\(\int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx\) [683]

Optimal result

Integrand size = 25, antiderivative size = 127 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx=\frac {b^2 x}{d^2}-\frac {2 (b c-3 d) \left (3 c d+b \left (c^2-2 d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^2 \left (c^2-d^2\right )^{3/2} f}+\frac {(b c-3 d)^2 \cos (e+f x)}{d \left (c^2-d^2\right ) f (c+d \sin (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2869, 2814, 2739, 632, 210} \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx=-\frac {2 (b c-a d) \left (a c d+b \left (c^2-2 d^2\right )\right ) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^2 f \left (c^2-d^2\right )^{3/2}}+\frac {(b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) (c+d \sin (e+f x))}+\frac {b^2 x}{d^2} \]

[In]

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

[Out]

(b^2*x)/d^2 - (2*(b*c - a*d)*(a*c*d + b*(c^2 - 2*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^2*
(c^2 - d^2)^(3/2)*f) + ((b*c - a*d)^2*Cos[e + f*x])/(d*(c^2 - d^2)*f*(c + d*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 2869

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

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

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx=\frac {b^2 (e+f x)-\frac {2 \left (-9 c d^2+6 b d^3+b^2 \left (c^3-2 c d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{3/2}}+\frac {(b c-3 d)^2 d \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}}{d^2 f} \]

[In]

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

[Out]

(b^2*(e + f*x) - (2*(-9*c*d^2 + 6*b*d^3 + b^2*(c^3 - 2*c*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]
])/(c^2 - d^2)^(3/2) + ((b*c - 3*d)^2*d*Cos[e + f*x])/((c - d)*(c + d)*(c + d*Sin[e + f*x])))/(d^2*f)

Maple [A] (verified)

Time = 1.39 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.72

method	result	size
derivativedivides	\(\frac {\frac {2 b^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{2}}+\frac {\frac {2 \left (\frac {d^{2} \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) c}+\frac {d \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right )}{c^{2}-d^{2}}\right )}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {2 \left (a^{2} c \,d^{2}-2 a b \,d^{3}-b^{2} c^{3}+2 b^{2} c \,d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{2}-d^{2}\right )^{\frac {3}{2}}}}{d^{2}}}{f}\)	\(218\)
default	\(\frac {\frac {2 b^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{2}}+\frac {\frac {2 \left (\frac {d^{2} \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) c}+\frac {d \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right )}{c^{2}-d^{2}}\right )}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {2 \left (a^{2} c \,d^{2}-2 a b \,d^{3}-b^{2} c^{3}+2 b^{2} c \,d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{2}-d^{2}\right )^{\frac {3}{2}}}}{d^{2}}}{f}\)	\(218\)
risch	\(\frac {b^{2} x}{d^{2}}-\frac {2 i \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) \left (i d +c \,{\mathrm e}^{i \left (f x +e \right )}\right )}{d^{2} \left (c^{2}-d^{2}\right ) f \left (-i d \,{\mathrm e}^{2 i \left (f x +e \right )}+i d +2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) a^{2} c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}+\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) a b}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) b^{2} c^{3}}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f \,d^{2}}-\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) b^{2} c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) a^{2} c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}-\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) a b}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) b^{2} c^{3}}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f \,d^{2}}+\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) b^{2} c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}\)	\(761\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (124) = 248\).

Time = 0.31 (sec) , antiderivative size = 677, normalized size of antiderivative = 5.33 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx=\left [\frac {2 \, {\left (b^{2} c^{4} d - 2 \, b^{2} c^{2} d^{3} + b^{2} d^{5}\right )} f x \sin \left (f x + e\right ) + 2 \, {\left (b^{2} c^{5} - 2 \, b^{2} c^{3} d^{2} + b^{2} c d^{4}\right )} f x - {\left (b^{2} c^{4} + 2 \, a b c d^{3} - {\left (a^{2} + 2 \, b^{2}\right )} c^{2} d^{2} + {\left (b^{2} c^{3} d + 2 \, a b d^{4} - {\left (a^{2} + 2 \, b^{2}\right )} c d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \log \left (-\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} - 2 \, {\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) + 2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + 2 \, a b c d^{4} - a^{2} d^{5} + {\left (a^{2} - b^{2}\right )} c^{2} d^{3}\right )} \cos \left (f x + e\right )}{2 \, {\left ({\left (c^{4} d^{3} - 2 \, c^{2} d^{5} + d^{7}\right )} f \sin \left (f x + e\right ) + {\left (c^{5} d^{2} - 2 \, c^{3} d^{4} + c d^{6}\right )} f\right )}}, \frac {{\left (b^{2} c^{4} d - 2 \, b^{2} c^{2} d^{3} + b^{2} d^{5}\right )} f x \sin \left (f x + e\right ) + {\left (b^{2} c^{5} - 2 \, b^{2} c^{3} d^{2} + b^{2} c d^{4}\right )} f x + {\left (b^{2} c^{4} + 2 \, a b c d^{3} - {\left (a^{2} + 2 \, b^{2}\right )} c^{2} d^{2} + {\left (b^{2} c^{3} d + 2 \, a b d^{4} - {\left (a^{2} + 2 \, b^{2}\right )} c d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) + {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + 2 \, a b c d^{4} - a^{2} d^{5} + {\left (a^{2} - b^{2}\right )} c^{2} d^{3}\right )} \cos \left (f x + e\right )}{{\left (c^{4} d^{3} - 2 \, c^{2} d^{5} + d^{7}\right )} f \sin \left (f x + e\right ) + {\left (c^{5} d^{2} - 2 \, c^{3} d^{4} + c d^{6}\right )} f}\right ] \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*sin(f*x+e))^2/(c+d*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*d^2-4*c^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.90 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx=\frac {\frac {{\left (f x + e\right )} b^{2}}{d^{2}} - \frac {2 \, {\left (b^{2} c^{3} - a^{2} c d^{2} - 2 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (c^{2} d^{2} - d^{4}\right )} \sqrt {c^{2} - d^{2}}} + \frac {2 \, {\left (b^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}}{{\left (c^{3} d - c d^{3}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}}}{f} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 16.61 (sec) , antiderivative size = 5776, normalized size of antiderivative = 45.48 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(d*(c^2 - d^2)) + (2*tan(e/2 + (f*x)/2)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/
(c*(c^2 - d^2)))/(f*(c + 2*d*tan(e/2 + (f*x)/2) + c*tan(e/2 + (f*x)/2)^2)) - (2*b^2*atan(((b^2*((b^2*((32*(b^2
*c*d^8 + a^2*c^3*d^6 - a^2*c^5*d^4 - b^2*c^3*d^6 - 2*a*b*c^2*d^7 + 2*a*b*c^4*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^2)
+ (32*tan(e/2 + (f*x)/2)*(2*a^2*c^2*d^8 - 2*a^2*c^4*d^6 + 4*b^2*c^2*d^8 - 6*b^2*c^4*d^6 + 2*b^2*c^6*d^4 - 4*a
*b*c*d^9 + 4*a*b*c^3*d^7))/(d^7 - 2*c^2*d^5 + c^4*d^3) - (b^2*((32*(c^2*d^9 - 2*c^4*d^7 + c^6*d^5))/(d^6 - 2*c
^2*d^4 + c^4*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^11 - 8*c^3*d^9 + 7*c^5*d^7 - 2*c^7*d^5))/(d^7 - 2*c^2*d^5 +
c^4*d^3))*1i)/d^2)*1i)/d^2 - (32*(b^4*c^6*d + b^4*c^2*d^5 - 2*b^4*c^4*d^3))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*
tan(e/2 + (f*x)/2)*(2*b^4*c^7*d - 2*b^4*c*d^7 + a^4*c^3*d^5 + 9*b^4*c^3*d^5 - 8*b^4*c^5*d^3 - 8*a*b^3*c^2*d^6
+ 4*a*b^3*c^4*d^4 + 4*a^2*b^2*c*d^7 - 4*a^3*b*c^2*d^6 + 4*a^2*b^2*c^3*d^5 - 2*a^2*b^2*c^5*d^3))/(d^7 - 2*c^2*d
^5 + c^4*d^3)))/d^2 - (b^2*((32*(b^4*c^6*d + b^4*c^2*d^5 - 2*b^4*c^4*d^3))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (b^2*
((32*(b^2*c*d^8 + a^2*c^3*d^6 - a^2*c^5*d^4 - b^2*c^3*d^6 - 2*a*b*c^2*d^7 + 2*a*b*c^4*d^5))/(d^6 - 2*c^2*d^4 +
c^4*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^2*c^2*d^8 - 2*a^2*c^4*d^6 + 4*b^2*c^2*d^8 - 6*b^2*c^4*d^6 + 2*b^2*c^6*
d^4 - 4*a*b*c*d^9 + 4*a*b*c^3*d^7))/(d^7 - 2*c^2*d^5 + c^4*d^3) + (b^2*((32*(c^2*d^9 - 2*c^4*d^7 + c^6*d^5))/(
d^6 - 2*c^2*d^4 + c^4*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^11 - 8*c^3*d^9 + 7*c^5*d^7 - 2*c^7*d^5))/(d^7 - 2*c
^2*d^5 + c^4*d^3))*1i)/d^2)*1i)/d^2 - (32*tan(e/2 + (f*x)/2)*(2*b^4*c^7*d - 2*b^4*c*d^7 + a^4*c^3*d^5 + 9*b^4*
c^3*d^5 - 8*b^4*c^5*d^3 - 8*a*b^3*c^2*d^6 + 4*a*b^3*c^4*d^4 + 4*a^2*b^2*c*d^7 - 4*a^3*b*c^2*d^6 + 4*a^2*b^2*c^
3*d^5 - 2*a^2*b^2*c^5*d^3))/(d^7 - 2*c^2*d^5 + c^4*d^3)))/d^2)/((b^2*((b^2*((32*(b^2*c*d^8 + a^2*c^3*d^6 - a^2
*c^5*d^4 - b^2*c^3*d^6 - 2*a*b*c^2*d^7 + 2*a*b*c^4*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*tan(e/2 + (f*x)/2)*
(2*a^2*c^2*d^8 - 2*a^2*c^4*d^6 + 4*b^2*c^2*d^8 - 6*b^2*c^4*d^6 + 2*b^2*c^6*d^4 - 4*a*b*c*d^9 + 4*a*b*c^3*d^7))
/(d^7 - 2*c^2*d^5 + c^4*d^3) - (b^2*((32*(c^2*d^9 - 2*c^4*d^7 + c^6*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*ta
n(e/2 + (f*x)/2)*(3*c*d^11 - 8*c^3*d^9 + 7*c^5*d^7 - 2*c^7*d^5))/(d^7 - 2*c^2*d^5 + c^4*d^3))*1i)/d^2)*1i)/d^2
- (32*(b^4*c^6*d + b^4*c^2*d^5 - 2*b^4*c^4*d^3))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*tan(e/2 + (f*x)/2)*(2*b^4*
c^7*d - 2*b^4*c*d^7 + a^4*c^3*d^5 + 9*b^4*c^3*d^5 - 8*b^4*c^5*d^3 - 8*a*b^3*c^2*d^6 + 4*a*b^3*c^4*d^4 + 4*a^2*
b^2*c*d^7 - 4*a^3*b*c^2*d^6 + 4*a^2*b^2*c^3*d^5 - 2*a^2*b^2*c^5*d^3))/(d^7 - 2*c^2*d^5 + c^4*d^3))*1i)/d^2 - (
64*(2*b^6*c^3*d^2 - a^2*b^4*c^5 - b^6*c^5 - 6*a*b^5*c^2*d^3 + 4*a^2*b^4*c*d^4 + 3*a^2*b^4*c^3*d^2 - 4*a^3*b^3*
c^2*d^3 + a^4*b^2*c^3*d^2 + 2*a*b^5*c^4*d))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (b^2*((32*(b^4*c^6*d + b^4*c^2*d^5 -
2*b^4*c^4*d^3))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (b^2*((32*(b^2*c*d^8 + a^2*c^3*d^6 - a^2*c^5*d^4 - b^2*c^3*d^6
- 2*a*b*c^2*d^7 + 2*a*b*c^4*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^2*c^2*d^8 - 2*a^2*
c^4*d^6 + 4*b^2*c^2*d^8 - 6*b^2*c^4*d^6 + 2*b^2*c^6*d^4 - 4*a*b*c*d^9 + 4*a*b*c^3*d^7))/(d^7 - 2*c^2*d^5 + c^4
*d^3) + (b^2*((32*(c^2*d^9 - 2*c^4*d^7 + c^6*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d
^11 - 8*c^3*d^9 + 7*c^5*d^7 - 2*c^7*d^5))/(d^7 - 2*c^2*d^5 + c^4*d^3))*1i)/d^2)*1i)/d^2 - (32*tan(e/2 + (f*x)/
2)*(2*b^4*c^7*d - 2*b^4*c*d^7 + a^4*c^3*d^5 + 9*b^4*c^3*d^5 - 8*b^4*c^5*d^3 - 8*a*b^3*c^2*d^6 + 4*a*b^3*c^4*d^
4 + 4*a^2*b^2*c*d^7 - 4*a^3*b*c^2*d^6 + 4*a^2*b^2*c^3*d^5 - 2*a^2*b^2*c^5*d^3))/(d^7 - 2*c^2*d^5 + c^4*d^3))*1
i)/d^2 + (64*tan(e/2 + (f*x)/2)*(2*b^6*c^6 + 4*b^6*c^2*d^4 - 6*b^6*c^4*d^2 + 4*a*b^5*c^3*d^3 + 2*a^2*b^4*c^2*d
^4 - 2*a^2*b^4*c^4*d^2 - 4*a*b^5*c*d^5))/(d^7 - 2*c^2*d^5 + c^4*d^3))))/(d^2*f) + (atan((((a*d - b*c)*(-(c + d
)^3*(c - d)^3)^(1/2)*((32*(b^4*c^6*d + b^4*c^2*d^5 - 2*b^4*c^4*d^3))/(d^6 - 2*c^2*d^4 + c^4*d^2) - (32*tan(e/2
+ (f*x)/2)*(2*b^4*c^7*d - 2*b^4*c*d^7 + a^4*c^3*d^5 + 9*b^4*c^3*d^5 - 8*b^4*c^5*d^3 - 8*a*b^3*c^2*d^6 + 4*a*b
^3*c^4*d^4 + 4*a^2*b^2*c*d^7 - 4*a^3*b*c^2*d^6 + 4*a^2*b^2*c^3*d^5 - 2*a^2*b^2*c^5*d^3))/(d^7 - 2*c^2*d^5 + c^
4*d^3) + ((a*d - b*c)*(-(c + d)^3*(c - d)^3)^(1/2)*((32*(b^2*c*d^8 + a^2*c^3*d^6 - a^2*c^5*d^4 - b^2*c^3*d^6 -
2*a*b*c^2*d^7 + 2*a*b*c^4*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^2*c^2*d^8 - 2*a^2*c
^4*d^6 + 4*b^2*c^2*d^8 - 6*b^2*c^4*d^6 + 2*b^2*c^6*d^4 - 4*a*b*c*d^9 + 4*a*b*c^3*d^7))/(d^7 - 2*c^2*d^5 + c^4*
d^3) + (((32*(c^2*d^9 - 2*c^4*d^7 + c^6*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^11 -
8*c^3*d^9 + 7*c^5*d^7 - 2*c^7*d^5))/(d^7 - 2*c^2*d^5 + c^4*d^3))*(a*d - b*c)*(-(c + d)^3*(c - d)^3)^(1/2)*(b*
c^2 - 2*b*d^2 + a*c*d))/(d^8 - 3*c^2*d^6 + 3*c^4*d^4 - c^6*d^2))*(b*c^2 - 2*b*d^2 + a*c*d))/(d^8 - 3*c^2*d^6 +
3*c^4*d^4 - c^6*d^2))*(b*c^2 - 2*b*d^2 + a*c*d)*1i)/(d^8 - 3*c^2*d^6 + 3*c^4*d^4 - c^6*d^2) - ((a*d - b*c)*(-
(c + d)^3*(c - d)^3)^(1/2)*((32*tan(e/2 + (f*x)/2)*(2*b^4*c^7*d - 2*b^4*c*d^7 + a^4*c^3*d^5 + 9*b^4*c^3*d^5 -
8*b^4*c^5*d^3 - 8*a*b^3*c^2*d^6 + 4*a*b^3*c^4*d^4 + 4*a^2*b^2*c*d^7 - 4*a^3*b*c^2*d^6 + 4*a^2*b^2*c^3*d^5 - 2*
a^2*b^2*c^5*d^3))/(d^7 - 2*c^2*d^5 + c^4*d^3) - (32*(b^4*c^6*d + b^4*c^2*d^5 - 2*b^4*c^4*d^3))/(d^6 - 2*c^2*d^
4 + c^4*d^2) + ((a*d - b*c)*(-(c + d)^3*(c - d)^3)^(1/2)*((32*(b^2*c*d^8 + a^2*c^3*d^6 - a^2*c^5*d^4 - b^2*c^3
*d^6 - 2*a*b*c^2*d^7 + 2*a*b*c^4*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^2*c^2*d^8 - 2
*a^2*c^4*d^6 + 4*b^2*c^2*d^8 - 6*b^2*c^4*d^6 + 2*b^2*c^6*d^4 - 4*a*b*c*d^9 + 4*a*b*c^3*d^7))/(d^7 - 2*c^2*d^5
+ c^4*d^3) - (((32*(c^2*d^9 - 2*c^4*d^7 + c^6*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*
d^11 - 8*c^3*d^9 + 7*c^5*d^7 - 2*c^7*d^5))/(d^7 - 2*c^2*d^5 + c^4*d^3))*(a*d - b*c)*(-(c + d)^3*(c - d)^3)^(1/
2)*(b*c^2 - 2*b*d^2 + a*c*d))/(d^8 - 3*c^2*d^6 + 3*c^4*d^4 - c^6*d^2))*(b*c^2 - 2*b*d^2 + a*c*d))/(d^8 - 3*c^2
*d^6 + 3*c^4*d^4 - c^6*d^2))*(b*c^2 - 2*b*d^2 + a*c*d)*1i)/(d^8 - 3*c^2*d^6 + 3*c^4*d^4 - c^6*d^2))/((64*tan(e
/2 + (f*x)/2)*(2*b^6*c^6 + 4*b^6*c^2*d^4 - 6*b^6*c^4*d^2 + 4*a*b^5*c^3*d^3 + 2*a^2*b^4*c^2*d^4 - 2*a^2*b^4*c^4
*d^2 - 4*a*b^5*c*d^5))/(d^7 - 2*c^2*d^5 + c^4*d^3) - (64*(2*b^6*c^3*d^2 - a^2*b^4*c^5 - b^6*c^5 - 6*a*b^5*c^2*
d^3 + 4*a^2*b^4*c*d^4 + 3*a^2*b^4*c^3*d^2 - 4*a^3*b^3*c^2*d^3 + a^4*b^2*c^3*d^2 + 2*a*b^5*c^4*d))/(d^6 - 2*c^2
*d^4 + c^4*d^2) + ((a*d - b*c)*(-(c + d)^3*(c - d)^3)^(1/2)*((32*(b^4*c^6*d + b^4*c^2*d^5 - 2*b^4*c^4*d^3))/(d
^6 - 2*c^2*d^4 + c^4*d^2) - (32*tan(e/2 + (f*x)/2)*(2*b^4*c^7*d - 2*b^4*c*d^7 + a^4*c^3*d^5 + 9*b^4*c^3*d^5 -
8*b^4*c^5*d^3 - 8*a*b^3*c^2*d^6 + 4*a*b^3*c^4*d^4 + 4*a^2*b^2*c*d^7 - 4*a^3*b*c^2*d^6 + 4*a^2*b^2*c^3*d^5 - 2*
a^2*b^2*c^5*d^3))/(d^7 - 2*c^2*d^5 + c^4*d^3) + ((a*d - b*c)*(-(c + d)^3*(c - d)^3)^(1/2)*((32*(b^2*c*d^8 + a^
2*c^3*d^6 - a^2*c^5*d^4 - b^2*c^3*d^6 - 2*a*b*c^2*d^7 + 2*a*b*c^4*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*tan(
e/2 + (f*x)/2)*(2*a^2*c^2*d^8 - 2*a^2*c^4*d^6 + 4*b^2*c^2*d^8 - 6*b^2*c^4*d^6 + 2*b^2*c^6*d^4 - 4*a*b*c*d^9 +
4*a*b*c^3*d^7))/(d^7 - 2*c^2*d^5 + c^4*d^3) + (((32*(c^2*d^9 - 2*c^4*d^7 + c^6*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^
2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^11 - 8*c^3*d^9 + 7*c^5*d^7 - 2*c^7*d^5))/(d^7 - 2*c^2*d^5 + c^4*d^3))*(a*d
- b*c)*(-(c + d)^3*(c - d)^3)^(1/2)*(b*c^2 - 2*b*d^2 + a*c*d))/(d^8 - 3*c^2*d^6 + 3*c^4*d^4 - c^6*d^2))*(b*c^2
- 2*b*d^2 + a*c*d))/(d^8 - 3*c^2*d^6 + 3*c^4*d^4 - c^6*d^2))*(b*c^2 - 2*b*d^2 + a*c*d))/(d^8 - 3*c^2*d^6 + 3*
c^4*d^4 - c^6*d^2) + ((a*d - b*c)*(-(c + d)^3*(c - d)^3)^(1/2)*((32*tan(e/2 + (f*x)/2)*(2*b^4*c^7*d - 2*b^4*c*
d^7 + a^4*c^3*d^5 + 9*b^4*c^3*d^5 - 8*b^4*c^5*d^3 - 8*a*b^3*c^2*d^6 + 4*a*b^3*c^4*d^4 + 4*a^2*b^2*c*d^7 - 4*a^
3*b*c^2*d^6 + 4*a^2*b^2*c^3*d^5 - 2*a^2*b^2*c^5*d^3))/(d^7 - 2*c^2*d^5 + c^4*d^3) - (32*(b^4*c^6*d + b^4*c^2*d
^5 - 2*b^4*c^4*d^3))/(d^6 - 2*c^2*d^4 + c^4*d^2) + ((a*d - b*c)*(-(c + d)^3*(c - d)^3)^(1/2)*((32*(b^2*c*d^8 +
a^2*c^3*d^6 - a^2*c^5*d^4 - b^2*c^3*d^6 - 2*a*b*c^2*d^7 + 2*a*b*c^4*d^5))/(d^6 - 2*c^2*d^4 + c^4*d^2) + (32*t
an(e/2 + (f*x)/2)*(2*a^2*c^2*d^8 - 2*a^2*c^4*d^6 + 4*b^2*c^2*d^8 - 6*b^2*c^4*d^6 + 2*b^2*c^6*d^4 - 4*a*b*c*d^9
+ 4*a*b*c^3*d^7))/(d^7 - 2*c^2*d^5 + c^4*d^3) - (((32*(c^2*d^9 - 2*c^4*d^7 + c^6*d^5))/(d^6 - 2*c^2*d^4 + c^4
*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^11 - 8*c^3*d^9 + 7*c^5*d^7 - 2*c^7*d^5))/(d^7 - 2*c^2*d^5 + c^4*d^3))*(a
*d - b*c)*(-(c + d)^3*(c - d)^3)^(1/2)*(b*c^2 - 2*b*d^2 + a*c*d))/(d^8 - 3*c^2*d^6 + 3*c^4*d^4 - c^6*d^2))*(b*
c^2 - 2*b*d^2 + a*c*d))/(d^8 - 3*c^2*d^6 + 3*c^4*d^4 - c^6*d^2))*(b*c^2 - 2*b*d^2 + a*c*d))/(d^8 - 3*c^2*d^6 +
3*c^4*d^4 - c^6*d^2)))*(a*d - b*c)*(-(c + d)^3*(c - d)^3)^(1/2)*(b*c^2 - 2*b*d^2 + a*c*d)*2i)/(f*(d^8 - 3*c^2
*d^6 + 3*c^4*d^4 - c^6*d^2))