3.2.6 \(\int \frac {\sin ^6(c+d x)}{(a+b \sin ^2(c+d x))^3} \, dx\) [106]

Optimal. Leaf size=148 \[ \frac {x}{b^3}-\frac {\sqrt {a} \left (8 a^2+20 a b+15 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 b^3 (a+b)^{5/2} d}+\frac {a \tan ^3(c+d x)}{4 b (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {a (4 a+7 b) \tan (c+d x)}{8 b^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )} \]

[Out]

x/b^3-1/8*(8*a^2+20*a*b+15*b^2)*arctan((a+b)^(1/2)*tan(d*x+c)/a^(1/2))*a^(1/2)/b^3/(a+b)^(5/2)/d+1/4*a*tan(d*x
+c)^3/b/(a+b)/d/(a+(a+b)*tan(d*x+c)^2)^2+1/8*a*(4*a+7*b)*tan(d*x+c)/b^2/(a+b)^2/d/(a+(a+b)*tan(d*x+c)^2)

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Rubi [A]
time = 0.19, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3266, 481, 592, 536, 209, 211} \begin {gather*} -\frac {\sqrt {a} \left (8 a^2+20 a b+15 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 b^3 d (a+b)^{5/2}}+\frac {a (4 a+7 b) \tan (c+d x)}{8 b^2 d (a+b)^2 \left ((a+b) \tan ^2(c+d x)+a\right )}+\frac {a \tan ^3(c+d x)}{4 b d (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )^2}+\frac {x}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^6/(a + b*Sin[c + d*x]^2)^3,x]

[Out]

x/b^3 - (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(8*b^3*(a + b)^(5/2)*d)
+ (a*Tan[c + d*x]^3)/(4*b*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2)^2) + (a*(4*a + 7*b)*Tan[c + d*x])/(8*b^2*(a
+ b)^2*d*(a + (a + b)*Tan[c + d*x]^2))

Rule 209

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

Rule 211

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

Rule 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 592

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c -
a*d)*(p + 1))), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

Rule 3266

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

Rubi steps

\begin {align*} \int \frac {\sin ^6(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {a \tan ^3(c+d x)}{4 b (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {x^2 \left (3 a+(-a-4 b) x^2\right )}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 b (a+b) d}\\ &=\frac {a \tan ^3(c+d x)}{4 b (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {a (4 a+7 b) \tan (c+d x)}{8 b^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {a (4 a+7 b)+\left (-4 a^2-9 a b-8 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{8 b^2 (a+b)^2 d}\\ &=\frac {a \tan ^3(c+d x)}{4 b (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {a (4 a+7 b) \tan (c+d x)}{8 b^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b^3 d}-\frac {\left (a \left (8 a^2+20 a b+15 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{8 b^3 (a+b)^2 d}\\ &=\frac {x}{b^3}-\frac {\sqrt {a} \left (8 a^2+20 a b+15 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 b^3 (a+b)^{5/2} d}+\frac {a \tan ^3(c+d x)}{4 b (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {a (4 a+7 b) \tan (c+d x)}{8 b^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 1.81, size = 134, normalized size = 0.91 \begin {gather*} \frac {8 (c+d x)-\frac {\sqrt {a} \left (8 a^2+20 a b+15 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{5/2}}+\frac {a b \left (8 a^2+20 a b+9 b^2-3 b (2 a+3 b) \cos (2 (c+d x))\right ) \sin (2 (c+d x))}{(a+b)^2 (2 a+b-b \cos (2 (c+d x)))^2}}{8 b^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^6/(a + b*Sin[c + d*x]^2)^3,x]

[Out]

(8*(c + d*x) - (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a + b)^(5/2) +
(a*b*(8*a^2 + 20*a*b + 9*b^2 - 3*b*(2*a + 3*b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/((a + b)^2*(2*a + b - b*Cos
[2*(c + d*x)])^2))/(8*b^3*d)

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Maple [A]
time = 0.34, size = 158, normalized size = 1.07

method	result	size
derivativedivides	\(\frac {\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b^{3}}-\frac {a \left (\frac {-\frac {\left (4 a +9 b \right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{8 \left (a +b \right )}-\frac {a b \left (4 a +7 b \right ) \tan \left (d x +c \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+b \left (\tan ^{2}\left (d x +c \right )\right )+a \right )^{2}}+\frac {\left (8 a^{2}+20 a b +15 b^{2}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a \left (a +b \right )}}\right )}{b^{3}}}{d}\)	\(158\)
default	\(\frac {\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b^{3}}-\frac {a \left (\frac {-\frac {\left (4 a +9 b \right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{8 \left (a +b \right )}-\frac {a b \left (4 a +7 b \right ) \tan \left (d x +c \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+b \left (\tan ^{2}\left (d x +c \right )\right )+a \right )^{2}}+\frac {\left (8 a^{2}+20 a b +15 b^{2}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a \left (a +b \right )}}\right )}{b^{3}}}{d}\)	\(158\)
risch	\(\frac {x}{b^{3}}-\frac {i a \left (-16 b \,{\mathrm e}^{6 i \left (d x +c \right )} a^{2}-28 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-9 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+48 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+120 b \,{\mathrm e}^{4 i \left (d x +c \right )} a^{2}+90 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+27 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-32 b \,{\mathrm e}^{2 i \left (d x +c \right )} a^{2}-68 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-27 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+6 a \,b^{2}+9 b^{3}\right )}{4 b^{3} \left (a +b \right )^{2} d \left (-b \,{\mathrm e}^{4 i \left (d x +c \right )}+4 a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}-b \right )^{2}}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right ) a^{2}}{2 \left (a +b \right )^{3} d \,b^{3}}-\frac {5 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right ) a}{4 \left (a +b \right )^{3} d \,b^{2}}-\frac {15 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{16 \left (a +b \right )^{3} d b}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right ) a^{2}}{2 \left (a +b \right )^{3} d \,b^{3}}+\frac {5 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right ) a}{4 \left (a +b \right )^{3} d \,b^{2}}+\frac {15 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{16 \left (a +b \right )^{3} d b}\)	\(551\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^6/(a+sin(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/b^3*arctan(tan(d*x+c))-a/b^3*((-1/8*(4*a+9*b)*b/(a+b)*tan(d*x+c)^3-1/8*a*b*(4*a+7*b)/(a^2+2*a*b+b^2)*ta
n(d*x+c))/(a*tan(d*x+c)^2+b*tan(d*x+c)^2+a)^2+1/8*(8*a^2+20*a*b+15*b^2)/(a^2+2*a*b+b^2)/(a*(a+b))^(1/2)*arctan
((a+b)*tan(d*x+c)/(a*(a+b))^(1/2))))

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Maxima [A]
time = 0.55, size = 234, normalized size = 1.58 \begin {gather*} -\frac {\frac {{\left (8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right )} \sqrt {{\left (a + b\right )} a}} - \frac {{\left (4 \, a^{3} + 13 \, a^{2} b + 9 \, a b^{2}\right )} \tan \left (d x + c\right )^{3} + {\left (4 \, a^{3} + 7 \, a^{2} b\right )} \tan \left (d x + c\right )}{a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} + {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right )} \tan \left (d x + c\right )^{2}} - \frac {8 \, {\left (d x + c\right )}}{b^{3}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a+b*sin(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-1/8*((8*a^3 + 20*a^2*b + 15*a*b^2)*arctan((a + b)*tan(d*x + c)/sqrt((a + b)*a))/((a^2*b^3 + 2*a*b^4 + b^5)*sq
rt((a + b)*a)) - ((4*a^3 + 13*a^2*b + 9*a*b^2)*tan(d*x + c)^3 + (4*a^3 + 7*a^2*b)*tan(d*x + c))/(a^4*b^2 + 2*a
^3*b^3 + a^2*b^4 + (a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*tan(d*x + c)^4 + 2*(a^4*b^2 + 3*a^3*b^3 +
3*a^2*b^4 + a*b^5)*tan(d*x + c)^2) - 8*(d*x + c)/b^3)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (134) = 268\).
time = 0.47, size = 950, normalized size = 6.42 \begin {gather*} \left [\frac {32 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} d x \cos \left (d x + c\right )^{4} - 64 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} d x \cos \left (d x + c\right )^{2} + 32 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x + {\left ({\left (8 \, a^{2} b^{2} + 20 \, a b^{3} + 15 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 8 \, a^{4} + 36 \, a^{3} b + 63 \, a^{2} b^{2} + 50 \, a b^{3} + 15 \, b^{4} - 2 \, {\left (8 \, a^{3} b + 28 \, a^{2} b^{2} + 35 \, a b^{3} + 15 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a}{a + b}} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (2 \, a^{2} + 3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {-\frac {a}{a + b}} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) - 4 \, {\left (3 \, {\left (2 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{3} b + 13 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{32 \, {\left ({\left (a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right )} d \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b^{3} + 4 \, a^{3} b^{4} + 6 \, a^{2} b^{5} + 4 \, a b^{6} + b^{7}\right )} d\right )}}, \frac {16 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} d x \cos \left (d x + c\right )^{4} - 32 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} d x \cos \left (d x + c\right )^{2} + 16 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x + {\left ({\left (8 \, a^{2} b^{2} + 20 \, a b^{3} + 15 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 8 \, a^{4} + 36 \, a^{3} b + 63 \, a^{2} b^{2} + 50 \, a b^{3} + 15 \, b^{4} - 2 \, {\left (8 \, a^{3} b + 28 \, a^{2} b^{2} + 35 \, a b^{3} + 15 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {a}{a + b}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {\frac {a}{a + b}}}{2 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) - 2 \, {\left (3 \, {\left (2 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{3} b + 13 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right )} d \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b^{3} + 4 \, a^{3} b^{4} + 6 \, a^{2} b^{5} + 4 \, a b^{6} + b^{7}\right )} d\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a+b*sin(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/32*(32*(a^2*b^2 + 2*a*b^3 + b^4)*d*x*cos(d*x + c)^4 - 64*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x*cos(d*x +
c)^2 + 32*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x + ((8*a^2*b^2 + 20*a*b^3 + 15*b^4)*cos(d*x + c)^4 +
8*a^4 + 36*a^3*b + 63*a^2*b^2 + 50*a*b^3 + 15*b^4 - 2*(8*a^3*b + 28*a^2*b^2 + 35*a*b^3 + 15*b^4)*cos(d*x + c)^
2)*sqrt(-a/(a + b))*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2
*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2
*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*(3*(2*a^2*b^2 + 3*a*b
^3)*cos(d*x + c)^3 - (4*a^3*b + 13*a^2*b^2 + 9*a*b^3)*cos(d*x + c))*sin(d*x + c))/((a^2*b^5 + 2*a*b^6 + b^7)*d
*cos(d*x + c)^4 - 2*(a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + b^7)*d*cos(d*x + c)^2 + (a^4*b^3 + 4*a^3*b^4 + 6*a^2*b^5
+ 4*a*b^6 + b^7)*d), 1/16*(16*(a^2*b^2 + 2*a*b^3 + b^4)*d*x*cos(d*x + c)^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 +
b^4)*d*x*cos(d*x + c)^2 + 16*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x + ((8*a^2*b^2 + 20*a*b^3 + 15*b^
4)*cos(d*x + c)^4 + 8*a^4 + 36*a^3*b + 63*a^2*b^2 + 50*a*b^3 + 15*b^4 - 2*(8*a^3*b + 28*a^2*b^2 + 35*a*b^3 + 1
5*b^4)*cos(d*x + c)^2)*sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*
x + c)*sin(d*x + c))) - 2*(3*(2*a^2*b^2 + 3*a*b^3)*cos(d*x + c)^3 - (4*a^3*b + 13*a^2*b^2 + 9*a*b^3)*cos(d*x +
c))*sin(d*x + c))/((a^2*b^5 + 2*a*b^6 + b^7)*d*cos(d*x + c)^4 - 2*(a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + b^7)*d*cos
(d*x + c)^2 + (a^4*b^3 + 4*a^3*b^4 + 6*a^2*b^5 + 4*a*b^6 + b^7)*d)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**6/(a+b*sin(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [A]
time = 0.56, size = 224, normalized size = 1.51 \begin {gather*} -\frac {\frac {{\left (8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{{\left (a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right )} \sqrt {a^{2} + a b}} - \frac {4 \, a^{3} \tan \left (d x + c\right )^{3} + 13 \, a^{2} b \tan \left (d x + c\right )^{3} + 9 \, a b^{2} \tan \left (d x + c\right )^{3} + 4 \, a^{3} \tan \left (d x + c\right ) + 7 \, a^{2} b \tan \left (d x + c\right )}{{\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}^{2}} - \frac {8 \, {\left (d x + c\right )}}{b^{3}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a+b*sin(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/8*((8*a^3 + 20*a^2*b + 15*a*b^2)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*
tan(d*x + c))/sqrt(a^2 + a*b)))/((a^2*b^3 + 2*a*b^4 + b^5)*sqrt(a^2 + a*b)) - (4*a^3*tan(d*x + c)^3 + 13*a^2*b
*tan(d*x + c)^3 + 9*a*b^2*tan(d*x + c)^3 + 4*a^3*tan(d*x + c) + 7*a^2*b*tan(d*x + c))/((a^2*b^2 + 2*a*b^3 + b^
4)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)^2) - 8*(d*x + c)/b^3)/d

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Mupad [B]
time = 17.96, size = 2500, normalized size = 16.89 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^6/(a + b*sin(c + d*x)^2)^3,x)

[Out]

((tan(c + d*x)^3*(9*a*b + 4*a^2))/(8*(a*b^2 + b^3)) + (a*tan(c + d*x)*(7*a*b + 4*a^2))/(8*(2*a*b^3 + b^4 + a^2
*b^2)))/(d*(tan(c + d*x)^4*(2*a*b + a^2 + b^2) + a^2 + tan(c + d*x)^2*(2*a*b + 2*a^2))) - atan(((((((7*a*b^10)
/2 + (25*a^2*b^9)/2 + (33*a^3*b^8)/2 + (19*a^4*b^7)/2 + 2*a^5*b^6)*1i)/(2*(3*a*b^8 + b^9 + 3*a^2*b^7 + a^3*b^6
)) - (tan(c + d*x)*(1792*a*b^11 + 256*b^12 + 5120*a^2*b^10 + 7680*a^3*b^9 + 6400*a^4*b^8 + 2816*a^5*b^7 + 512*
a^6*b^6))/(128*b^3*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)))/(2*b^3) + (tan(c + d*x)*(384*a*b^5 + 704*a^5*b + 12
8*a^6 + 64*b^6 + 1185*a^2*b^4 + 1880*a^3*b^3 + 1600*a^4*b^2))/(64*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)))/b^3
- (((((7*a*b^10)/2 + (25*a^2*b^9)/2 + (33*a^3*b^8)/2 + (19*a^4*b^7)/2 + 2*a^5*b^6)*1i)/(2*(3*a*b^8 + b^9 + 3*a
^2*b^7 + a^3*b^6)) + (tan(c + d*x)*(1792*a*b^11 + 256*b^12 + 5120*a^2*b^10 + 7680*a^3*b^9 + 6400*a^4*b^8 + 281
6*a^5*b^7 + 512*a^6*b^6))/(128*b^3*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)))/(2*b^3) - (tan(c + d*x)*(384*a*b^5
+ 704*a^5*b + 128*a^6 + 64*b^6 + 1185*a^2*b^4 + 1880*a^3*b^3 + 1600*a^4*b^2))/(64*(3*a*b^6 + b^7 + 3*a^2*b^5 +
a^3*b^4)))/b^3)/(((15*a*b^4)/4 + (19*a^4*b)/4 + a^5 + (295*a^2*b^3)/32 + (19*a^3*b^2)/2)/(3*a*b^8 + b^9 + 3*a
^2*b^7 + a^3*b^6) + ((((((7*a*b^10)/2 + (25*a^2*b^9)/2 + (33*a^3*b^8)/2 + (19*a^4*b^7)/2 + 2*a^5*b^6)*1i)/(2*(
3*a*b^8 + b^9 + 3*a^2*b^7 + a^3*b^6)) - (tan(c + d*x)*(1792*a*b^11 + 256*b^12 + 5120*a^2*b^10 + 7680*a^3*b^9 +
6400*a^4*b^8 + 2816*a^5*b^7 + 512*a^6*b^6))/(128*b^3*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)))*1i)/(2*b^3) + (t
an(c + d*x)*(384*a*b^5 + 704*a^5*b + 128*a^6 + 64*b^6 + 1185*a^2*b^4 + 1880*a^3*b^3 + 1600*a^4*b^2)*1i)/(64*(3
*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)))/b^3 + ((((((7*a*b^10)/2 + (25*a^2*b^9)/2 + (33*a^3*b^8)/2 + (19*a^4*b^7)
/2 + 2*a^5*b^6)*1i)/(2*(3*a*b^8 + b^9 + 3*a^2*b^7 + a^3*b^6)) + (tan(c + d*x)*(1792*a*b^11 + 256*b^12 + 5120*a
^2*b^10 + 7680*a^3*b^9 + 6400*a^4*b^8 + 2816*a^5*b^7 + 512*a^6*b^6))/(128*b^3*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3
*b^4)))*1i)/(2*b^3) - (tan(c + d*x)*(384*a*b^5 + 704*a^5*b + 128*a^6 + 64*b^6 + 1185*a^2*b^4 + 1880*a^3*b^3 +
1600*a^4*b^2)*1i)/(64*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)))/b^3))/(b^3*d) - (atan((((-a*(a + b)^5)^(1/2)*((t
an(c + d*x)*(384*a*b^5 + 704*a^5*b + 128*a^6 + 64*b^6 + 1185*a^2*b^4 + 1880*a^3*b^3 + 1600*a^4*b^2))/(32*(3*a*
b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)) - ((-a*(a + b)^5)^(1/2)*(((7*a*b^10)/2 + (25*a^2*b^9)/2 + (33*a^3*b^8)/2 + (
19*a^4*b^7)/2 + 2*a^5*b^6)/(3*a*b^8 + b^9 + 3*a^2*b^7 + a^3*b^6) - (tan(c + d*x)*(-a*(a + b)^5)^(1/2)*(20*a*b
+ 8*a^2 + 15*b^2)*(1792*a*b^11 + 256*b^12 + 5120*a^2*b^10 + 7680*a^3*b^9 + 6400*a^4*b^8 + 2816*a^5*b^7 + 512*a
^6*b^6))/(512*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)*(5*a*b^7 + b^8 + 10*a^2*b^6 + 10*a^3*b^5 + 5*a^4*b^4 + a^5
*b^3)))*(20*a*b + 8*a^2 + 15*b^2))/(16*(5*a*b^7 + b^8 + 10*a^2*b^6 + 10*a^3*b^5 + 5*a^4*b^4 + a^5*b^3)))*(20*a
*b + 8*a^2 + 15*b^2)*1i)/(16*(5*a*b^7 + b^8 + 10*a^2*b^6 + 10*a^3*b^5 + 5*a^4*b^4 + a^5*b^3)) + ((-a*(a + b)^5
)^(1/2)*((tan(c + d*x)*(384*a*b^5 + 704*a^5*b + 128*a^6 + 64*b^6 + 1185*a^2*b^4 + 1880*a^3*b^3 + 1600*a^4*b^2)
)/(32*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)) + ((-a*(a + b)^5)^(1/2)*(((7*a*b^10)/2 + (25*a^2*b^9)/2 + (33*a^3
*b^8)/2 + (19*a^4*b^7)/2 + 2*a^5*b^6)/(3*a*b^8 + b^9 + 3*a^2*b^7 + a^3*b^6) + (tan(c + d*x)*(-a*(a + b)^5)^(1/
2)*(20*a*b + 8*a^2 + 15*b^2)*(1792*a*b^11 + 256*b^12 + 5120*a^2*b^10 + 7680*a^3*b^9 + 6400*a^4*b^8 + 2816*a^5*
b^7 + 512*a^6*b^6))/(512*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)*(5*a*b^7 + b^8 + 10*a^2*b^6 + 10*a^3*b^5 + 5*a^
4*b^4 + a^5*b^3)))*(20*a*b + 8*a^2 + 15*b^2))/(16*(5*a*b^7 + b^8 + 10*a^2*b^6 + 10*a^3*b^5 + 5*a^4*b^4 + a^5*b
^3)))*(20*a*b + 8*a^2 + 15*b^2)*1i)/(16*(5*a*b^7 + b^8 + 10*a^2*b^6 + 10*a^3*b^5 + 5*a^4*b^4 + a^5*b^3)))/(((1
5*a*b^4)/4 + (19*a^4*b)/4 + a^5 + (295*a^2*b^3)/32 + (19*a^3*b^2)/2)/(3*a*b^8 + b^9 + 3*a^2*b^7 + a^3*b^6) - (
(-a*(a + b)^5)^(1/2)*((tan(c + d*x)*(384*a*b^5 + 704*a^5*b + 128*a^6 + 64*b^6 + 1185*a^2*b^4 + 1880*a^3*b^3 +
1600*a^4*b^2))/(32*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)) - ((-a*(a + b)^5)^(1/2)*(((7*a*b^10)/2 + (25*a^2*b^9
)/2 + (33*a^3*b^8)/2 + (19*a^4*b^7)/2 + 2*a^5*b^6)/(3*a*b^8 + b^9 + 3*a^2*b^7 + a^3*b^6) - (tan(c + d*x)*(-a*(
a + b)^5)^(1/2)*(20*a*b + 8*a^2 + 15*b^2)*(1792*a*b^11 + 256*b^12 + 5120*a^2*b^10 + 7680*a^3*b^9 + 6400*a^4*b^
8 + 2816*a^5*b^7 + 512*a^6*b^6))/(512*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)*(5*a*b^7 + b^8 + 10*a^2*b^6 + 10*a
^3*b^5 + 5*a^4*b^4 + a^5*b^3)))*(20*a*b + 8*a^2 + 15*b^2))/(16*(5*a*b^7 + b^8 + 10*a^2*b^6 + 10*a^3*b^5 + 5*a^
4*b^4 + a^5*b^3)))*(20*a*b + 8*a^2 + 15*b^2))/(16*(5*a*b^7 + b^8 + 10*a^2*b^6 + 10*a^3*b^5 + 5*a^4*b^4 + a^5*b
^3)) + ((-a*(a + b)^5)^(1/2)*((tan(c + d*x)*(384*a*b^5 + 704*a^5*b + 128*a^6 + 64*b^6 + 1185*a^2*b^4 + 1880*a^
3*b^3 + 1600*a^4*b^2))/(32*(3*a*b^6 + b^7 + 3*a^2*b^5 + a^3*b^4)) + ((-a*(a + b)^5)^(1/2)*(((7*a*b^10)/2 + (25
*a^2*b^9)/2 + (33*a^3*b^8)/2 + (19*a^4*b^7)/2 + 2*a^5*b^6)/(3*a*b^8 + b^9 + 3*a^2*b^7 + a^3*b^6) + (tan(c + d*
x)*(-a*(a + b)^5)^(1/2)*(20*a*b + 8*a^2 + 15*b^...

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