3.3.35 \(\int \frac {\sin ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx\) [235]

Optimal. Leaf size=129 \[ -\frac {19 A x}{2 a^3}-\frac {4 A \cos (c+d x)}{a^3 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {41 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}-\frac {199 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))} \]

[Out]

-19/2*A*x/a^3-4*A*cos(d*x+c)/a^3/d+1/2*A*cos(d*x+c)*sin(d*x+c)/a^3/d-2/5*A*cos(d*x+c)/a^3/d/(1+sin(d*x+c))^3+4
1/15*A*cos(d*x+c)/a^3/d/(1+sin(d*x+c))^2-199/15*A*cos(d*x+c)/a^3/d/(1+sin(d*x+c))

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Rubi [A]
time = 0.15, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3045, 2718, 2715, 8, 2729, 2727} \begin {gather*} -\frac {4 A \cos (c+d x)}{a^3 d}+\frac {A \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {199 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}+\frac {41 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}-\frac {2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac {19 A x}{2 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sin[c + d*x]^4*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]

[Out]

(-19*A*x)/(2*a^3) - (4*A*Cos[c + d*x])/(a^3*d) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (2*A*Cos[c + d*x])/
(5*a^3*d*(1 + Sin[c + d*x])^3) + (41*A*Cos[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x])^2) - (199*A*Cos[c + d*x])/(1
5*a^3*d*(1 + Sin[c + d*x]))

Rule 8

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2729

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d
*(2*n + 1))), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 3045

Int[sin[(e_.) + (f_.)*(x_)]^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; Fr
eeQ[{a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {\sin ^4(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx &=\int \left (-\frac {9 A}{a^3}+\frac {4 A \sin (c+d x)}{a^3}-\frac {A \sin ^2(c+d x)}{a^3}+\frac {2 A}{a^3 (1+\sin (c+d x))^3}-\frac {9 A}{a^3 (1+\sin (c+d x))^2}+\frac {16 A}{a^3 (1+\sin (c+d x))}\right ) \, dx\\ &=-\frac {9 A x}{a^3}-\frac {A \int \sin ^2(c+d x) \, dx}{a^3}+\frac {(2 A) \int \frac {1}{(1+\sin (c+d x))^3} \, dx}{a^3}+\frac {(4 A) \int \sin (c+d x) \, dx}{a^3}-\frac {(9 A) \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{a^3}+\frac {(16 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3}\\ &=-\frac {9 A x}{a^3}-\frac {4 A \cos (c+d x)}{a^3 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {3 A \cos (c+d x)}{a^3 d (1+\sin (c+d x))^2}-\frac {16 A \cos (c+d x)}{a^3 d (1+\sin (c+d x))}-\frac {A \int 1 \, dx}{2 a^3}+\frac {(4 A) \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{5 a^3}-\frac {(3 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3}\\ &=-\frac {19 A x}{2 a^3}-\frac {4 A \cos (c+d x)}{a^3 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {41 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}-\frac {13 A \cos (c+d x)}{a^3 d (1+\sin (c+d x))}+\frac {(4 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{15 a^3}\\ &=-\frac {19 A x}{2 a^3}-\frac {4 A \cos (c+d x)}{a^3 d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {41 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}-\frac {199 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 0.66, size = 254, normalized size = 1.97 \begin {gather*} \frac {A \left (-11400 d x \cos \left (\frac {d x}{2}\right )+12060 \cos \left (c+\frac {d x}{2}\right )-14090 \cos \left (c+\frac {3 d x}{2}\right )+5700 d x \cos \left (2 c+\frac {3 d x}{2}\right )+1140 d x \cos \left (2 c+\frac {5 d x}{2}\right )+1050 \cos \left (3 c+\frac {5 d x}{2}\right )+165 \cos \left (3 c+\frac {7 d x}{2}\right )+15 \cos \left (5 c+\frac {9 d x}{2}\right )+19780 \sin \left (\frac {d x}{2}\right )-11400 d x \sin \left (c+\frac {d x}{2}\right )-5700 d x \sin \left (c+\frac {3 d x}{2}\right )+1830 \sin \left (2 c+\frac {3 d x}{2}\right )-4234 \sin \left (2 c+\frac {5 d x}{2}\right )+1140 d x \sin \left (3 c+\frac {5 d x}{2}\right )+165 \sin \left (4 c+\frac {7 d x}{2}\right )-15 \sin \left (4 c+\frac {9 d x}{2}\right )\right )}{480 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sin[c + d*x]^4*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]

[Out]

(A*(-11400*d*x*Cos[(d*x)/2] + 12060*Cos[c + (d*x)/2] - 14090*Cos[c + (3*d*x)/2] + 5700*d*x*Cos[2*c + (3*d*x)/2
] + 1140*d*x*Cos[2*c + (5*d*x)/2] + 1050*Cos[3*c + (5*d*x)/2] + 165*Cos[3*c + (7*d*x)/2] + 15*Cos[5*c + (9*d*x
)/2] + 19780*Sin[(d*x)/2] - 11400*d*x*Sin[c + (d*x)/2] - 5700*d*x*Sin[c + (3*d*x)/2] + 1830*Sin[2*c + (3*d*x)/
2] - 4234*Sin[2*c + (5*d*x)/2] + 1140*d*x*Sin[3*c + (5*d*x)/2] + 165*Sin[4*c + (7*d*x)/2] - 15*Sin[4*c + (9*d*
x)/2]))/(480*a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)

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Maple [A]
time = 0.40, size = 154, normalized size = 1.19

method	result	size
derivativedivides	\(\frac {32 A \left (-\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+4}{16 \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {19 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {1}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {5}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {9}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\right )}{d \,a^{3}}\)	\(154\)
default	\(\frac {32 A \left (-\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+4}{16 \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {19 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {1}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {5}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {9}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\right )}{d \,a^{3}}\)	\(154\)
risch	\(-\frac {19 A x}{2 a^{3}}-\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {2 A \,{\mathrm e}^{i \left (d x +c \right )}}{a^{3} d}-\frac {2 A \,{\mathrm e}^{-i \left (d x +c \right )}}{a^{3} d}+\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {2 \left (825 i A \,{\mathrm e}^{3 i \left (d x +c \right )}+240 A \,{\mathrm e}^{4 i \left (d x +c \right )}-755 i A \,{\mathrm e}^{i \left (d x +c \right )}-1165 A \,{\mathrm e}^{2 i \left (d x +c \right )}+199 A \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5}}\)	\(159\)
norman	\(\frac {-\frac {19 A \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2232 A \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {1235 A x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {665 A x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {391 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a d}-\frac {285 A x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {95 A x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {19 A x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {2300 A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {5599 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {448 A}{15 a d}-\frac {6979 A \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {7192 A \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {5117 A \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {19 A x}{2 a}-\frac {285 A x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {665 A x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1235 A x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1919 A x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {2565 A x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {2945 A x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {2945 A x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {2565 A x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1919 A x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1308 A \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {353 A \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {5751 A \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}-\frac {1900 A \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {836 A \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {95 A \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {95 A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\)	\(596\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^4*(A-A*sin(d*x+c))/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

32/d*A/a^3*(-1/16*(1/2*tan(1/2*d*x+1/2*c)^3+4*tan(1/2*d*x+1/2*c)^2-1/2*tan(1/2*d*x+1/2*c)+4)/(1+tan(1/2*d*x+1/
2*c)^2)^2-19/32*arctan(tan(1/2*d*x+1/2*c))-1/10/(tan(1/2*d*x+1/2*c)+1)^5+1/4/(tan(1/2*d*x+1/2*c)+1)^4+1/24/(ta
n(1/2*d*x+1/2*c)+1)^3-5/16/(tan(1/2*d*x+1/2*c)+1)^2-9/16/(tan(1/2*d*x+1/2*c)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (119) = 238\).
time = 0.52, size = 715, normalized size = 5.54 \begin {gather*} -\frac {A {\left (\frac {\frac {1325 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2673 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {4329 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3575 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {2275 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {975 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {195 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 304}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {20 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {26 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {26 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}} + \frac {195 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + 6 \, A {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {189 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {160 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {75 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 24}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {11 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {11 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{15 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^4*(A-A*sin(d*x+c))/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/15*(A*((1325*sin(d*x + c)/(cos(d*x + c) + 1) + 2673*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3805*sin(d*x + c)
^3/(cos(d*x + c) + 1)^3 + 4329*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 3575*sin(d*x + c)^5/(cos(d*x + c) + 1)^5
+ 2275*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 975*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 195*sin(d*x + c)^8/(cos
(d*x + c) + 1)^8 + 304)/(a^3 + 5*a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 12*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1
)^2 + 20*a^3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 26*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 26*a^3*sin(d*x
+ c)^5/(cos(d*x + c) + 1)^5 + 20*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 12*a^3*sin(d*x + c)^7/(cos(d*x + c
) + 1)^7 + 5*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + a^3*sin(d*x + c)^9/(cos(d*x + c) + 1)^9) + 195*arctan(s
in(d*x + c)/(cos(d*x + c) + 1))/a^3) + 6*A*((105*sin(d*x + c)/(cos(d*x + c) + 1) + 189*sin(d*x + c)^2/(cos(d*x
+ c) + 1)^2 + 200*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 160*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 75*sin(d*x
+ c)^5/(cos(d*x + c) + 1)^5 + 15*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 24)/(a^3 + 5*a^3*sin(d*x + c)/(cos(d*x
+ c) + 1) + 11*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a^3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*a^3*s
in(d*x + c)^4/(cos(d*x + c) + 1)^4 + 11*a^3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*a^3*sin(d*x + c)^6/(cos(d*
x + c) + 1)^6 + a^3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7) + 15*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (119) = 238\).
time = 0.38, size = 248, normalized size = 1.92 \begin {gather*} -\frac {15 \, A \cos \left (d x + c\right )^{5} + 90 \, A \cos \left (d x + c\right )^{4} + {\left (285 \, A d x + 683 \, A\right )} \cos \left (d x + c\right )^{3} - 1140 \, A d x + {\left (855 \, A d x - 526 \, A\right )} \cos \left (d x + c\right )^{2} - 6 \, {\left (95 \, A d x + 191 \, A\right )} \cos \left (d x + c\right ) - {\left (15 \, A \cos \left (d x + c\right )^{4} - 75 \, A \cos \left (d x + c\right )^{3} + 1140 \, A d x - 19 \, {\left (15 \, A d x - 32 \, A\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (95 \, A d x + 189 \, A\right )} \cos \left (d x + c\right ) - 12 \, A\right )} \sin \left (d x + c\right ) - 12 \, A}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^4*(A-A*sin(d*x+c))/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/30*(15*A*cos(d*x + c)^5 + 90*A*cos(d*x + c)^4 + (285*A*d*x + 683*A)*cos(d*x + c)^3 - 1140*A*d*x + (855*A*d*
x - 526*A)*cos(d*x + c)^2 - 6*(95*A*d*x + 191*A)*cos(d*x + c) - (15*A*cos(d*x + c)^4 - 75*A*cos(d*x + c)^3 + 1
140*A*d*x - 19*(15*A*d*x - 32*A)*cos(d*x + c)^2 + 6*(95*A*d*x + 189*A)*cos(d*x + c) - 12*A)*sin(d*x + c) - 12*
A)/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d + (a^3*d*cos(d*x + c)^2 - 2
*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3614 vs. \(2 (126) = 252\).
time = 27.94, size = 3614, normalized size = 28.02 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**4*(A-A*sin(d*x+c))/(a+a*sin(d*x+c))**3,x)

[Out]

Piecewise((-285*A*d*x*tan(c/2 + d*x/2)**9/(30*a**3*d*tan(c/2 + d*x/2)**9 + 150*a**3*d*tan(c/2 + d*x/2)**8 + 36
0*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 + d*x/2)**6 + 780*a**3*d*tan(c/2 + d*x/2)**5 + 780*a**3*d*ta
n(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3 + 360*a**3*d*tan(c/2 + d*x/2)**2 + 150*a**3*d*tan(c/2 + d*x
/2) + 30*a**3*d) - 1425*A*d*x*tan(c/2 + d*x/2)**8/(30*a**3*d*tan(c/2 + d*x/2)**9 + 150*a**3*d*tan(c/2 + d*x/2)
**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 + d*x/2)**6 + 780*a**3*d*tan(c/2 + d*x/2)**5 + 780*a
**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3 + 360*a**3*d*tan(c/2 + d*x/2)**2 + 150*a**3*d*tan(c
/2 + d*x/2) + 30*a**3*d) - 3420*A*d*x*tan(c/2 + d*x/2)**7/(30*a**3*d*tan(c/2 + d*x/2)**9 + 150*a**3*d*tan(c/2
+ d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 + d*x/2)**6 + 780*a**3*d*tan(c/2 + d*x/2)**5
+ 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3 + 360*a**3*d*tan(c/2 + d*x/2)**2 + 150*a**3
*d*tan(c/2 + d*x/2) + 30*a**3*d) - 5700*A*d*x*tan(c/2 + d*x/2)**6/(30*a**3*d*tan(c/2 + d*x/2)**9 + 150*a**3*d*
tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 + d*x/2)**6 + 780*a**3*d*tan(c/2 + d
*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3 + 360*a**3*d*tan(c/2 + d*x/2)**2 +
150*a**3*d*tan(c/2 + d*x/2) + 30*a**3*d) - 7410*A*d*x*tan(c/2 + d*x/2)**5/(30*a**3*d*tan(c/2 + d*x/2)**9 + 150
*a**3*d*tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 + d*x/2)**6 + 780*a**3*d*tan
(c/2 + d*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3 + 360*a**3*d*tan(c/2 + d*x/
2)**2 + 150*a**3*d*tan(c/2 + d*x/2) + 30*a**3*d) - 7410*A*d*x*tan(c/2 + d*x/2)**4/(30*a**3*d*tan(c/2 + d*x/2)*
*9 + 150*a**3*d*tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 + d*x/2)**6 + 780*a*
*3*d*tan(c/2 + d*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3 + 360*a**3*d*tan(c/
2 + d*x/2)**2 + 150*a**3*d*tan(c/2 + d*x/2) + 30*a**3*d) - 5700*A*d*x*tan(c/2 + d*x/2)**3/(30*a**3*d*tan(c/2 +
d*x/2)**9 + 150*a**3*d*tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 + d*x/2)**6
+ 780*a**3*d*tan(c/2 + d*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3 + 360*a**3*
d*tan(c/2 + d*x/2)**2 + 150*a**3*d*tan(c/2 + d*x/2) + 30*a**3*d) - 3420*A*d*x*tan(c/2 + d*x/2)**2/(30*a**3*d*t
an(c/2 + d*x/2)**9 + 150*a**3*d*tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 + d*
x/2)**6 + 780*a**3*d*tan(c/2 + d*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3 + 3
60*a**3*d*tan(c/2 + d*x/2)**2 + 150*a**3*d*tan(c/2 + d*x/2) + 30*a**3*d) - 1425*A*d*x*tan(c/2 + d*x/2)/(30*a**
3*d*tan(c/2 + d*x/2)**9 + 150*a**3*d*tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2
+ d*x/2)**6 + 780*a**3*d*tan(c/2 + d*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**
3 + 360*a**3*d*tan(c/2 + d*x/2)**2 + 150*a**3*d*tan(c/2 + d*x/2) + 30*a**3*d) - 285*A*d*x/(30*a**3*d*tan(c/2 +
d*x/2)**9 + 150*a**3*d*tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 + d*x/2)**6
+ 780*a**3*d*tan(c/2 + d*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3 + 360*a**3*
d*tan(c/2 + d*x/2)**2 + 150*a**3*d*tan(c/2 + d*x/2) + 30*a**3*d) - 570*A*tan(c/2 + d*x/2)**8/(30*a**3*d*tan(c/
2 + d*x/2)**9 + 150*a**3*d*tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 + d*x/2)*
*6 + 780*a**3*d*tan(c/2 + d*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3 + 360*a*
*3*d*tan(c/2 + d*x/2)**2 + 150*a**3*d*tan(c/2 + d*x/2) + 30*a**3*d) - 2850*A*tan(c/2 + d*x/2)**7/(30*a**3*d*ta
n(c/2 + d*x/2)**9 + 150*a**3*d*tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 + d*x
/2)**6 + 780*a**3*d*tan(c/2 + d*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3 + 36
0*a**3*d*tan(c/2 + d*x/2)**2 + 150*a**3*d*tan(c/2 + d*x/2) + 30*a**3*d) - 6650*A*tan(c/2 + d*x/2)**6/(30*a**3*
d*tan(c/2 + d*x/2)**9 + 150*a**3*d*tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(c/2 +
d*x/2)**6 + 780*a**3*d*tan(c/2 + d*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**3
+ 360*a**3*d*tan(c/2 + d*x/2)**2 + 150*a**3*d*tan(c/2 + d*x/2) + 30*a**3*d) - 10450*A*tan(c/2 + d*x/2)**5/(30*
a**3*d*tan(c/2 + d*x/2)**9 + 150*a**3*d*tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d*tan(
c/2 + d*x/2)**6 + 780*a**3*d*tan(c/2 + d*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2
)**3 + 360*a**3*d*tan(c/2 + d*x/2)**2 + 150*a**3*d*tan(c/2 + d*x/2) + 30*a**3*d) - 12846*A*tan(c/2 + d*x/2)**4
/(30*a**3*d*tan(c/2 + d*x/2)**9 + 150*a**3*d*tan(c/2 + d*x/2)**8 + 360*a**3*d*tan(c/2 + d*x/2)**7 + 600*a**3*d
*tan(c/2 + d*x/2)**6 + 780*a**3*d*tan(c/2 + d*x/2)**5 + 780*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 +
d*x/2)**3 + 360*a**3*d*tan(c/2 + d*x/2)**2 + 15...

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Giac [A]
time = 0.49, size = 156, normalized size = 1.21 \begin {gather*} -\frac {\frac {285 \, {\left (d x + c\right )} A}{a^{3}} + \frac {30 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, A\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} + \frac {4 \, {\left (135 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 615 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1025 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 685 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 164 \, A\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{30 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^4*(A-A*sin(d*x+c))/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/30*(285*(d*x + c)*A/a^3 + 30*(A*tan(1/2*d*x + 1/2*c)^3 + 8*A*tan(1/2*d*x + 1/2*c)^2 - A*tan(1/2*d*x + 1/2*c
) + 8*A)/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^3) + 4*(135*A*tan(1/2*d*x + 1/2*c)^4 + 615*A*tan(1/2*d*x + 1/2*c)^3
+ 1025*A*tan(1/2*d*x + 1/2*c)^2 + 685*A*tan(1/2*d*x + 1/2*c) + 164*A)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^5))/d

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Mupad [B]
time = 17.08, size = 326, normalized size = 2.53 \begin {gather*} \frac {\left (\frac {95\,A\,\left (c+d\,x\right )}{2}-\frac {A\,\left (1425\,c+1425\,d\,x+570\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\left (114\,A\,\left (c+d\,x\right )-\frac {A\,\left (3420\,c+3420\,d\,x+2850\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (190\,A\,\left (c+d\,x\right )-\frac {A\,\left (5700\,c+5700\,d\,x+6650\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (247\,A\,\left (c+d\,x\right )-\frac {A\,\left (7410\,c+7410\,d\,x+10450\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (247\,A\,\left (c+d\,x\right )-\frac {A\,\left (7410\,c+7410\,d\,x+12846\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (190\,A\,\left (c+d\,x\right )-\frac {A\,\left (5700\,c+5700\,d\,x+11270\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (114\,A\,\left (c+d\,x\right )-\frac {A\,\left (3420\,c+3420\,d\,x+7902\right )}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (\frac {95\,A\,\left (c+d\,x\right )}{2}-\frac {A\,\left (1425\,c+1425\,d\,x+3910\right )}{30}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {19\,A\,\left (c+d\,x\right )}{2}-\frac {A\,\left (285\,c+285\,d\,x+896\right )}{30}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}-\frac {19\,A\,x}{2\,a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sin(c + d*x)^4*(A - A*sin(c + d*x)))/(a + a*sin(c + d*x))^3,x)

[Out]

(tan(c/2 + (d*x)/2)*((95*A*(c + d*x))/2 - (A*(1425*c + 1425*d*x + 3910))/30) + tan(c/2 + (d*x)/2)^8*((95*A*(c
+ d*x))/2 - (A*(1425*c + 1425*d*x + 570))/30) + tan(c/2 + (d*x)/2)^7*(114*A*(c + d*x) - (A*(3420*c + 3420*d*x
+ 2850))/30) + tan(c/2 + (d*x)/2)^2*(114*A*(c + d*x) - (A*(3420*c + 3420*d*x + 7902))/30) + tan(c/2 + (d*x)/2)
^6*(190*A*(c + d*x) - (A*(5700*c + 5700*d*x + 6650))/30) + tan(c/2 + (d*x)/2)^3*(190*A*(c + d*x) - (A*(5700*c
+ 5700*d*x + 11270))/30) + tan(c/2 + (d*x)/2)^5*(247*A*(c + d*x) - (A*(7410*c + 7410*d*x + 10450))/30) + tan(c
/2 + (d*x)/2)^4*(247*A*(c + d*x) - (A*(7410*c + 7410*d*x + 12846))/30) + (19*A*(c + d*x))/2 - (A*(285*c + 285*
d*x + 896))/30)/(a^3*d*(tan(c/2 + (d*x)/2) + 1)^5*(tan(c/2 + (d*x)/2)^2 + 1)^2) - (19*A*x)/(2*a^3)

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