3.3.44 \(\int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^4} \, dx\) [244]

Optimal. Leaf size=67 \[ \frac {a^2 c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}+\frac {a^2 c \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^5} \]

[Out]

1/7*a^2*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^6+1/35*a^2*c*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^5

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Rubi [A]
time = 0.09, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751, 2750} \begin {gather*} \frac {a^2 c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}+\frac {a^2 c \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^4,x]

[Out]

(a^2*c^2*Cos[e + f*x]^5)/(7*f*(c - c*Sin[e + f*x])^6) + (a^2*c*Cos[e + f*x]^5)/(35*f*(c - c*Sin[e + f*x])^5)

Rule 2750

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] &&  !ILtQ[p, 0]

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 2815

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0,
n, m] || LtQ[m, n, 0]))

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^4} \, dx &=\left (a^2 c^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^6} \, dx\\ &=\frac {a^2 c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}+\frac {1}{7} \left (a^2 c\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^5} \, dx\\ &=\frac {a^2 c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}+\frac {a^2 c \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^5}\\ \end {align*}

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Mathematica [A]
time = 0.43, size = 117, normalized size = 1.75 \begin {gather*} -\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-35 \cos \left (\frac {1}{2} (e+f x)\right )+14 \cos \left (\frac {3}{2} (e+f x)\right )+\cos \left (\frac {7}{2} (e+f x)\right )-70 \sin \left (\frac {1}{2} (e+f x)\right )-35 \sin \left (\frac {3}{2} (e+f x)\right )+7 \sin \left (\frac {5}{2} (e+f x)\right )\right )}{140 c^4 f (-1+\sin (e+f x))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^4,x]

[Out]

-1/140*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(-35*Cos[(e + f*x)/2] + 14*Cos[(3*(e + f*x))/2] + Cos[(7*(e
+ f*x))/2] - 70*Sin[(e + f*x)/2] - 35*Sin[(3*(e + f*x))/2] + 7*Sin[(5*(e + f*x))/2]))/(c^4*f*(-1 + Sin[e + f*x
])^4)

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Maple [A]
time = 0.39, size = 118, normalized size = 1.76

method	result	size
risch	\(-\frac {2 i a^{2} \left (35 i {\mathrm e}^{4 i \left (f x +e \right )}+35 \,{\mathrm e}^{5 i \left (f x +e \right )}-14 i {\mathrm e}^{2 i \left (f x +e \right )}-70 \,{\mathrm e}^{3 i \left (f x +e \right )}-i+7 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{35 f \,c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7}}\)	\(87\)
derivativedivides	\(\frac {2 a^{2} \left (-\frac {24}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {32}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {14}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}\right )}{f \,c^{4}}\)	\(118\)
default	\(\frac {2 a^{2} \left (-\frac {24}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {32}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {14}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}\right )}{f \,c^{4}}\)	\(118\)
norman	\(\frac {\frac {2 a^{2} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {12 a^{2}}{35 c f}-\frac {2 a^{2} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {2 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5 c f}+\frac {8 a^{2} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {12 a^{2} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {24 a^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}+\frac {52 a^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}-\frac {116 a^{2} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}-\frac {206 a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 c f}-\frac {656 a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\)	\(263\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^4,x,method=_RETURNVERBOSE)

[Out]

2/f*a^2/c^4*(-24/(tan(1/2*f*x+1/2*e)-1)^4-32/7/(tan(1/2*f*x+1/2*e)-1)^7-1/(tan(1/2*f*x+1/2*e)-1)-16/(tan(1/2*f
*x+1/2*e)-1)^6-14/(tan(1/2*f*x+1/2*e)-1)^3-128/5/(tan(1/2*f*x+1/2*e)-1)^5-5/(tan(1/2*f*x+1/2*e)-1)^2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 888 vs. \(2 (69) = 138\).
time = 0.32, size = 888, normalized size = 13.25 \begin {gather*} \frac {2 \, {\left (\frac {2 \, a^{2} {\left (\frac {91 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {168 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {280 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {175 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - 13\right )}}{c^{4} - \frac {7 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {21 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}} - \frac {3 \, a^{2} {\left (\frac {49 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {147 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {210 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {210 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {35 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 12\right )}}{c^{4} - \frac {7 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {21 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}} - \frac {4 \, a^{2} {\left (\frac {14 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {42 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 2\right )}}{c^{4} - \frac {7 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {21 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}}\right )}}{105 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^4,x, algorithm="maxima")

[Out]

2/105*(2*a^2*(91*sin(f*x + e)/(cos(f*x + e) + 1) - 168*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 280*sin(f*x + e)^
3/(cos(f*x + e) + 1)^3 - 175*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 105*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 1
3)/(c^4 - 7*c^4*sin(f*x + e)/(cos(f*x + e) + 1) + 21*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 35*c^4*sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3 + 35*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 21*c^4*sin(f*x + e)^5/(cos(f*x + e)
+ 1)^5 + 7*c^4*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) - 3*a^2*(49*sin
(f*x + e)/(cos(f*x + e) + 1) - 147*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 210*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3 - 210*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 105*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 35*sin(f*x + e)^6/(co
s(f*x + e) + 1)^6 - 12)/(c^4 - 7*c^4*sin(f*x + e)/(cos(f*x + e) + 1) + 21*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1
)^2 - 35*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 35*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 21*c^4*sin(f*x
+ e)^5/(cos(f*x + e) + 1)^5 + 7*c^4*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - c^4*sin(f*x + e)^7/(cos(f*x + e) +
1)^7) - 4*a^2*(14*sin(f*x + e)/(cos(f*x + e) + 1) - 42*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 35*sin(f*x + e)^3
/(cos(f*x + e) + 1)^3 - 35*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 2)/(c^4 - 7*c^4*sin(f*x + e)/(cos(f*x + e) +
1) + 21*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 35*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 35*c^4*sin(f*x
+ e)^4/(cos(f*x + e) + 1)^4 - 21*c^4*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 7*c^4*sin(f*x + e)^6/(cos(f*x + e)
+ 1)^6 - c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7))/f

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (69) = 138\).
time = 0.33, size = 238, normalized size = 3.55 \begin {gather*} -\frac {a^{2} \cos \left (f x + e\right )^{4} + 4 \, a^{2} \cos \left (f x + e\right )^{3} + 13 \, a^{2} \cos \left (f x + e\right )^{2} - 10 \, a^{2} \cos \left (f x + e\right ) - 20 \, a^{2} - {\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 10 \, a^{2} \cos \left (f x + e\right ) + 20 \, a^{2}\right )} \sin \left (f x + e\right )}{35 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^4,x, algorithm="fricas")

[Out]

-1/35*(a^2*cos(f*x + e)^4 + 4*a^2*cos(f*x + e)^3 + 13*a^2*cos(f*x + e)^2 - 10*a^2*cos(f*x + e) - 20*a^2 - (a^2
*cos(f*x + e)^3 - 3*a^2*cos(f*x + e)^2 + 10*a^2*cos(f*x + e) + 20*a^2)*sin(f*x + e))/(c^4*f*cos(f*x + e)^4 - 3
*c^4*f*cos(f*x + e)^3 - 8*c^4*f*cos(f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^4*f + (c^4*f*cos(f*x + e)^3 + 4*c^
4*f*cos(f*x + e)^2 - 4*c^4*f*cos(f*x + e) - 8*c^4*f)*sin(f*x + e))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1074 vs. \(2 (58) = 116\).
time = 10.86, size = 1074, normalized size = 16.03 \begin {gather*} \begin {cases} - \frac {70 a^{2} \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 1225 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1225 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 c^{4} f} + \frac {70 a^{2} \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 1225 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1225 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 c^{4} f} - \frac {280 a^{2} \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 1225 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1225 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 c^{4} f} + \frac {140 a^{2} \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 1225 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1225 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 c^{4} f} - \frac {182 a^{2} \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 1225 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1225 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 c^{4} f} + \frac {14 a^{2} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 1225 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1225 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 c^{4} f} - \frac {12 a^{2}}{35 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 1225 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1225 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 c^{4} f} & \text {for}\: f \neq 0 \\\frac {x \left (a \sin {\left (e \right )} + a\right )^{2}}{\left (- c \sin {\left (e \right )} + c\right )^{4}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**2/(c-c*sin(f*x+e))**4,x)

[Out]

Piecewise((-70*a**2*tan(e/2 + f*x/2)**6/(35*c**4*f*tan(e/2 + f*x/2)**7 - 245*c**4*f*tan(e/2 + f*x/2)**6 + 735*
c**4*f*tan(e/2 + f*x/2)**5 - 1225*c**4*f*tan(e/2 + f*x/2)**4 + 1225*c**4*f*tan(e/2 + f*x/2)**3 - 735*c**4*f*ta
n(e/2 + f*x/2)**2 + 245*c**4*f*tan(e/2 + f*x/2) - 35*c**4*f) + 70*a**2*tan(e/2 + f*x/2)**5/(35*c**4*f*tan(e/2
+ f*x/2)**7 - 245*c**4*f*tan(e/2 + f*x/2)**6 + 735*c**4*f*tan(e/2 + f*x/2)**5 - 1225*c**4*f*tan(e/2 + f*x/2)**
4 + 1225*c**4*f*tan(e/2 + f*x/2)**3 - 735*c**4*f*tan(e/2 + f*x/2)**2 + 245*c**4*f*tan(e/2 + f*x/2) - 35*c**4*f
) - 280*a**2*tan(e/2 + f*x/2)**4/(35*c**4*f*tan(e/2 + f*x/2)**7 - 245*c**4*f*tan(e/2 + f*x/2)**6 + 735*c**4*f*
tan(e/2 + f*x/2)**5 - 1225*c**4*f*tan(e/2 + f*x/2)**4 + 1225*c**4*f*tan(e/2 + f*x/2)**3 - 735*c**4*f*tan(e/2 +
f*x/2)**2 + 245*c**4*f*tan(e/2 + f*x/2) - 35*c**4*f) + 140*a**2*tan(e/2 + f*x/2)**3/(35*c**4*f*tan(e/2 + f*x/
2)**7 - 245*c**4*f*tan(e/2 + f*x/2)**6 + 735*c**4*f*tan(e/2 + f*x/2)**5 - 1225*c**4*f*tan(e/2 + f*x/2)**4 + 12
25*c**4*f*tan(e/2 + f*x/2)**3 - 735*c**4*f*tan(e/2 + f*x/2)**2 + 245*c**4*f*tan(e/2 + f*x/2) - 35*c**4*f) - 18
2*a**2*tan(e/2 + f*x/2)**2/(35*c**4*f*tan(e/2 + f*x/2)**7 - 245*c**4*f*tan(e/2 + f*x/2)**6 + 735*c**4*f*tan(e/
2 + f*x/2)**5 - 1225*c**4*f*tan(e/2 + f*x/2)**4 + 1225*c**4*f*tan(e/2 + f*x/2)**3 - 735*c**4*f*tan(e/2 + f*x/2
)**2 + 245*c**4*f*tan(e/2 + f*x/2) - 35*c**4*f) + 14*a**2*tan(e/2 + f*x/2)/(35*c**4*f*tan(e/2 + f*x/2)**7 - 24
5*c**4*f*tan(e/2 + f*x/2)**6 + 735*c**4*f*tan(e/2 + f*x/2)**5 - 1225*c**4*f*tan(e/2 + f*x/2)**4 + 1225*c**4*f*
tan(e/2 + f*x/2)**3 - 735*c**4*f*tan(e/2 + f*x/2)**2 + 245*c**4*f*tan(e/2 + f*x/2) - 35*c**4*f) - 12*a**2/(35*
c**4*f*tan(e/2 + f*x/2)**7 - 245*c**4*f*tan(e/2 + f*x/2)**6 + 735*c**4*f*tan(e/2 + f*x/2)**5 - 1225*c**4*f*tan
(e/2 + f*x/2)**4 + 1225*c**4*f*tan(e/2 + f*x/2)**3 - 735*c**4*f*tan(e/2 + f*x/2)**2 + 245*c**4*f*tan(e/2 + f*x
/2) - 35*c**4*f), Ne(f, 0)), (x*(a*sin(e) + a)**2/(-c*sin(e) + c)**4, True))

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Giac [A]
time = 0.45, size = 128, normalized size = 1.91 \begin {gather*} -\frac {2 \, {\left (35 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 35 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 140 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 70 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 91 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a^{2}\right )}}{35 \, c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^4,x, algorithm="giac")

[Out]

-2/35*(35*a^2*tan(1/2*f*x + 1/2*e)^6 - 35*a^2*tan(1/2*f*x + 1/2*e)^5 + 140*a^2*tan(1/2*f*x + 1/2*e)^4 - 70*a^2
*tan(1/2*f*x + 1/2*e)^3 + 91*a^2*tan(1/2*f*x + 1/2*e)^2 - 7*a^2*tan(1/2*f*x + 1/2*e) + 6*a^2)/(c^4*f*(tan(1/2*
f*x + 1/2*e) - 1)^7)

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Mupad [B]
time = 7.30, size = 99, normalized size = 1.48 \begin {gather*} \frac {\sqrt {2}\,a^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {5\,\cos \left (3\,e+3\,f\,x\right )}{8}-\frac {105\,\sin \left (e+f\,x\right )}{8}-\frac {27\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {121\,\cos \left (e+f\,x\right )}{8}+\frac {7\,\sin \left (2\,e+2\,f\,x\right )}{2}+\frac {7\,\sin \left (3\,e+3\,f\,x\right )}{8}+\frac {109}{4}\right )}{280\,c^4\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^4,x)

[Out]

(2^(1/2)*a^2*cos(e/2 + (f*x)/2)*((5*cos(3*e + 3*f*x))/8 - (105*sin(e + f*x))/8 - (27*cos(2*e + 2*f*x))/4 - (12
1*cos(e + f*x))/8 + (7*sin(2*e + 2*f*x))/2 + (7*sin(3*e + 3*f*x))/8 + 109/4))/(280*c^4*f*cos(e/2 + pi/4 + (f*x
)/2)^7)

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