1}{4\,da}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( \sin \left ( dx+c \right ) +\sqrt [4]{{\frac{a}{b}}} \right ) \left ( \sin \left ( dx+c \right ) -\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{a}{2\,{b}^{2}d}\arctan \left ({\sin \left ( dx+c \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3}{2\,bd}\arctan \left ({\sin \left ( dx+c \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{a}{4\,{b}^{2}d}\ln \left ({ \left ( \sin \left ( dx+c \right ) +\sqrt [4]{{\frac{a}{b}}} \right ) \left ( \sin \left ( dx+c \right ) -\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{3}{4\,bd}\ln \left ({ \left ( \sin \left ( dx+c \right ) +\sqrt [4]{{\frac{a}{b}}} \right ) \left ( \sin \left ( dx+c \right ) -\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x)

[Out]

1/3*sin(d*x+c)^3/b/d-3*sin(d*x+c)/b/d+3/2/d/b*(a/b)^(1/4)*arctan(sin(d*x+c)/(a/b)^(1/4))+1/2/d*(a/b)^(1/4)/a*a
rctan(sin(d*x+c)/(a/b)^(1/4))+3/4/d/b*(a/b)^(1/4)*ln((sin(d*x+c)+(a/b)^(1/4))/(sin(d*x+c)-(a/b)^(1/4)))+1/4/d*
(a/b)^(1/4)/a*ln((sin(d*x+c)+(a/b)^(1/4))/(sin(d*x+c)-(a/b)^(1/4)))+1/2/d/b^2/(a/b)^(1/4)*a*arctan(sin(d*x+c)/
(a/b)^(1/4))+3/2/d/b/(a/b)^(1/4)*arctan(sin(d*x+c)/(a/b)^(1/4))-1/4/d/b^2/(a/b)^(1/4)*a*ln((sin(d*x+c)+(a/b)^(
1/4))/(sin(d*x+c)-(a/b)^(1/4)))-3/4/d/b/(a/b)^(1/4)*ln((sin(d*x+c)+(a/b)^(1/4))/(sin(d*x+c)-(a/b)^(1/4)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.49865, size = 3235, normalized size = 24.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

1/12*(3*b*d*sqrt(-(a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/
(a^3*b^7*d^4)) + 6*a^2 + 20*a*b + 6*b^2)/(a*b^3*d^2))*log(1/2*(a^6 + 12*a^5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a
*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 2
55*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - (3*a^5*b^2 + 46*a^4*b^3 + 60*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sq
rt(-(a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4))
+ 6*a^2 + 20*a*b + 6*b^2)/(a*b^3*d^2))) - 3*b*d*sqrt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*
b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))*log(1/2*(a^6 + 12*a^
5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a
^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + (3*a^5*b^2 + 46*a^4*b^3 + 60
*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sqrt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b
^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))) - 3*b*d*sqrt(-(a*b^3*d^2*sqrt((a^6
+ 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + 6*a^2 + 20*a*b + 6*b^
2)/(a*b^3*d^2))*log(-1/2*(a^6 + 12*a^5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*
b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7
*d^4)) - (3*a^5*b^2 + 46*a^4*b^3 + 60*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sqrt(-(a*b^3*d^2*sqrt((a^6 + 30*a^5*b +
255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + 6*a^2 + 20*a*b + 6*b^2)/(a*b^3*d^2
))) + 3*b*d*sqrt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(
a^3*b^7*d^4)) - 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))*log(-1/2*(a^6 + 12*a^5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a
*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 2
55*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + (3*a^5*b^2 + 46*a^4*b^3 + 60*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sq
rt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4))
- 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))) - 4*(cos(d*x + c)^2 + 8)*sin(d*x + c))/(b*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(cos(d*x+c)**7/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

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Giac [B]  time = 6.4702, size = 486, normalized size = 3.71 \begin{align*} \frac{\frac{8 \,{\left (b^{2} \sin \left (d x + c\right )^{3} - 9 \, b^{2} \sin \left (d x + c\right )\right )}}{b^{3}} - \frac{6 \, \sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{3}{4}}{\left (a + 3 \, b\right )} - \left (-a b^{3}\right )^{\frac{1}{4}}{\left (3 \, a b^{2} + b^{3}\right )}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{a b^{4}} - \frac{6 \, \sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{3}{4}}{\left (a + 3 \, b\right )} - \left (-a b^{3}\right )^{\frac{1}{4}}{\left (3 \, a b^{2} + b^{3}\right )}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{a b^{4}} + \frac{3 \, \sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{3}{4}}{\left (a + 3 \, b\right )} + \left (-a b^{3}\right )^{\frac{1}{4}}{\left (3 \, a b^{2} + b^{3}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}} \sin \left (d x + c\right ) + \sqrt{-\frac{a}{b}}\right )}{a b^{4}} - \frac{3 \, \sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{3}{4}}{\left (a + 3 \, b\right )} + \left (-a b^{3}\right )^{\frac{1}{4}}{\left (3 \, a b^{2} + b^{3}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}} \sin \left (d x + c\right ) + \sqrt{-\frac{a}{b}}\right )}{a b^{4}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

1/24*(8*(b^2*sin(d*x + c)^3 - 9*b^2*sin(d*x + c))/b^3 - 6*sqrt(2)*((-a*b^3)^(3/4)*(a + 3*b) - (-a*b^3)^(1/4)*(
3*a*b^2 + b^3))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b^4) - 6*sqrt(2)*(
(-a*b^3)^(3/4)*(a + 3*b) - (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d
*x + c))/(-a/b)^(1/4))/(a*b^4) + 3*sqrt(2)*((-a*b^3)^(3/4)*(a + 3*b) + (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin
(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^4) - 3*sqrt(2)*((-a*b^3)^(3/4)*(a + 3*b) +
(-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^4))/
d