3.215 \(\int \frac {\sin ^3(c+d x)}{(a-b \sin ^4(c+d x))^2} \, dx\)

Optimal. Leaf size=186 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} b^{3/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} b^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )} \]

[Out]

-1/4*cos(d*x+c)*(2-cos(d*x+c)^2)/(a-b)/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)-1/8*arctan(b^(1/4)*cos(d*x+c)/(
a^(1/2)-b^(1/2))^(1/2))/b^(3/4)/d/a^(1/2)/(a^(1/2)-b^(1/2))^(3/2)+1/8*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1
/2))^(1/2))/b^(3/4)/d/a^(1/2)/(a^(1/2)+b^(1/2))^(3/2)

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Rubi [A]  time = 0.18, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3215, 1178, 1166, 205, 208} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} b^{3/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} b^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4)^2,x]

[Out]

-ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(8*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/4)*d) + ArcT
anh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(8*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/4)*d) - (Cos[c +
d*x]*(2 - Cos[c + d*x]^2))/(4*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))

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]

Rule 1166

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

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\sin ^3(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {-4 a b+2 a b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{8 a (a-b) b d}\\ &=-\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right ) d}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right ) d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{3/4} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{3/4} d}-\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [C]  time = 0.34, size = 345, normalized size = 1.85 \[ \frac {\frac {16 (\cos (3 (c+d x))-5 \cos (c+d x))}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-i \text {RootSum}\left [\text {$\#$1}^8 b-4 \text {$\#$1}^6 b-16 \text {$\#$1}^4 a+6 \text {$\#$1}^4 b-4 \text {$\#$1}^2 b+b\& ,\frac {2 \text {$\#$1}^6 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-14 \text {$\#$1}^4 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-7 i \text {$\#$1}^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+i \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+14 \text {$\#$1}^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \text {$\#$1}^6 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+7 i \text {$\#$1}^4 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )}{\text {$\#$1}^7 b-3 \text {$\#$1}^5 b-8 \text {$\#$1}^3 a+3 \text {$\#$1}^3 b-\text {$\#$1} b}\& \right ]}{32 d (a-b)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4)^2,x]

[Out]

((16*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) - I*RootSum
[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*
Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 14*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (7*I)*Log[1 - 2*Cos[c +
d*x]*#1 + #1^2]*#1^2 - 14*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (7*I)*Log[1 - 2*Cos[c + d*x]*#1 + #
1^2]*#1^4 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#
1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(32*(a - b)*d)

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fricas [B]  time = 0.69, size = 2049, normalized size = 11.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

-1/16*(4*cos(d*x + c)^3 - ((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)
*d)*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a
^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4
)*d^2))*log((a + 3*b)*cos(d*x + c) - ((a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*d^3*sqrt((a^2 + 6*a*b + 9*b^2)
/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 2*(a^2*b + 3*a*b^2)
*d)*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a
^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4
)*d^2))) + ((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(((a^4*
b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*
b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 3*a - b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log((a +
3*b)*cos(d*x + c) - ((a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*d^3*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a
^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 2*(a^2*b + 3*a*b^2)*d)*sqrt(((a^4*b
- 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b
^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 3*a - b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))) + ((a*b
- b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(-((a^4*b - 3*a^3*b^2 +
3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7
- 6*a^2*b^8 + a*b^9)*d^4)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log(-(a + 3*b)*cos(d*x +
c) - ((a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*d^3*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5
*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 2*(a^2*b + 3*a*b^2)*d)*sqrt(-((a^4*b - 3*a^3*b^2 +
3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^
7 - 6*a^2*b^8 + a*b^9)*d^4)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))) - ((a*b - b^2)*d*cos(d
*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b
^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 +
a*b^9)*d^4)) - 3*a - b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log(-(a + 3*b)*cos(d*x + c) - ((a^5*b^2
- 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*d^3*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b
^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 2*(a^2*b + 3*a*b^2)*d)*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b
^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 +
a*b^9)*d^4)) - 3*a - b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))) - 8*cos(d*x + c))/((a*b - b^2)*d*cos(d
*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)

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giac [B]  time = 1.04, size = 605, normalized size = 3.25 \[ -\frac {\frac {\cos \left (d x + c\right )^{3}}{d} - \frac {2 \, \cos \left (d x + c\right )}{d}}{4 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} - a + b\right )} {\left (a - b\right )}} + \frac {{\left ({\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} d^{4} - 2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {-b^{2} + \sqrt {a b} b} d^{2} {\left | -a d^{2} + b d^{2} \right |} + {\left (a d^{2} - b d^{2}\right )}^{2} \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a\right )} \arctan \left (\frac {\cos \left (d x + c\right )}{d \sqrt {-\frac {a b d^{2} - b^{2} d^{2} + \sqrt {{\left (a b d^{2} - b^{2} d^{2}\right )}^{2} + {\left (a b d^{4} - b^{2} d^{4}\right )} {\left (a^{2} - 2 \, a b + b^{2}\right )}}}{a b d^{4} - b^{2} d^{4}}}}\right )}{8 \, {\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right )} d^{3} {\left | -a d^{2} + b d^{2} \right |} {\left | b \right |}} - \frac {{\left ({\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} d^{4} + 2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {-b^{2} - \sqrt {a b} b} d^{2} {\left | -a d^{2} + b d^{2} \right |} + {\left (a d^{2} - b d^{2}\right )}^{2} \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a\right )} \arctan \left (\frac {\cos \left (d x + c\right )}{d \sqrt {-\frac {a b d^{2} - b^{2} d^{2} - \sqrt {{\left (a b d^{2} - b^{2} d^{2}\right )}^{2} + {\left (a b d^{4} - b^{2} d^{4}\right )} {\left (a^{2} - 2 \, a b + b^{2}\right )}}}{a b d^{4} - b^{2} d^{4}}}}\right )}{8 \, {\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right )} d^{3} {\left | -a d^{2} + b d^{2} \right |} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^2,x, algorithm="giac")

[Out]

-1/4*(cos(d*x + c)^3/d - 2*cos(d*x + c)/d)/((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 - a + b)*(a - b)) + 1/8*((a
^2*b - 2*a*b^2 + b^3)*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*d^4 - 2*(a^2*b - a*b^2)*sqrt(-b^2 + sqrt(a*b)*b)*d^2*
abs(-a*d^2 + b*d^2) + (a*d^2 - b*d^2)^2*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a)*arctan(cos(d*x + c)/(d*sqrt(-(a*
b*d^2 - b^2*d^2 + sqrt((a*b*d^2 - b^2*d^2)^2 + (a*b*d^4 - b^2*d^4)*(a^2 - 2*a*b + b^2)))/(a*b*d^4 - b^2*d^4)))
)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^3*abs(-a*d^2 + b*d^2)*abs(b)) - 1/8*((a^2*b - 2*a*b^2 + b^3)*sqrt
(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*d^4 + 2*(a^2*b - a*b^2)*sqrt(-b^2 - sqrt(a*b)*b)*d^2*abs(-a*d^2 + b*d^2) + (a*d
^2 - b*d^2)^2*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a)*arctan(cos(d*x + c)/(d*sqrt(-(a*b*d^2 - b^2*d^2 - sqrt((a*
b*d^2 - b^2*d^2)^2 + (a*b*d^4 - b^2*d^4)*(a^2 - 2*a*b + b^2)))/(a*b*d^4 - b^2*d^4))))/((a^4*b - 3*a^3*b^2 + 3*
a^2*b^3 - a*b^4)*d^3*abs(-a*d^2 + b*d^2)*abs(b))

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maple [A]  time = 0.26, size = 213, normalized size = 1.15 \[ \frac {\sqrt {a b}\, \cos \left (d x +c \right )}{8 d a \left (\sqrt {a b}+b \right ) \left (-b \left (\cos ^{2}\left (d x +c \right )\right )+\sqrt {a b}+b \right )}+\frac {\sqrt {a b}\, \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{8 d a \left (\sqrt {a b}+b \right ) \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {\sqrt {a b}\, \cos \left (d x +c \right )}{8 d a \left (\sqrt {a b}-b \right ) \left (b \left (\cos ^{2}\left (d x +c \right )\right )-b +\sqrt {a b}\right )}-\frac {\sqrt {a b}\, \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{8 d a \left (\sqrt {a b}-b \right ) \sqrt {\left (\sqrt {a b}-b \right ) b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^2,x)

[Out]

1/8/d*(a*b)^(1/2)/a*cos(d*x+c)/((a*b)^(1/2)+b)/(-b*cos(d*x+c)^2+(a*b)^(1/2)+b)+1/8/d*(a*b)^(1/2)/a/((a*b)^(1/2
)+b)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))-1/8/d*(a*b)^(1/2)/a*cos(d*x+c)/
((a*b)^(1/2)-b)/(b*cos(d*x+c)^2-b+(a*b)^(1/2))-1/8/d*(a*b)^(1/2)/a/((a*b)^(1/2)-b)/(((a*b)^(1/2)-b)*b)^(1/2)*a
rctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 15.60, size = 3060, normalized size = 16.45 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^3/(a - b*sin(c + d*x)^4)^2,x)

[Out]

(cos(c + d*x)^3/(4*(a - b)) - cos(c + d*x)/(2*(a - b)))/(d*(a - b + 2*b*cos(c + d*x)^2 - b*cos(c + d*x)^4)) -
(atan(((((512*a*b^4 - 512*a^2*b^3)/(64*(a^2 - 2*a*b + b^2)) - (cos(c + d*x)*((a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3
)^(1/2) + a*b^3 + 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2)*(256*a*b^6 - 512*a^2*b^5
+ 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*((a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) + a*b^3 + 3*a^2*b^2)/(256*
(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2) + (cos(c + d*x)*(a*b^2 + b^3))/(4*(a^2 - 2*a*b + b^2)))*((
a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) + a*b^3 + 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3))
)^(1/2)*1i - (((512*a*b^4 - 512*a^2*b^3)/(64*(a^2 - 2*a*b + b^2)) + (cos(c + d*x)*((a*(a^3*b^3)^(1/2) + 3*b*(a
^3*b^3)^(1/2) + a*b^3 + 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2)*(256*a*b^6 - 512*a
^2*b^5 + 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*((a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) + a*b^3 + 3*a^2*b^2)
/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2) - (cos(c + d*x)*(a*b^2 + b^3))/(4*(a^2 - 2*a*b + b^2
)))*((a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) + a*b^3 + 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5
*b^3)))^(1/2)*1i)/(b/(32*(a^2 - 2*a*b + b^2)) + (((512*a*b^4 - 512*a^2*b^3)/(64*(a^2 - 2*a*b + b^2)) - (cos(c
+ d*x)*((a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) + a*b^3 + 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 -
a^5*b^3)))^(1/2)*(256*a*b^6 - 512*a^2*b^5 + 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*((a*(a^3*b^3)^(1/2) + 3*b*(
a^3*b^3)^(1/2) + a*b^3 + 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2) + (cos(c + d*x)*(
a*b^2 + b^3))/(4*(a^2 - 2*a*b + b^2)))*((a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) + a*b^3 + 3*a^2*b^2)/(256*(a^
2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2) + (((512*a*b^4 - 512*a^2*b^3)/(64*(a^2 - 2*a*b + b^2)) + (cos
(c + d*x)*((a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) + a*b^3 + 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4
- a^5*b^3)))^(1/2)*(256*a*b^6 - 512*a^2*b^5 + 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*((a*(a^3*b^3)^(1/2) + 3*
b*(a^3*b^3)^(1/2) + a*b^3 + 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2) - (cos(c + d*x
)*(a*b^2 + b^3))/(4*(a^2 - 2*a*b + b^2)))*((a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) + a*b^3 + 3*a^2*b^2)/(256*
(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2)))*((a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) + a*b^3 + 3*a^
2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2)*2i)/d - (atan(((((512*a*b^4 - 512*a^2*b^3)/(64
*(a^2 - 2*a*b + b^2)) - (cos(c + d*x)*(-(a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) - a*b^3 - 3*a^2*b^2)/(256*(a^
2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2)*(256*a*b^6 - 512*a^2*b^5 + 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^
2)))*(-(a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) - a*b^3 - 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a
^5*b^3)))^(1/2) + (cos(c + d*x)*(a*b^2 + b^3))/(4*(a^2 - 2*a*b + b^2)))*(-(a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(
1/2) - a*b^3 - 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2)*1i - (((512*a*b^4 - 512*a^2
*b^3)/(64*(a^2 - 2*a*b + b^2)) + (cos(c + d*x)*(-(a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) - a*b^3 - 3*a^2*b^2)
/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2)*(256*a*b^6 - 512*a^2*b^5 + 256*a^3*b^4))/(4*(a^2 - 2
*a*b + b^2)))*(-(a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) - a*b^3 - 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^
4*b^4 - a^5*b^3)))^(1/2) - (cos(c + d*x)*(a*b^2 + b^3))/(4*(a^2 - 2*a*b + b^2)))*(-(a*(a^3*b^3)^(1/2) + 3*b*(a
^3*b^3)^(1/2) - a*b^3 - 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2)*1i)/(b/(32*(a^2 -
2*a*b + b^2)) + (((512*a*b^4 - 512*a^2*b^3)/(64*(a^2 - 2*a*b + b^2)) - (cos(c + d*x)*(-(a*(a^3*b^3)^(1/2) + 3*
b*(a^3*b^3)^(1/2) - a*b^3 - 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2)*(256*a*b^6 - 5
12*a^2*b^5 + 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*(-(a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) - a*b^3 - 3*a^2
*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2) + (cos(c + d*x)*(a*b^2 + b^3))/(4*(a^2 - 2*a*b
+ b^2)))*(-(a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) - a*b^3 - 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4
- a^5*b^3)))^(1/2) + (((512*a*b^4 - 512*a^2*b^3)/(64*(a^2 - 2*a*b + b^2)) + (cos(c + d*x)*(-(a*(a^3*b^3)^(1/2
) + 3*b*(a^3*b^3)^(1/2) - a*b^3 - 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2)*(256*a*b
^6 - 512*a^2*b^5 + 256*a^3*b^4))/(4*(a^2 - 2*a*b + b^2)))*(-(a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) - a*b^3 -
3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2) - (cos(c + d*x)*(a*b^2 + b^3))/(4*(a^2 -
2*a*b + b^2)))*(-(a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) - a*b^3 - 3*a^2*b^2)/(256*(a^2*b^6 - 3*a^3*b^5 + 3*a
^4*b^4 - a^5*b^3)))^(1/2)))*(-(a*(a^3*b^3)^(1/2) + 3*b*(a^3*b^3)^(1/2) - a*b^3 - 3*a^2*b^2)/(256*(a^2*b^6 - 3*
a^3*b^5 + 3*a^4*b^4 - a^5*b^3)))^(1/2)*2i)/d

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**3/(a-b*sin(d*x+c)**4)**2,x)

[Out]

Timed out

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