3.3.79 \(\int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx\) [279]

Optimal. Leaf size=161 \[ -\frac {63 c^5 x}{2 a^3}-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )} \]

[Out]

-63/2*c^5*x/a^3-63/2*c^5*cos(f*x+e)/a^3/f-2/5*a^4*c^5*cos(f*x+e)^9/f/(a+a*sin(f*x+e))^7+6/5*a^2*c^5*cos(f*x+e)
^7/f/(a+a*sin(f*x+e))^5-42/5*c^5*cos(f*x+e)^5/f/(a+a*sin(f*x+e))^3-21/2*c^5*cos(f*x+e)^3/f/(a^3+a^3*sin(f*x+e)
)

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Rubi [A]
time = 0.20, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 7, 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^4 c^5 \cos ^9(e+f x)}{5 f (a \sin (e+f x)+a)^7}-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3 \sin (e+f x)+a^3\right )}-\frac {63 c^5 x}{2 a^3}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - c*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^3,x]

[Out]

(-63*c^5*x)/(2*a^3) - (63*c^5*Cos[e + f*x])/(2*a^3*f) - (2*a^4*c^5*Cos[e + f*x]^9)/(5*f*(a + a*Sin[e + f*x])^7
) + (6*a^2*c^5*Cos[e + f*x]^7)/(5*f*(a + a*Sin[e + f*x])^5) - (42*c^5*Cos[e + f*x]^5)/(5*f*(a + a*Sin[e + f*x]
)^3) - (21*c^5*Cos[e + f*x]^3)/(2*f*(a^3 + a^3*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))^5}{(a+a \sin (e+f x))^3} \, dx &=\left (a^5 c^5\right ) \int \frac {\cos ^{10}(e+f x)}{(a+a \sin (e+f x))^8} \, dx\\ &=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}-\frac {1}{5} \left (9 a^3 c^5\right ) \int \frac {\cos ^8(e+f x)}{(a+a \sin (e+f x))^6} \, dx\\ &=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}+\frac {1}{5} \left (21 a c^5\right ) \int \frac {\cos ^6(e+f x)}{(a+a \sin (e+f x))^4} \, dx\\ &=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {\left (21 c^5\right ) \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^2} \, dx}{a}\\ &=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {\left (63 c^5\right ) \int \frac {\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{2 a^2}\\ &=-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {\left (63 c^5\right ) \int 1 \, dx}{2 a^3}\\ &=-\frac {63 c^5 x}{2 a^3}-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.57, size = 303, normalized size = 1.88 \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))^5 \left (256 \sin \left (\frac {1}{2} (e+f x)\right )-128 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-896 \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+448 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+2304 \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 )^4-630 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-160 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sin (2 (e+f x))\right )}{20 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10} (a+a \sin (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c - c*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^3,x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^5*(256*Sin[(e + f*x)/2] - 128*(Cos[(e + f*x)/2] +
Sin[(e + f*x)/2]) - 896*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 448*(Cos[(e + f*x)/2] + Sin
[(e + f*x)/2])^3 + 2304*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 630*(e + f*x)*(Cos[(e + f*x
)/2] + Sin[(e + f*x)/2])^5 - 160*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 5*(Cos[(e + f*x)/2] +
Sin[(e + f*x)/2])^5*Sin[2*(e + f*x)]))/(20*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(a + a*Sin[e + f*x])^3)

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Maple [A]
time = 0.44, size = 156, normalized size = 0.97

method	result	size
derivativedivides	\(\frac {2 c^{5} \left (-\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+8 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+8}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {63 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {32}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\)	\(156\)
default	\(\frac {2 c^{5} \left (-\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+8 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+8}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {63 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {32}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\)	\(156\)
risch	\(-\frac {63 c^{5} x}{2 a^{3}}-\frac {i c^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{8 f \,a^{3}}-\frac {4 c^{5} {\mathrm e}^{i \left (f x +e \right )}}{a^{3} f}-\frac {4 c^{5} {\mathrm e}^{-i \left (f x +e \right )}}{a^{3} f}+\frac {i c^{5} {\mathrm e}^{-2 i \left (f x +e \right )}}{8 f \,a^{3}}-\frac {32 \left (-105 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}+75 i c^{5} {\mathrm e}^{3 i \left (f x +e \right )}+25 c^{5} {\mathrm e}^{4 i \left (f x +e \right )}-65 i c^{5} {\mathrm e}^{i \left (f x +e \right )}+18 c^{5}\right )}{5 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\)	\(179\)
norman	\(\frac {-\frac {8505 c^{5} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {6363 c^{5} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {63 c^{5} x}{2 a}-\frac {2036 c^{5} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2516 c^{5} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {4412 c^{5} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {7915 c^{5} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {7736 c^{5} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {4095 c^{5} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {8505 c^{5} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {9765 c^{5} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {956 c^{5} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {5477 c^{5} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {19741 c^{5} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a f}-\frac {945 c^{5} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {309 c^{5} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {65 c^{5} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {9765 c^{5} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {4095 c^{5} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {945 c^{5} x \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {1179 c^{5} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {6071 c^{5} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {7640 c^{5} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2205 c^{5} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {6363 c^{5} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {431 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {315 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}-\frac {2205 c^{5} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {315 c^{5} x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {63 c^{5} x \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {496 c^{5}}{5 a f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\)	\(658\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

2/f*c^5/a^3*(-(1/2*tan(1/2*f*x+1/2*e)^3+8*tan(1/2*f*x+1/2*e)^2-1/2*tan(1/2*f*x+1/2*e)+8)/(1+tan(1/2*f*x+1/2*e)
^2)^2-63/2*arctan(tan(1/2*f*x+1/2*e))-128/5/(tan(1/2*f*x+1/2*e)+1)^5+64/(tan(1/2*f*x+1/2*e)+1)^4-32/(tan(1/2*f
*x+1/2*e)+1)^3-16/(tan(1/2*f*x+1/2*e)+1)^2-32/(tan(1/2*f*x+1/2*e)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1628 vs. \(2 (158) = 316\).
time = 0.57, size = 1628, normalized size = 10.11 \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))^5/(a+a*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

-1/15*(c^5*((1325*sin(f*x + e)/(cos(f*x + e) + 1) + 2673*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3805*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 + 4329*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3575*sin(f*x + e)^5/(cos(f*x + e) + 1)^
5 + 2275*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 975*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 195*sin(f*x + e)^8/(c
os(f*x + e) + 1)^8 + 304)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 12*a^3*sin(f*x + e)^2/(cos(f*x + e) +
1)^2 + 20*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 26*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 26*a^3*sin(f
*x + e)^5/(cos(f*x + e) + 1)^5 + 20*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 12*a^3*sin(f*x + e)^7/(cos(f*x +
e) + 1)^7 + 5*a^3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a^3*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 195*arctan
(sin(f*x + e)/(cos(f*x + e) + 1))/a^3) + 30*c^5*((105*sin(f*x + e)/(cos(f*x + e) + 1) + 189*sin(f*x + e)^2/(co
s(f*x + e) + 1)^2 + 200*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 160*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 75*sin
(f*x + e)^5/(cos(f*x + e) + 1)^5 + 15*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 24)/(a^3 + 5*a^3*sin(f*x + e)/(cos
(f*x + e) + 1) + 11*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 15*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*
a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 11*a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5*a^3*sin(f*x + e)^6/(c
os(f*x + e) + 1)^6 + a^3*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^3
) + 20*c^5*((95*sin(f*x + e)/(cos(f*x + e) + 1) + 145*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 75*sin(f*x + e)^3/
(cos(f*x + e) + 1)^3 + 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 22)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) +
1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x +
e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 15*arctan(sin(f*x + e)/(cos(f*x + e) +
1))/a^3) + 2*c^5*(20*sin(f*x + e)/(cos(f*x + e) + 1) + 40*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 30*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 + 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 7)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e
) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f
*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 40*c^5*(5*sin(f*x + e)/(cos(f*x +
e) + 1) + 10*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*si
n(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x
+ e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) - 30*c^5*(5*sin(f*x + e)/(cos(f*x + e) + 1) + 5*sin(f*
x + e)^2/(cos(f*x + e) + 1)^2 + 5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x
+ e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*si
n(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5))/f

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Fricas [A]
time = 0.33, size = 301, normalized size = 1.87 \begin {gather*} -\frac {5 \, c^{5} \cos \left (f x + e\right )^{5} + 70 \, c^{5} \cos \left (f x + e\right )^{4} - 1260 \, c^{5} f x - 64 \, c^{5} + 7 \, {\left (45 \, c^{5} f x + 113 \, c^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (945 \, c^{5} f x - 502 \, c^{5}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (315 \, c^{5} f x + 646 \, c^{5}\right )} \cos \left (f x + e\right ) - {\left (5 \, c^{5} \cos \left (f x + e\right )^{4} - 65 \, c^{5} \cos \left (f x + e\right )^{3} + 1260 \, c^{5} f x - 64 \, c^{5} - 3 \, {\left (105 \, c^{5} f x - 242 \, c^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (315 \, c^{5} f x + 614 \, c^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{10 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} 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))^5/(a+a*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/10*(5*c^5*cos(f*x + e)^5 + 70*c^5*cos(f*x + e)^4 - 1260*c^5*f*x - 64*c^5 + 7*(45*c^5*f*x + 113*c^5)*cos(f*x
+ e)^3 + (945*c^5*f*x - 502*c^5)*cos(f*x + e)^2 - 2*(315*c^5*f*x + 646*c^5)*cos(f*x + e) - (5*c^5*cos(f*x + e
)^4 - 65*c^5*cos(f*x + e)^3 + 1260*c^5*f*x - 64*c^5 - 3*(105*c^5*f*x - 242*c^5)*cos(f*x + e)^2 + 2*(315*c^5*f*
x + 614*c^5)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e)
- 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3643 vs. \(2 (153) = 306\).
time = 29.23, size = 3643, normalized size = 22.63 \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))**5/(a+a*sin(f*x+e))**3,x)

[Out]

Piecewise((-315*c**5*f*x*tan(e/2 + f*x/2)**9/(10*a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 +
120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*
tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*
x/2) + 10*a**3*f) - 1575*c**5*f*x*tan(e/2 + f*x/2)**8/(10*a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x
/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 26
0*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan
(e/2 + f*x/2) + 10*a**3*f) - 3780*c**5*f*x*tan(e/2 + f*x/2)**7/(10*a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(
e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2
)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a
**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 6300*c**5*f*x*tan(e/2 + f*x/2)**6/(10*a**3*f*tan(e/2 + f*x/2)**9 + 50*a*
*3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/
2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)**3 + 120*a**3*f*tan(e/2 + f*x/2)*
*2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 8190*c**5*f*x*tan(e/2 + f*x/2)**5/(10*a**3*f*tan(e/2 + f*x/2)**
9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan(e/2 + f*x/2)**6 + 260*a**3
*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)**3 + 120*a**3*f*tan(e/2
+ f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 8190*c**5*f*x*tan(e/2 + f*x/2)**4/(10*a**3*f*tan(e/2 +
f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan(e/2 + f*x/2)**6 +
260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)**3 + 120*a**3*f
*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 6300*c**5*f*x*tan(e/2 + f*x/2)**3/(10*a**3*f*
tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan(e/2 + f*
x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)**3 + 1
20*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 3780*c**5*f*x*tan(e/2 + f*x/2)**2/(1
0*a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan
(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/
2)**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 1575*c**5*f*x*tan(e/2 + f*x
/2)/(10*a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3
*f*tan(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2
+ f*x/2)**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 315*c**5*f*x/(10*a**3
*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan(e/2 +
f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)**3
+ 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 650*c**5*tan(e/2 + f*x/2)**8/(10*
a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*tan(e
/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f*x/2)
**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 3090*c**5*tan(e/2 + f*x/2)**7
/(10*a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a**3*f*
tan(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/2 + f
*x/2)**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 7610*c**5*tan(e/2 + f*x/
2)**6/(10*a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 + 200*a*
*3*f*tan(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*tan(e/
2 + f*x/2)**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 11090*c**5*tan(e/2
+ f*x/2)**5/(10*a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)**7 +
200*a**3*f*tan(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a**3*f*
tan(e/2 + f*x/2)**3 + 120*a**3*f*tan(e/2 + f*x/2)**2 + 50*a**3*f*tan(e/2 + f*x/2) + 10*a**3*f) - 14702*c**5*ta
n(e/2 + f*x/2)**4/(10*a**3*f*tan(e/2 + f*x/2)**9 + 50*a**3*f*tan(e/2 + f*x/2)**8 + 120*a**3*f*tan(e/2 + f*x/2)
**7 + 200*a**3*f*tan(e/2 + f*x/2)**6 + 260*a**3*f*tan(e/2 + f*x/2)**5 + 260*a**3*f*tan(e/2 + f*x/2)**4 + 200*a
**3*f*tan(e/2 + f*x/2)**3 + 120*a**3*f*tan(e/2 ...

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Giac [A]
time = 0.48, size = 186, normalized size = 1.16 \begin {gather*} -\frac {\frac {315 \, {\left (f x + e\right )} c^{5}}{a^{3}} + \frac {10 \, {\left (c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 16 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 16 \, c^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a^{3}} + \frac {64 \, {\left (10 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 45 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 85 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 55 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13 \, c^{5}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{10 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x, algorithm="giac")

[Out]

-1/10*(315*(f*x + e)*c^5/a^3 + 10*(c^5*tan(1/2*f*x + 1/2*e)^3 + 16*c^5*tan(1/2*f*x + 1/2*e)^2 - c^5*tan(1/2*f*
x + 1/2*e) + 16*c^5)/((tan(1/2*f*x + 1/2*e)^2 + 1)^2*a^3) + 64*(10*c^5*tan(1/2*f*x + 1/2*e)^4 + 45*c^5*tan(1/2
*f*x + 1/2*e)^3 + 85*c^5*tan(1/2*f*x + 1/2*e)^2 + 55*c^5*tan(1/2*f*x + 1/2*e) + 13*c^5)/(a^3*(tan(1/2*f*x + 1/
2*e) + 1)^5))/f

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Mupad [B]
time = 11.14, size = 364, normalized size = 2.26 \begin {gather*} \frac {\frac {63\,c^5\,\left (e+f\,x\right )}{2}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (1575\,e+1575\,f\,x+4310\right )}{10}\right )-\frac {c^5\,\left (315\,e+315\,f\,x+992\right )}{10}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (1575\,e+1575\,f\,x+650\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (378\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+3090\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (378\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+8814\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (6300\,e+6300\,f\,x+7610\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (6300\,e+6300\,f\,x+12230\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (819\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (8190\,e+8190\,f\,x+11090\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (819\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (8190\,e+8190\,f\,x+14702\right )}{10}\right )}{a^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^5\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^2}-\frac {63\,c^5\,x}{2\,a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - c*sin(e + f*x))^5/(a + a*sin(e + f*x))^3,x)

[Out]

((63*c^5*(e + f*x))/2 + tan(e/2 + (f*x)/2)*((315*c^5*(e + f*x))/2 - (c^5*(1575*e + 1575*f*x + 4310))/10) - (c^
5*(315*e + 315*f*x + 992))/10 + tan(e/2 + (f*x)/2)^8*((315*c^5*(e + f*x))/2 - (c^5*(1575*e + 1575*f*x + 650))/
10) + tan(e/2 + (f*x)/2)^7*(378*c^5*(e + f*x) - (c^5*(3780*e + 3780*f*x + 3090))/10) + tan(e/2 + (f*x)/2)^2*(3
78*c^5*(e + f*x) - (c^5*(3780*e + 3780*f*x + 8814))/10) + tan(e/2 + (f*x)/2)^6*(630*c^5*(e + f*x) - (c^5*(6300
*e + 6300*f*x + 7610))/10) + tan(e/2 + (f*x)/2)^3*(630*c^5*(e + f*x) - (c^5*(6300*e + 6300*f*x + 12230))/10) +
tan(e/2 + (f*x)/2)^5*(819*c^5*(e + f*x) - (c^5*(8190*e + 8190*f*x + 11090))/10) + tan(e/2 + (f*x)/2)^4*(819*c
^5*(e + f*x) - (c^5*(8190*e + 8190*f*x + 14702))/10))/(a^3*f*(tan(e/2 + (f*x)/2) + 1)^5*(tan(e/2 + (f*x)/2)^2
+ 1)^2) - (63*c^5*x)/(2*a^3)

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