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

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

[Out]

cos(d*x+c)/b/d-1/3*cos(d*x+c)^3/b/d-1/2*a*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/b^(7/4)/d/(a^(1/2
)-b^(1/2))^(1/2)+1/2*a*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/b^(7/4)/d/(a^(1/2)+b^(1/2))^(1/2)

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Rubi [A]  time = 0.18, antiderivative size = 148, normalized size of antiderivative = 1.00, 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 = {3215, 1170, 1166, 205, 208} \[ -\frac {a \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{7/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{7/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\cos ^3(c+d x)}{3 b d}+\frac {\cos (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 1170

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), 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] && IntegerQ[q]

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

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Mathematica [C]  time = 0.29, size = 310, normalized size = 2.09 \[ \frac {-3 i a \text {RootSum}\left [\text {$\#$1}^8 b-4 \text {$\#$1}^6 b-16 \text {$\#$1}^4 a+6 \text {$\#$1}^4 b-4 \text {$\#$1}^2 b+b\& ,\frac {2 \text {$\#$1}^6 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-6 \text {$\#$1}^4 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-3 i \text {$\#$1}^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+i \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+6 \text {$\#$1}^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \text {$\#$1}^6 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+3 i \text {$\#$1}^4 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )}{\text {$\#$1}^7 b-3 \text {$\#$1}^5 b-8 \text {$\#$1}^3 a+3 \text {$\#$1}^3 b-\text {$\#$1} b}\& \right ]+18 \cos (c+d x)-2 \cos (3 (c+d x))}{24 b d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(18*Cos[c + d*x] - 2*Cos[3*(c + d*x)] - (3*I)*a*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] + 6*ArcTan[Sin[c + d*
x]/(Cos[c + d*x] - #1)]*#1^2 - (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 6*ArcTan[Sin[c + d*x]/(Cos[c + d
*x] - #1)]*#1^4 + (3*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) & ])/(24*b*
d)

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fricas [B]  time = 0.56, size = 849, normalized size = 5.74 \[ \frac {3 \, b d \sqrt {-\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} + a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}} \log \left (a^{3} \cos \left (d x + c\right ) + {\left (a^{2} b^{2} d - {\left (a b^{5} - b^{6}\right )} d^{3} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} + a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}}\right ) - 3 \, b d \sqrt {\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} - a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}} \log \left (a^{3} \cos \left (d x + c\right ) - {\left (a^{2} b^{2} d + {\left (a b^{5} - b^{6}\right )} d^{3} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} - a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}}\right ) - 3 \, b d \sqrt {-\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} + a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}} \log \left (-a^{3} \cos \left (d x + c\right ) + {\left (a^{2} b^{2} d - {\left (a b^{5} - b^{6}\right )} d^{3} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} + a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}}\right ) + 3 \, b d \sqrt {\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} - a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}} \log \left (-a^{3} \cos \left (d x + c\right ) - {\left (a^{2} b^{2} d + {\left (a b^{5} - b^{6}\right )} d^{3} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{5}}{{\left (a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right )} d^{4}}} - a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}}\right ) - 4 \, \cos \left (d x + c\right )^{3} + 12 \, \cos \left (d x + c\right )}{12 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b*d*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log
(a^3*cos(d*x + c) + (a^2*b^2*d - (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(-((a*b^3 -
b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) - 3*b*d*sqrt(((a*b^3 - b^4)*d^
2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(a^3*cos(d*x + c) - (a^2*b^2*d + (a
*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^
8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))) - 3*b*d*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^
9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(-a^3*cos(d*x + c) + (a^2*b^2*d - (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7
- 2*a*b^8 + b^9)*d^4)))*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b
^4)*d^2))) + 3*b*d*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2
))*log(-a^3*cos(d*x + c) - (a^2*b^2*d + (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(((a*
b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))) - 4*cos(d*x + c)^3 + 12*
cos(d*x + c))/(b*d)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the
root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[76,51]Warning,
need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done
assuming [a,b]=[11,92]-2/d*(-6*(1-cos(c+d*x))/(1+cos(c+d*x))-2)*1/3/b/((1-cos(c+d*x))/(1+cos(c+d*x))+1)^3+2/d*
4*a/b*2/d*(1/(8*b*2)*(c+d*x)+((2*a^4*b-8*a^3*b^2-2*a^3*a*b+3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+10*a^
2*b^3+8*a^2*b*a*b-12*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-4*a*b^4-10*a*b^2*a*b+11*a*b^2*sqrt(a*b)*sqr
t(a^2-a*b+sqrt(a*b)*(a-b))+4*b^3*a*b+2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)*b^2+(-4*a^4*b^2-3
*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+8*a^3*b^3+9*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*a^3*b*a*b-4*a^2*b^4-5
*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-8*a^2*b^2*a*b-a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*a*b^3*a*b)*abs(a-b)
*abs(b)+(2*a^4*b^3-4*a^3*b^4-2*a^3*b^2*a*b+3*a^3*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+2*a^2*b^5+4*a^2*b
^3*a*b-6*a^2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-2*a*b^4*a*b-a*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a
-b)))*abs(a-b))/((96*a^6*b^2-480*a^5*b^3+832*a^4*b^4-576*a^3*b^5+96*a^2*b^6+32*a*b^7)*abs(b))*(atan(tan(c+d*x)
/sqrt(-(32*a*b+sqrt(32*a*b*32*a*b+4*(-16*a*b+16*b^2)*16*a*b))/2/(-16*a*b+16*b^2)))+pi*floor((c+d*x)/pi+1/2))-(
(2*a^4*b-8*a^3*b^2-2*a^3*a*b+3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+10*a^2*b^3+8*a^2*b*a*b-12*a^2*b*sq
rt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-4*a*b^4-10*a*b^2*a*b+11*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+
4*b^3*a*b+2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)*b^2+(-4*a^4*b^2+3*a^4*b*sqrt(a^2-a*b+sqrt(a
*b)*(-a+b))+8*a^3*b^3-9*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+4*a^3*b*a*b-4*a^2*b^4+5*a^2*b^3*sqrt(a^2-a*b+sq
rt(a*b)*(-a+b))-8*a^2*b^2*a*b+a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+4*a*b^3*a*b)*abs(a-b)*abs(b)+(2*a^4*b^3-4*a
^3*b^4-2*a^3*b^2*a*b+3*a^3*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+2*a^2*b^5+4*a^2*b^3*a*b-6*a^2*b^3*sqrt
(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-2*a*b^4*a*b-a*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b))/((9
6*a^6*b^2-480*a^5*b^3+832*a^4*b^4-576*a^3*b^5+96*a^2*b^6+32*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(32*a*b-sqrt
(32*a*b*32*a*b+4*(-16*a*b+16*b^2)*16*a*b))/2/(-16*a*b+16*b^2)))+pi*floor((c+d*x)/pi+1/2)))

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maple [A]  time = 0.31, size = 115, normalized size = 0.78 \[ -\frac {\cos ^{3}\left (d x +c \right )}{3 b d}+\frac {\cos \left (d x +c \right )}{b d}+\frac {a \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 d b \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {a \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 d b \sqrt {\left (\sqrt {a b}-b \right ) b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(12*b*d*integrate(-2*(12*a*b*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) - 4*a*b*cos(d*x + c)*sin(2*d*x + 2*c) + 4
*a*b*cos(2*d*x + 2*c)*sin(d*x + c) - a*b*sin(d*x + c) + (a*b*sin(7*d*x + 7*c) - 3*a*b*sin(5*d*x + 5*c) + 3*a*b
*sin(3*d*x + 3*c) - a*b*sin(d*x + c))*cos(8*d*x + 8*c) + 2*(2*a*b*sin(6*d*x + 6*c) + 2*a*b*sin(2*d*x + 2*c) +
(8*a^2 - 3*a*b)*sin(4*d*x + 4*c))*cos(7*d*x + 7*c) + 4*(3*a*b*sin(5*d*x + 5*c) - 3*a*b*sin(3*d*x + 3*c) + a*b*
sin(d*x + c))*cos(6*d*x + 6*c) - 6*(2*a*b*sin(2*d*x + 2*c) + (8*a^2 - 3*a*b)*sin(4*d*x + 4*c))*cos(5*d*x + 5*c
) - 2*(3*(8*a^2 - 3*a*b)*sin(3*d*x + 3*c) - (8*a^2 - 3*a*b)*sin(d*x + c))*cos(4*d*x + 4*c) - (a*b*cos(7*d*x +
7*c) - 3*a*b*cos(5*d*x + 5*c) + 3*a*b*cos(3*d*x + 3*c) - a*b*cos(d*x + c))*sin(8*d*x + 8*c) - (4*a*b*cos(6*d*x
+ 6*c) + 4*a*b*cos(2*d*x + 2*c) - a*b + 2*(8*a^2 - 3*a*b)*cos(4*d*x + 4*c))*sin(7*d*x + 7*c) - 4*(3*a*b*cos(5
*d*x + 5*c) - 3*a*b*cos(3*d*x + 3*c) + a*b*cos(d*x + c))*sin(6*d*x + 6*c) + 3*(4*a*b*cos(2*d*x + 2*c) - a*b +
2*(8*a^2 - 3*a*b)*cos(4*d*x + 4*c))*sin(5*d*x + 5*c) + 2*(3*(8*a^2 - 3*a*b)*cos(3*d*x + 3*c) - (8*a^2 - 3*a*b)
*cos(d*x + c))*sin(4*d*x + 4*c) - 3*(4*a*b*cos(2*d*x + 2*c) - a*b)*sin(3*d*x + 3*c))/(b^3*cos(8*d*x + 8*c)^2 +
16*b^3*cos(6*d*x + 6*c)^2 + 16*b^3*cos(2*d*x + 2*c)^2 + b^3*sin(8*d*x + 8*c)^2 + 16*b^3*sin(6*d*x + 6*c)^2 +
16*b^3*sin(2*d*x + 2*c)^2 - 8*b^3*cos(2*d*x + 2*c) + b^3 + 4*(64*a^2*b - 48*a*b^2 + 9*b^3)*cos(4*d*x + 4*c)^2
+ 4*(64*a^2*b - 48*a*b^2 + 9*b^3)*sin(4*d*x + 4*c)^2 + 16*(8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c)
- 2*(4*b^3*cos(6*d*x + 6*c) + 4*b^3*cos(2*d*x + 2*c) - b^3 + 2*(8*a*b^2 - 3*b^3)*cos(4*d*x + 4*c))*cos(8*d*x +
8*c) + 8*(4*b^3*cos(2*d*x + 2*c) - b^3 + 2*(8*a*b^2 - 3*b^3)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) - 4*(8*a*b^2
- 3*b^3 - 4*(8*a*b^2 - 3*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(2*b^3*sin(6*d*x + 6*c) + 2*b^3*sin(2*d*x
+ 2*c) + (8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2*b^3*sin(2*d*x + 2*c) + (8*a*b^2 - 3*b^3
)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x) + cos(3*d*x + 3*c) - 9*cos(d*x + c))/(b*d)

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mupad [B]  time = 14.27, size = 1119, normalized size = 7.56 \[ \frac {\cos \left (c+d\,x\right )}{b\,d}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,b\,d}+\frac {\mathrm {atan}\left (-\frac {a^3\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}}{16\,\left (a\,b^7-b^8\right )}-\frac {a^2\,b^4}{16\,\left (a\,b^7-b^8\right )}}\,8{}\mathrm {i}}{\frac {2\,a^4}{b^2}+\frac {2\,a^4\,b^6}{a\,b^7-b^8}+\frac {2\,a^2\,b^2\,\sqrt {a^5\,b^7}}{a\,b^7-b^8}}+\frac {a^3\,b^8\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}}{16\,\left (a\,b^7-b^8\right )}-\frac {a^2\,b^4}{16\,\left (a\,b^7-b^8\right )}}\,8{}\mathrm {i}}{2\,a^4\,b^6-2\,a^5\,b^5+\frac {2\,a^4\,b^{14}}{a\,b^7-b^8}-\frac {2\,a^5\,b^{13}}{a\,b^7-b^8}+\frac {2\,a^2\,b^{10}\,\sqrt {a^5\,b^7}}{a\,b^7-b^8}-\frac {2\,a^3\,b^9\,\sqrt {a^5\,b^7}}{a\,b^7-b^8}}+\frac {a\,b^4\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}}{16\,\left (a\,b^7-b^8\right )}-\frac {a^2\,b^4}{16\,\left (a\,b^7-b^8\right )}}\,\sqrt {a^5\,b^7}\,8{}\mathrm {i}}{2\,a^4\,b^6-2\,a^5\,b^5+\frac {2\,a^4\,b^{14}}{a\,b^7-b^8}-\frac {2\,a^5\,b^{13}}{a\,b^7-b^8}+\frac {2\,a^2\,b^{10}\,\sqrt {a^5\,b^7}}{a\,b^7-b^8}-\frac {2\,a^3\,b^9\,\sqrt {a^5\,b^7}}{a\,b^7-b^8}}\right )\,\sqrt {-\frac {\sqrt {a^5\,b^7}+a^2\,b^4}{16\,\left (a\,b^7-b^8\right )}}\,2{}\mathrm {i}}{d}-\frac {\mathrm {atan}\left (\frac {a^3\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}}{16\,\left (a\,b^7-b^8\right )}-\frac {a^2\,b^4}{16\,\left (a\,b^7-b^8\right )}}\,8{}\mathrm {i}}{\frac {2\,a^4}{b^2}+\frac {2\,a^4\,b^6}{a\,b^7-b^8}-\frac {2\,a^2\,b^2\,\sqrt {a^5\,b^7}}{a\,b^7-b^8}}-\frac {a^3\,b^8\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}}{16\,\left (a\,b^7-b^8\right )}-\frac {a^2\,b^4}{16\,\left (a\,b^7-b^8\right )}}\,8{}\mathrm {i}}{2\,a^4\,b^6-2\,a^5\,b^5+\frac {2\,a^4\,b^{14}}{a\,b^7-b^8}-\frac {2\,a^5\,b^{13}}{a\,b^7-b^8}-\frac {2\,a^2\,b^{10}\,\sqrt {a^5\,b^7}}{a\,b^7-b^8}+\frac {2\,a^3\,b^9\,\sqrt {a^5\,b^7}}{a\,b^7-b^8}}+\frac {a\,b^4\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}}{16\,\left (a\,b^7-b^8\right )}-\frac {a^2\,b^4}{16\,\left (a\,b^7-b^8\right )}}\,\sqrt {a^5\,b^7}\,8{}\mathrm {i}}{2\,a^4\,b^6-2\,a^5\,b^5+\frac {2\,a^4\,b^{14}}{a\,b^7-b^8}-\frac {2\,a^5\,b^{13}}{a\,b^7-b^8}-\frac {2\,a^2\,b^{10}\,\sqrt {a^5\,b^7}}{a\,b^7-b^8}+\frac {2\,a^3\,b^9\,\sqrt {a^5\,b^7}}{a\,b^7-b^8}}\right )\,\sqrt {\frac {\sqrt {a^5\,b^7}-a^2\,b^4}{16\,\left (a\,b^7-b^8\right )}}\,2{}\mathrm {i}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(c + d*x)/(b*d) - cos(c + d*x)^3/(3*b*d) + (atan((a^3*b^8*cos(c + d*x)*(- (a^5*b^7)^(1/2)/(16*(a*b^7 - b^8)
) - (a^2*b^4)/(16*(a*b^7 - b^8)))^(1/2)*8i)/(2*a^4*b^6 - 2*a^5*b^5 + (2*a^4*b^14)/(a*b^7 - b^8) - (2*a^5*b^13)
/(a*b^7 - b^8) + (2*a^2*b^10*(a^5*b^7)^(1/2))/(a*b^7 - b^8) - (2*a^3*b^9*(a^5*b^7)^(1/2))/(a*b^7 - b^8)) - (a^
3*cos(c + d*x)*(- (a^5*b^7)^(1/2)/(16*(a*b^7 - b^8)) - (a^2*b^4)/(16*(a*b^7 - b^8)))^(1/2)*8i)/((2*a^4)/b^2 +
(2*a^4*b^6)/(a*b^7 - b^8) + (2*a^2*b^2*(a^5*b^7)^(1/2))/(a*b^7 - b^8)) + (a*b^4*cos(c + d*x)*(- (a^5*b^7)^(1/2
)/(16*(a*b^7 - b^8)) - (a^2*b^4)/(16*(a*b^7 - b^8)))^(1/2)*(a^5*b^7)^(1/2)*8i)/(2*a^4*b^6 - 2*a^5*b^5 + (2*a^4
*b^14)/(a*b^7 - b^8) - (2*a^5*b^13)/(a*b^7 - b^8) + (2*a^2*b^10*(a^5*b^7)^(1/2))/(a*b^7 - b^8) - (2*a^3*b^9*(a
^5*b^7)^(1/2))/(a*b^7 - b^8)))*(-((a^5*b^7)^(1/2) + a^2*b^4)/(16*(a*b^7 - b^8)))^(1/2)*2i)/d - (atan((a^3*cos(
c + d*x)*((a^5*b^7)^(1/2)/(16*(a*b^7 - b^8)) - (a^2*b^4)/(16*(a*b^7 - b^8)))^(1/2)*8i)/((2*a^4)/b^2 + (2*a^4*b
^6)/(a*b^7 - b^8) - (2*a^2*b^2*(a^5*b^7)^(1/2))/(a*b^7 - b^8)) - (a^3*b^8*cos(c + d*x)*((a^5*b^7)^(1/2)/(16*(a
*b^7 - b^8)) - (a^2*b^4)/(16*(a*b^7 - b^8)))^(1/2)*8i)/(2*a^4*b^6 - 2*a^5*b^5 + (2*a^4*b^14)/(a*b^7 - b^8) - (
2*a^5*b^13)/(a*b^7 - b^8) - (2*a^2*b^10*(a^5*b^7)^(1/2))/(a*b^7 - b^8) + (2*a^3*b^9*(a^5*b^7)^(1/2))/(a*b^7 -
b^8)) + (a*b^4*cos(c + d*x)*((a^5*b^7)^(1/2)/(16*(a*b^7 - b^8)) - (a^2*b^4)/(16*(a*b^7 - b^8)))^(1/2)*(a^5*b^7
)^(1/2)*8i)/(2*a^4*b^6 - 2*a^5*b^5 + (2*a^4*b^14)/(a*b^7 - b^8) - (2*a^5*b^13)/(a*b^7 - b^8) - (2*a^2*b^10*(a^
5*b^7)^(1/2))/(a*b^7 - b^8) + (2*a^3*b^9*(a^5*b^7)^(1/2))/(a*b^7 - b^8)))*(((a^5*b^7)^(1/2) - a^2*b^4)/(16*(a*
b^7 - b^8)))^(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)**7/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

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