3.8.7 \(\int \frac {(c+d \sin (e+f x))^3}{(a+b \sin (e+f x))^2} \, dx\) [707]

Optimal. Leaf size=205 \[ \frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {2 (b c-a d)^2 \left (a b c+2 a^2 d-3 b^2 d\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \left (a^2-b^2\right )^{3/2} f}+\frac {d \left (2 a b c d-2 a^2 d^2-b^2 \left (c^2-d^2\right )\right ) \cos (e+f x)}{b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))} \]

[Out]

d^2*(-2*a*d+3*b*c)*x/b^3+2*(-a*d+b*c)^2*(2*a^2*d+a*b*c-3*b^2*d)*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2
))/b^3/(a^2-b^2)^(3/2)/f+d*(2*a*b*c*d-2*a^2*d^2-b^2*(c^2-d^2))*cos(f*x+e)/b^2/(a^2-b^2)/f+(-a*d+b*c)^2*cos(f*x
+e)*(c+d*sin(f*x+e))/b/(a^2-b^2)/f/(a+b*sin(f*x+e))

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Rubi [A]
time = 0.33, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2871, 3102, 2814, 2739, 632, 210} \begin {gather*} \frac {2 (b c-a d)^2 \left (2 a^2 d+a b c-3 b^2 d\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^3 f \left (a^2-b^2\right )^{3/2}}+\frac {d \left (-2 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-d^2\right )\right )\right ) \cos (e+f x)}{b^2 f \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}+\frac {d^2 x (3 b c-2 a d)}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(3*b*c - 2*a*d)*x)/b^3 + (2*(b*c - a*d)^2*(a*b*c + 2*a^2*d - 3*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqr
t[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*f) + (d*(2*a*b*c*d - 2*a^2*d^2 - b^2*(c^2 - d^2))*Cos[e + f*x])/(b^2*(a^
2 - b^2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x]))

Rule 210

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2871

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/
(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e
+ f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 +
c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 +
d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 1.16, size = 151, normalized size = 0.74 \begin {gather*} \frac {d^2 (3 b c-2 a d) (e+f x)+\frac {2 (b c-a d)^2 \left (a b c+2 a^2 d-3 b^2 d\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-b d^3 \cos (e+f x)+\frac {b (b c-a d)^3 \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}}{b^3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(3*b*c - 2*a*d)*(e + f*x) + (2*(b*c - a*d)^2*(a*b*c + 2*a^2*d - 3*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/
Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - b*d^3*Cos[e + f*x] + (b*(b*c - a*d)^3*Cos[e + f*x])/((a - b)*(a + b)*(a
+ b*Sin[e + f*x])))/(b^3*f)

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Maple [A]
time = 0.40, size = 303, normalized size = 1.48

method	result	size
derivativedivides	\(\frac {-\frac {2 d^{2} \left (\frac {b d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (2 a d -3 b c \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{b^{3}}+\frac {\frac {2 \left (-\frac {b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) a}-\frac {b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{a^{2}-b^{2}}\right )}{a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {2 \left (2 a^{4} d^{3}-3 a^{3} b c \,d^{2}-3 a^{2} b^{2} d^{3}+a \,b^{3} c^{3}+6 a \,b^{3} c \,d^{2}-3 b^{4} c^{2} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}}{b^{3}}}{f}\)	\(303\)
default	\(\frac {-\frac {2 d^{2} \left (\frac {b d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (2 a d -3 b c \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{b^{3}}+\frac {\frac {2 \left (-\frac {b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) a}-\frac {b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{a^{2}-b^{2}}\right )}{a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {2 \left (2 a^{4} d^{3}-3 a^{3} b c \,d^{2}-3 a^{2} b^{2} d^{3}+a \,b^{3} c^{3}+6 a \,b^{3} c \,d^{2}-3 b^{4} c^{2} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}}{b^{3}}}{f}\)	\(303\)
risch	\(-\frac {2 d^{3} x a}{b^{3}}+\frac {3 d^{2} x c}{b^{2}}-\frac {d^{3} {\mathrm e}^{i \left (f x +e \right )}}{2 b^{2} f}-\frac {d^{3} {\mathrm e}^{-i \left (f x +e \right )}}{2 b^{2} f}+\frac {2 i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (i b +a \,{\mathrm e}^{i \left (f x +e \right )}\right )}{b^{3} \left (a^{2}-b^{2}\right ) f \left (-i b \,{\mathrm e}^{2 i \left (f x +e \right )}+i b +2 a \,{\mathrm e}^{i \left (f x +e \right )}\right )}-\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{4} d^{3}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f \,b^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{3} c \,d^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f \,b^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{2} d^{3}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f b}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a \,c^{3}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}-\frac {6 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a c \,d^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}+\frac {3 b \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) c^{2} d}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}+\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{4} d^{3}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f \,b^{3}}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{3} c \,d^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f \,b^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{2} d^{3}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f b}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a \,c^{3}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}+\frac {6 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a c \,d^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}-\frac {3 b \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) c^{2} d}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}\)	\(1185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(-2*d^2/b^3*(b*d/(1+tan(1/2*f*x+1/2*e)^2)+(2*a*d-3*b*c)*arctan(tan(1/2*f*x+1/2*e)))+2/b^3*((-b^2*(a^3*d^3-
3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(a^2-b^2)/a*tan(1/2*f*x+1/2*e)-b*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3
*c^3)/(a^2-b^2))/(a*tan(1/2*f*x+1/2*e)^2+2*b*tan(1/2*f*x+1/2*e)+a)+(2*a^4*d^3-3*a^3*b*c*d^2-3*a^2*b^2*d^3+a*b^
3*c^3+6*a*b^3*c*d^2-3*b^4*c^2*d)/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (205) = 410\).
time = 0.45, size = 1025, normalized size = 5.00 \begin {gather*} \left [\frac {2 \, {\left (3 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} c d^{2} - 2 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d^{3}\right )} f x - {\left (a^{2} b^{3} c^{3} - 3 \, a b^{4} c^{2} d - 3 \, {\left (a^{4} b - 2 \, a^{2} b^{3}\right )} c d^{2} + {\left (2 \, a^{5} - 3 \, a^{3} b^{2}\right )} d^{3} + {\left (a b^{4} c^{3} - 3 \, b^{5} c^{2} d - 3 \, {\left (a^{3} b^{2} - 2 \, a b^{4}\right )} c d^{2} + {\left (2 \, a^{4} b - 3 \, a^{2} b^{3}\right )} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + b \cos \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left ({\left (a^{2} b^{4} - b^{6}\right )} c^{3} - 3 \, {\left (a^{3} b^{3} - a b^{5}\right )} c^{2} d + 3 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} c d^{2} - {\left (2 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right )} d^{3}\right )} \cos \left (f x + e\right ) - 2 \, {\left ({\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} d^{3} \cos \left (f x + e\right ) - {\left (3 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} c d^{2} - 2 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} d^{3}\right )} f x\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} f \sin \left (f x + e\right ) + {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} f\right )}}, \frac {{\left (3 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} c d^{2} - 2 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d^{3}\right )} f x - {\left (a^{2} b^{3} c^{3} - 3 \, a b^{4} c^{2} d - 3 \, {\left (a^{4} b - 2 \, a^{2} b^{3}\right )} c d^{2} + {\left (2 \, a^{5} - 3 \, a^{3} b^{2}\right )} d^{3} + {\left (a b^{4} c^{3} - 3 \, b^{5} c^{2} d - 3 \, {\left (a^{3} b^{2} - 2 \, a b^{4}\right )} c d^{2} + {\left (2 \, a^{4} b - 3 \, a^{2} b^{3}\right )} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (f x + e\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (f x + e\right )}\right ) + {\left ({\left (a^{2} b^{4} - b^{6}\right )} c^{3} - 3 \, {\left (a^{3} b^{3} - a b^{5}\right )} c^{2} d + 3 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} c d^{2} - {\left (2 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right )} d^{3}\right )} \cos \left (f x + e\right ) - {\left ({\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} d^{3} \cos \left (f x + e\right ) - {\left (3 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} c d^{2} - 2 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} d^{3}\right )} f x\right )} \sin \left (f x + e\right )}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} f \sin \left (f x + e\right ) + {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*(3*(a^5*b - 2*a^3*b^3 + a*b^5)*c*d^2 - 2*(a^6 - 2*a^4*b^2 + a^2*b^4)*d^3)*f*x - (a^2*b^3*c^3 - 3*a*b^4
*c^2*d - 3*(a^4*b - 2*a^2*b^3)*c*d^2 + (2*a^5 - 3*a^3*b^2)*d^3 + (a*b^4*c^3 - 3*b^5*c^2*d - 3*(a^3*b^2 - 2*a*b
^4)*c*d^2 + (2*a^4*b - 3*a^2*b^3)*d^3)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*
b*sin(f*x + e) - a^2 - b^2 + 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x +
e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + 2*((a^2*b^4 - b^6)*c^3 - 3*(a^3*b^3 - a*b^5)*c^2*d + 3*(a^4*b^2 - a
^2*b^4)*c*d^2 - (2*a^5*b - 3*a^3*b^3 + a*b^5)*d^3)*cos(f*x + e) - 2*((a^4*b^2 - 2*a^2*b^4 + b^6)*d^3*cos(f*x +
e) - (3*(a^4*b^2 - 2*a^2*b^4 + b^6)*c*d^2 - 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d^3)*f*x)*sin(f*x + e))/((a^4*b^4 -
2*a^2*b^6 + b^8)*f*sin(f*x + e) + (a^5*b^3 - 2*a^3*b^5 + a*b^7)*f), ((3*(a^5*b - 2*a^3*b^3 + a*b^5)*c*d^2 - 2
*(a^6 - 2*a^4*b^2 + a^2*b^4)*d^3)*f*x - (a^2*b^3*c^3 - 3*a*b^4*c^2*d - 3*(a^4*b - 2*a^2*b^3)*c*d^2 + (2*a^5 -
3*a^3*b^2)*d^3 + (a*b^4*c^3 - 3*b^5*c^2*d - 3*(a^3*b^2 - 2*a*b^4)*c*d^2 + (2*a^4*b - 3*a^2*b^3)*d^3)*sin(f*x +
e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + ((a^2*b^4 - b^6)*c^3 - 3*(
a^3*b^3 - a*b^5)*c^2*d + 3*(a^4*b^2 - a^2*b^4)*c*d^2 - (2*a^5*b - 3*a^3*b^3 + a*b^5)*d^3)*cos(f*x + e) - ((a^4
*b^2 - 2*a^2*b^4 + b^6)*d^3*cos(f*x + e) - (3*(a^4*b^2 - 2*a^2*b^4 + b^6)*c*d^2 - 2*(a^5*b - 2*a^3*b^3 + a*b^5
)*d^3)*f*x)*sin(f*x + e))/((a^4*b^4 - 2*a^2*b^6 + b^8)*f*sin(f*x + e) + (a^5*b^3 - 2*a^3*b^5 + a*b^7)*f)]

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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((c+d*sin(f*x+e))**3/(a+b*sin(f*x+e))**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (205) = 410\).
time = 0.46, size = 579, normalized size = 2.82 \begin {gather*} \frac {\frac {2 \, {\left (a b^{3} c^{3} - 3 \, b^{4} c^{2} d - 3 \, a^{3} b c d^{2} + 6 \, a b^{3} c d^{2} + 2 \, a^{4} d^{3} - 3 \, a^{2} b^{2} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{3} - b^{5}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, {\left (b^{4} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a b^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{2} b^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - a^{3} b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a b^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, a^{2} b^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, a^{3} b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{2} b^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + b^{4} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a b^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a^{2} b^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a^{3} b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a b^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3} + a^{2} b^{2} d^{3}\right )}}{{\left (a^{3} b^{2} - a b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a\right )}} + \frac {{\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} {\left (f x + e\right )}}{b^{3}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a*b^3*c^3 - 3*b^4*c^2*d - 3*a^3*b*c*d^2 + 6*a*b^3*c*d^2 + 2*a^4*d^3 - 3*a^2*b^2*d^3)*(pi*floor(1/2*(f*x +
e)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*f*x + 1/2*e) + b)/sqrt(a^2 - b^2)))/((a^2*b^3 - b^5)*sqrt(a^2 - b^2))
+ 2*(b^4*c^3*tan(1/2*f*x + 1/2*e)^3 - 3*a*b^3*c^2*d*tan(1/2*f*x + 1/2*e)^3 + 3*a^2*b^2*c*d^2*tan(1/2*f*x + 1/2
*e)^3 - a^3*b*d^3*tan(1/2*f*x + 1/2*e)^3 + a*b^3*c^3*tan(1/2*f*x + 1/2*e)^2 - 3*a^2*b^2*c^2*d*tan(1/2*f*x + 1/
2*e)^2 + 3*a^3*b*c*d^2*tan(1/2*f*x + 1/2*e)^2 - 2*a^4*d^3*tan(1/2*f*x + 1/2*e)^2 + a^2*b^2*d^3*tan(1/2*f*x + 1
/2*e)^2 + b^4*c^3*tan(1/2*f*x + 1/2*e) - 3*a*b^3*c^2*d*tan(1/2*f*x + 1/2*e) + 3*a^2*b^2*c*d^2*tan(1/2*f*x + 1/
2*e) - 3*a^3*b*d^3*tan(1/2*f*x + 1/2*e) + 2*a*b^3*d^3*tan(1/2*f*x + 1/2*e) + a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a
^3*b*c*d^2 - 2*a^4*d^3 + a^2*b^2*d^3)/((a^3*b^2 - a*b^4)*(a*tan(1/2*f*x + 1/2*e)^4 + 2*b*tan(1/2*f*x + 1/2*e)^
3 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e) + a)) + (3*b*c*d^2 - 2*a*d^3)*(f*x + e)/b^3)/f

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Mupad [B]
time = 17.83, size = 2500, normalized size = 12.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*(b^3*c^3 - 2*a^3*d^3 + a*b^2*d^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2))/(b^2*(a^2 - b^2)) + (2*tan(e/2 + (f*x)/
2)^2*(b^3*c^3 - 2*a^3*d^3 + a*b^2*d^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2))/(b^2*(a^2 - b^2)) + (2*tan(e/2 + (f*x)
/2)*(b^3*c^3 - 3*a^3*d^3 + 2*a*b^2*d^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2))/(a*b*(a^2 - b^2)) - (2*tan(e/2 + (f*x
)/2)^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*b*(a^2 - b^2)))/(f*(a + 2*b*tan(e/2 + (f*x)/2)
+ 2*a*tan(e/2 + (f*x)/2)^2 + a*tan(e/2 + (f*x)/2)^4 + 2*b*tan(e/2 + (f*x)/2)^3)) + (2*d^2*atan(((d^2*(2*a*d -
3*b*c)*((32*(4*a^4*b^6*d^6 - 8*a^6*b^4*d^6 + 4*a^8*b^2*d^6 - 12*a^3*b^7*c*d^5 + 24*a^5*b^5*c*d^5 - 12*a^7*b^3*
c*d^5 + 9*a^2*b^8*c^2*d^4 - 18*a^4*b^6*c^2*d^4 + 9*a^6*b^4*c^2*d^4))/(b^9 - 2*a^2*b^7 + a^4*b^5) - (32*tan(e/2
+ (f*x)/2)*(a^3*b^8*c^6 - 8*a^3*b^8*d^6 + 29*a^5*b^6*d^6 - 28*a^7*b^4*d^6 + 8*a^9*b^2*d^6 - 18*a*b^10*c^2*d^4
+ 9*a*b^10*c^4*d^2 + 24*a^2*b^9*c*d^5 - 6*a^2*b^9*c^5*d - 96*a^4*b^7*c*d^5 + 90*a^6*b^5*c*d^5 - 24*a^8*b^3*c*
d^5 - 36*a^2*b^9*c^3*d^3 + 99*a^3*b^8*c^2*d^4 + 12*a^3*b^8*c^4*d^2 + 12*a^4*b^7*c^3*d^3 - 84*a^5*b^6*c^2*d^4 -
6*a^5*b^6*c^4*d^2 + 4*a^6*b^5*c^3*d^3 + 18*a^7*b^4*c^2*d^4))/(b^10 - 2*a^2*b^8 + a^4*b^6) + (d^2*(2*a*d - 3*b
*c)*((32*tan(e/2 + (f*x)/2)*(2*a^2*b^11*c^3 - 2*a^4*b^9*c^3 - 6*a^3*b^10*d^3 + 10*a^5*b^8*d^3 - 4*a^7*b^6*d^3
+ 12*a^2*b^11*c*d^2 + 6*a^3*b^10*c^2*d - 18*a^4*b^9*c*d^2 + 6*a^6*b^7*c*d^2 - 6*a*b^12*c^2*d))/(b^10 - 2*a^2*b
^8 + a^4*b^6) - (32*(a^5*b^7*c^3 - a^3*b^9*c^3 + 2*a^2*b^10*d^3 - 3*a^4*b^8*d^3 + a^6*b^6*d^3 + 3*a^2*b^10*c^2
*d + 3*a^3*b^9*c*d^2 - 3*a^4*b^8*c^2*d - 3*a*b^11*c*d^2))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (d^2*((32*(a^2*b^12 -
2*a^4*b^10 + a^6*b^8))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (32*tan(e/2 + (f*x)/2)*(3*a*b^14 - 8*a^3*b^12 + 7*a^5*b^1
0 - 2*a^7*b^8))/(b^10 - 2*a^2*b^8 + a^4*b^6))*(2*a*d - 3*b*c)*1i)/b^3)*1i)/b^3))/b^3 + (d^2*(2*a*d - 3*b*c)*((
32*(4*a^4*b^6*d^6 - 8*a^6*b^4*d^6 + 4*a^8*b^2*d^6 - 12*a^3*b^7*c*d^5 + 24*a^5*b^5*c*d^5 - 12*a^7*b^3*c*d^5 + 9
*a^2*b^8*c^2*d^4 - 18*a^4*b^6*c^2*d^4 + 9*a^6*b^4*c^2*d^4))/(b^9 - 2*a^2*b^7 + a^4*b^5) - (32*tan(e/2 + (f*x)/
2)*(a^3*b^8*c^6 - 8*a^3*b^8*d^6 + 29*a^5*b^6*d^6 - 28*a^7*b^4*d^6 + 8*a^9*b^2*d^6 - 18*a*b^10*c^2*d^4 + 9*a*b^
10*c^4*d^2 + 24*a^2*b^9*c*d^5 - 6*a^2*b^9*c^5*d - 96*a^4*b^7*c*d^5 + 90*a^6*b^5*c*d^5 - 24*a^8*b^3*c*d^5 - 36*
a^2*b^9*c^3*d^3 + 99*a^3*b^8*c^2*d^4 + 12*a^3*b^8*c^4*d^2 + 12*a^4*b^7*c^3*d^3 - 84*a^5*b^6*c^2*d^4 - 6*a^5*b^
6*c^4*d^2 + 4*a^6*b^5*c^3*d^3 + 18*a^7*b^4*c^2*d^4))/(b^10 - 2*a^2*b^8 + a^4*b^6) + (d^2*(2*a*d - 3*b*c)*((32*
(a^5*b^7*c^3 - a^3*b^9*c^3 + 2*a^2*b^10*d^3 - 3*a^4*b^8*d^3 + a^6*b^6*d^3 + 3*a^2*b^10*c^2*d + 3*a^3*b^9*c*d^2
- 3*a^4*b^8*c^2*d - 3*a*b^11*c*d^2))/(b^9 - 2*a^2*b^7 + a^4*b^5) - (32*tan(e/2 + (f*x)/2)*(2*a^2*b^11*c^3 - 2
*a^4*b^9*c^3 - 6*a^3*b^10*d^3 + 10*a^5*b^8*d^3 - 4*a^7*b^6*d^3 + 12*a^2*b^11*c*d^2 + 6*a^3*b^10*c^2*d - 18*a^4
*b^9*c*d^2 + 6*a^6*b^7*c*d^2 - 6*a*b^12*c^2*d))/(b^10 - 2*a^2*b^8 + a^4*b^6) + (d^2*((32*(a^2*b^12 - 2*a^4*b^1
0 + a^6*b^8))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (32*tan(e/2 + (f*x)/2)*(3*a*b^14 - 8*a^3*b^12 + 7*a^5*b^10 - 2*a^7
*b^8))/(b^10 - 2*a^2*b^8 + a^4*b^6))*(2*a*d - 3*b*c)*1i)/b^3)*1i)/b^3))/b^3)/((64*(6*a^6*b^2*d^9 - 4*a^8*d^9 -
27*a*b^7*c^5*d^4 - 39*a^5*b^3*c*d^8 + 4*a^7*b*c^3*d^6 + 99*a^2*b^6*c^4*d^5 + 18*a^2*b^6*c^6*d^3 - 144*a^3*b^5
*c^3*d^6 - 39*a^3*b^5*c^5*d^4 - 3*a^3*b^5*c^7*d^2 + 105*a^4*b^4*c^2*d^7 + 3*a^4*b^4*c^4*d^5 + 2*a^4*b^4*c^6*d^
3 + 55*a^5*b^3*c^3*d^6 + 9*a^5*b^3*c^5*d^4 - 57*a^6*b^2*c^2*d^7 - 12*a^6*b^2*c^4*d^5 + 24*a^7*b*c*d^8))/(b^9 -
2*a^2*b^7 + a^4*b^5) + (64*tan(e/2 + (f*x)/2)*(40*a^7*b^2*d^9 - 24*a^5*b^4*d^9 - 16*a^9*d^9 - 54*a*b^8*c^4*d^
5 + 120*a^4*b^5*c*d^8 - 192*a^6*b^3*c*d^8 + 180*a^2*b^7*c^3*d^6 + 18*a^2*b^7*c^5*d^4 - 222*a^3*b^6*c^2*d^7 + 3
0*a^3*b^6*c^4*d^5 - 226*a^4*b^5*c^3*d^6 - 18*a^4*b^5*c^5*d^4 + 330*a^5*b^4*c^2*d^7 + 24*a^5*b^4*c^4*d^5 + 46*a
^6*b^3*c^3*d^6 - 108*a^7*b^2*c^2*d^7 + 72*a^8*b*c*d^8))/(b^10 - 2*a^2*b^8 + a^4*b^6) + (d^2*(2*a*d - 3*b*c)*((
32*(4*a^4*b^6*d^6 - 8*a^6*b^4*d^6 + 4*a^8*b^2*d^6 - 12*a^3*b^7*c*d^5 + 24*a^5*b^5*c*d^5 - 12*a^7*b^3*c*d^5 + 9
*a^2*b^8*c^2*d^4 - 18*a^4*b^6*c^2*d^4 + 9*a^6*b^4*c^2*d^4))/(b^9 - 2*a^2*b^7 + a^4*b^5) - (32*tan(e/2 + (f*x)/
2)*(a^3*b^8*c^6 - 8*a^3*b^8*d^6 + 29*a^5*b^6*d^6 - 28*a^7*b^4*d^6 + 8*a^9*b^2*d^6 - 18*a*b^10*c^2*d^4 + 9*a*b^
10*c^4*d^2 + 24*a^2*b^9*c*d^5 - 6*a^2*b^9*c^5*d - 96*a^4*b^7*c*d^5 + 90*a^6*b^5*c*d^5 - 24*a^8*b^3*c*d^5 - 36*
a^2*b^9*c^3*d^3 + 99*a^3*b^8*c^2*d^4 + 12*a^3*b^8*c^4*d^2 + 12*a^4*b^7*c^3*d^3 - 84*a^5*b^6*c^2*d^4 - 6*a^5*b^
6*c^4*d^2 + 4*a^6*b^5*c^3*d^3 + 18*a^7*b^4*c^2*d^4))/(b^10 - 2*a^2*b^8 + a^4*b^6) + (d^2*(2*a*d - 3*b*c)*((32*
tan(e/2 + (f*x)/2)*(2*a^2*b^11*c^3 - 2*a^4*b^9*c^3 - 6*a^3*b^10*d^3 + 10*a^5*b^8*d^3 - 4*a^7*b^6*d^3 + 12*a^2*
b^11*c*d^2 + 6*a^3*b^10*c^2*d - 18*a^4*b^9*c*d^2 + 6*a^6*b^7*c*d^2 - 6*a*b^12*c^2*d))/(b^10 - 2*a^2*b^8 + a^4*
b^6) - (32*(a^5*b^7*c^3 - a^3*b^9*c^3 + 2*a^2*b^10*d^3 - 3*a^4*b^8*d^3 + a^6*b^6*d^3 + 3*a^2*b^10*c^2*d + 3*a^
3*b^9*c*d^2 - 3*a^4*b^8*c^2*d - 3*a*b^11*c*d^2)...

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