3.3.70 \(\int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx\) [270]

Optimal. Leaf size=135 \[ \frac {35 c^4 x}{2 a^2}+\frac {35 c^4 \cos (e+f x)}{2 a^2 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {14 a^4 c^4 \cos ^5(e+f x)}{3 f \left (a^2+a^2 \sin (e+f x)\right )^3}+\frac {35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )} \]

[Out]

35/2*c^4*x/a^2+35/2*c^4*cos(f*x+e)/a^2/f-2/3*a^3*c^4*cos(f*x+e)^7/f/(a+a*sin(f*x+e))^5+14/3*a^4*c^4*cos(f*x+e)
^5/f/(a^2+a^2*sin(f*x+e))^3+35/6*c^4*cos(f*x+e)^3/f/(a^2+a^2*sin(f*x+e))

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Rubi [A]
time = 0.16, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2815, 2759, 2758, 2761, 8} \begin {gather*} -\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a \sin (e+f x)+a)^5}+\frac {35 c^4 \cos (e+f x)}{2 a^2 f}+\frac {35 c^4 \cos ^3(e+f x)}{6 f \left (a^2 \sin (e+f x)+a^2\right )}+\frac {35 c^4 x}{2 a^2}+\frac {14 a^4 c^4 \cos ^5(e+f x)}{3 f \left (a^2 \sin (e+f x)+a^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(35*c^4*x)/(2*a^2) + (35*c^4*Cos[e + f*x])/(2*a^2*f) - (2*a^3*c^4*Cos[e + f*x]^7)/(3*f*(a + a*Sin[e + f*x])^5)
+ (14*a^4*c^4*Cos[e + f*x]^5)/(3*f*(a^2 + a^2*Sin[e + f*x])^3) + (35*c^4*Cos[e + f*x]^3)/(6*f*(a^2 + a^2*Sin[
e + f*x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2758

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

Rule 2759

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

Rule 2761

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*((g*Cos[e
+ f*x])^(p - 1)/(b*f*(p - 1))), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

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 {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx &=\left (a^4 c^4\right ) \int \frac {\cos ^8(e+f x)}{(a+a \sin (e+f x))^6} \, dx\\ &=-\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}-\frac {1}{3} \left (7 a^2 c^4\right ) \int \frac {\cos ^6(e+f x)}{(a+a \sin (e+f x))^4} \, dx\\ &=-\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac {1}{3} \left (35 c^4\right ) \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^2} \, dx\\ &=-\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac {35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {\left (35 c^4\right ) \int \frac {\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{2 a}\\ &=\frac {35 c^4 \cos (e+f x)}{2 a^2 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac {35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {\left (35 c^4\right ) \int 1 \, dx}{2 a^2}\\ &=\frac {35 c^4 x}{2 a^2}+\frac {35 c^4 \cos (e+f x)}{2 a^2 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac {35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 243, normalized size = 1.80 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^4 \left (128 \sin \left (\frac {1}{2} (e+f x)\right )-64 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-640 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+210 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+72 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sin (2 (e+f x))\right )}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8 (a+a \sin (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^4*(128*Sin[(e + f*x)/2] - 64*(Cos[(e + f*x)/2] + S
in[(e + f*x)/2]) - 640*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 210*(e + f*x)*(Cos[(e + f*x)
/2] + Sin[(e + f*x)/2])^3 + 72*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 3*(Cos[(e + f*x)/2] + Si
n[(e + f*x)/2])^3*Sin[2*(e + f*x)]))/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(a + a*Sin[e + f*x])^2)

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Maple [A]
time = 0.40, size = 125, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {2 c^{4} \left (-\frac {32}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {16}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}+\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+6 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+6}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {35 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{f \,a^{2}}\)	\(125\)
default	\(\frac {2 c^{4} \left (-\frac {32}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {16}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}+\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+6 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+6}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {35 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{f \,a^{2}}\)	\(125\)
risch	\(\frac {35 c^{4} x}{2 a^{2}}+\frac {i c^{4} {\mathrm e}^{2 i \left (f x +e \right )}}{8 a^{2} f}+\frac {3 c^{4} {\mathrm e}^{i \left (f x +e \right )}}{a^{2} f}+\frac {3 c^{4} {\mathrm e}^{-i \left (f x +e \right )}}{a^{2} f}-\frac {i c^{4} {\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{2} f}+\frac {96 i c^{4} {\mathrm e}^{i \left (f x +e \right )}+64 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}-\frac {160 c^{4}}{3}}{f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\)	\(150\)
norman	\(\frac {\frac {111 c^{4} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {131 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {500 c^{4} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {35 c^{4} x}{2 a}+\frac {164 c^{4}}{3 a f}+\frac {105 c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}+\frac {245 c^{4} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {455 c^{4} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {315 c^{4} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {385 c^{4} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {385 c^{4} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {315 c^{4} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {455 c^{4} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {245 c^{4} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {105 c^{4} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {35 c^{4} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {33 c^{4} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {460 c^{4} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {524 c^{4} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {718 c^{4} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {632 c^{4} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {1454 c^{4} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {815 c^{4} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4} a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\)	\(490\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f*c^4/a^2*(-32/3/(tan(1/2*f*x+1/2*e)+1)^3+16/(tan(1/2*f*x+1/2*e)+1)^2+16/(tan(1/2*f*x+1/2*e)+1)+(1/2*tan(1/2
*f*x+1/2*e)^3+6*tan(1/2*f*x+1/2*e)^2-1/2*tan(1/2*f*x+1/2*e)+6)/(1+tan(1/2*f*x+1/2*e)^2)^2+35/2*arctan(tan(1/2*
f*x+1/2*e)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (132) = 264\).
time = 0.55, size = 981, normalized size = 7.27 \begin {gather*} \frac {c^{4} {\left (\frac {\frac {75 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {97 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {126 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {98 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {63 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {21 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + 32}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {7 \, a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {7 \, a^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {5 \, a^{2} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {a^{2} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}} + \frac {21 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} + 16 \, c^{4} {\left (\frac {\frac {12 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {11 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {9 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 5}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {4 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {4 \, a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} + 12 \, c^{4} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 4}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} - \frac {2 \, c^{4} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {8 \, c^{4} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}}{3 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(c^4*((75*sin(f*x + e)/(cos(f*x + e) + 1) + 97*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 126*sin(f*x + e)^3/(c
os(f*x + e) + 1)^3 + 98*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 63*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 21*sin(
f*x + e)^6/(cos(f*x + e) + 1)^6 + 32)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 5*a^2*sin(f*x + e)^2/(cos
(f*x + e) + 1)^2 + 7*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 7*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5*a
^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 3*a^2*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a^2*sin(f*x + e)^7/(cos(f
*x + e) + 1)^7) + 21*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 16*c^4*((12*sin(f*x + e)/(cos(f*x + e) + 1
) + 11*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 9*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^4/(cos(f*x
+ e) + 1)^4 + 5)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 4*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4
*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^2*sin(f*x + e)^5/(cos
(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 12*c^4*((9*sin(f*x + e)/(cos(f*x + e) + 1
) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x +
e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1
))/a^2) - 2*c^4*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2)/(a^2 + 3*a^2*s
in(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3) + 8*c^4*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 1)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*s
in(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3))/f

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Fricas [A]
time = 0.32, size = 222, normalized size = 1.64 \begin {gather*} -\frac {3 \, c^{4} \cos \left (f x + e\right )^{4} - 30 \, c^{4} \cos \left (f x + e\right )^{3} + 210 \, c^{4} f x - 32 \, c^{4} - {\left (105 \, c^{4} f x - 193 \, c^{4}\right )} \cos \left (f x + e\right )^{2} + {\left (105 \, c^{4} f x + 194 \, c^{4}\right )} \cos \left (f x + e\right ) + {\left (3 \, c^{4} \cos \left (f x + e\right )^{3} + 210 \, c^{4} f x + 33 \, c^{4} \cos \left (f x + e\right )^{2} + 32 \, c^{4} + {\left (105 \, c^{4} f x + 226 \, c^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*c^4*cos(f*x + e)^4 - 30*c^4*cos(f*x + e)^3 + 210*c^4*f*x - 32*c^4 - (105*c^4*f*x - 193*c^4)*cos(f*x +
e)^2 + (105*c^4*f*x + 194*c^4)*cos(f*x + e) + (3*c^4*cos(f*x + e)^3 + 210*c^4*f*x + 33*c^4*cos(f*x + e)^2 + 32
*c^4 + (105*c^4*f*x + 226*c^4)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*
f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2312 vs. \(2 (128) = 256\).
time = 8.82, size = 2312, normalized size = 17.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((105*c**4*f*x*tan(e/2 + f*x/2)**7/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30
*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/
2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 315*c**4*f*x*tan(e/2 + f*x/2)**6/(6*a**2*f*tan(e/2 +
f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42
*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 525*c**
4*f*x*tan(e/2 + f*x/2)**5/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 +
f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18
*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 735*c**4*f*x*tan(e/2 + f*x/2)**4/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**
2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 +
f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 735*c**4*f*x*tan(e/2 + f*
x/2)**3/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**
2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 +
f*x/2) + 6*a**2*f) + 525*c**4*f*x*tan(e/2 + f*x/2)**2/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/
2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**
2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 315*c**4*f*x*tan(e/2 + f*x/2)/(6*a**2*f*tan
(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)*
*4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) +
105*c**4*f*x/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 4
2*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e
/2 + f*x/2) + 6*a**2*f) + 198*c**4*tan(e/2 + f*x/2)**6/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x
/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a*
*2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 666*c**4*tan(e/2 + f*x/2)**5/(6*a**2*f*tan
(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)*
*4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) +
868*c**4*tan(e/2 + f*x/2)**4/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2
+ f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 +
18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 1428*c**4*tan(e/2 + f*x/2)**3/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**
2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 +
f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 974*c**4*tan(e/2 + f*x/2)
**2/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*
tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/
2) + 6*a**2*f) + 786*c**4*tan(e/2 + f*x/2)/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan(e/2 + f*x/2)**6 + 30*
a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)**3 + 30*a**2*f*tan(e/2
+ f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 328*c**4/(6*a**2*f*tan(e/2 + f*x/2)**7 + 18*a**2*f*tan
(e/2 + f*x/2)**6 + 30*a**2*f*tan(e/2 + f*x/2)**5 + 42*a**2*f*tan(e/2 + f*x/2)**4 + 42*a**2*f*tan(e/2 + f*x/2)*
*3 + 30*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f), Ne(f, 0)), (x*(-c*sin(e) + c)**4/
(a*sin(e) + a)**2, True))

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Giac [A]
time = 0.46, size = 152, normalized size = 1.13 \begin {gather*} \frac {\frac {105 \, {\left (f x + e\right )} c^{4}}{a^{2}} + \frac {6 \, {\left (c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 12 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, c^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a^{2}} + \frac {64 \, {\left (3 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, c^{4}\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{6 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(105*(f*x + e)*c^4/a^2 + 6*(c^4*tan(1/2*f*x + 1/2*e)^3 + 12*c^4*tan(1/2*f*x + 1/2*e)^2 - c^4*tan(1/2*f*x +
1/2*e) + 12*c^4)/((tan(1/2*f*x + 1/2*e)^2 + 1)^2*a^2) + 64*(3*c^4*tan(1/2*f*x + 1/2*e)^2 + 9*c^4*tan(1/2*f*x
+ 1/2*e) + 4*c^4)/(a^2*(tan(1/2*f*x + 1/2*e) + 1)^3))/f

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Mupad [B]
time = 10.09, size = 291, normalized size = 2.16 \begin {gather*} \frac {35\,c^4\,x}{2\,a^2}-\frac {\frac {35\,c^4\,\left (e+f\,x\right )}{2}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {105\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (315\,e+315\,f\,x+786\right )}{6}\right )-\frac {c^4\,\left (105\,e+105\,f\,x+328\right )}{6}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {105\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (315\,e+315\,f\,x+198\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {175\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (525\,e+525\,f\,x+666\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {175\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (525\,e+525\,f\,x+974\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {245\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (735\,e+735\,f\,x+868\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {245\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (735\,e+735\,f\,x+1428\right )}{6}\right )}{a^2\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(35*c^4*x)/(2*a^2) - ((35*c^4*(e + f*x))/2 + tan(e/2 + (f*x)/2)*((105*c^4*(e + f*x))/2 - (c^4*(315*e + 315*f*x
+ 786))/6) - (c^4*(105*e + 105*f*x + 328))/6 + tan(e/2 + (f*x)/2)^6*((105*c^4*(e + f*x))/2 - (c^4*(315*e + 31
5*f*x + 198))/6) + tan(e/2 + (f*x)/2)^5*((175*c^4*(e + f*x))/2 - (c^4*(525*e + 525*f*x + 666))/6) + tan(e/2 +
(f*x)/2)^2*((175*c^4*(e + f*x))/2 - (c^4*(525*e + 525*f*x + 974))/6) + tan(e/2 + (f*x)/2)^4*((245*c^4*(e + f*x
))/2 - (c^4*(735*e + 735*f*x + 868))/6) + tan(e/2 + (f*x)/2)^3*((245*c^4*(e + f*x))/2 - (c^4*(735*e + 735*f*x
+ 1428))/6))/(a^2*f*(tan(e/2 + (f*x)/2) + 1)^3*(tan(e/2 + (f*x)/2)^2 + 1)^2)

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