3.5.80 \(\int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx\) [480]

Optimal. Leaf size=75 \[ -\frac {(A-B) \cos (x)}{7 (1+\sin (x))^4}-\frac {(3 A+4 B) \cos (x)}{35 (1+\sin (x))^3}-\frac {2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))^2}-\frac {2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))} \]

[Out]

-1/7*(A-B)*cos(x)/(1+sin(x))^4-1/35*(3*A+4*B)*cos(x)/(1+sin(x))^3-2/105*(3*A+4*B)*cos(x)/(1+sin(x))^2-2/105*(3
*A+4*B)*cos(x)/(1+sin(x))

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Rubi [A]
time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2829, 2729, 2727} \begin {gather*} -\frac {2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)}-\frac {2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)^2}-\frac {(3 A+4 B) \cos (x)}{35 (\sin (x)+1)^3}-\frac {(A-B) \cos (x)}{7 (\sin (x)+1)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Sin[x])/(1 + Sin[x])^4,x]

[Out]

-1/7*((A - B)*Cos[x])/(1 + Sin[x])^4 - ((3*A + 4*B)*Cos[x])/(35*(1 + Sin[x])^3) - (2*(3*A + 4*B)*Cos[x])/(105*
(1 + Sin[x])^2) - (2*(3*A + 4*B)*Cos[x])/(105*(1 + Sin[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 2829

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

Rubi steps

\begin {align*} \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx &=-\frac {(A-B) \cos (x)}{7 (1+\sin (x))^4}+\frac {1}{7} (3 A+4 B) \int \frac {1}{(1+\sin (x))^3} \, dx\\ &=-\frac {(A-B) \cos (x)}{7 (1+\sin (x))^4}-\frac {(3 A+4 B) \cos (x)}{35 (1+\sin (x))^3}+\frac {1}{35} (2 (3 A+4 B)) \int \frac {1}{(1+\sin (x))^2} \, dx\\ &=-\frac {(A-B) \cos (x)}{7 (1+\sin (x))^4}-\frac {(3 A+4 B) \cos (x)}{35 (1+\sin (x))^3}-\frac {2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))^2}+\frac {1}{105} (2 (3 A+4 B)) \int \frac {1}{1+\sin (x)} \, dx\\ &=-\frac {(A-B) \cos (x)}{7 (1+\sin (x))^4}-\frac {(3 A+4 B) \cos (x)}{35 (1+\sin (x))^3}-\frac {2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))^2}-\frac {2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 55, normalized size = 0.73 \begin {gather*} -\frac {\cos (x) \left (36 A+13 B+13 (3 A+4 B) \sin (x)+8 (3 A+4 B) \sin ^2(x)+(6 A+8 B) \sin ^3(x)\right )}{105 (1+\sin (x))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Sin[x])/(1 + Sin[x])^4,x]

[Out]

-1/105*(Cos[x]*(36*A + 13*B + 13*(3*A + 4*B)*Sin[x] + 8*(3*A + 4*B)*Sin[x]^2 + (6*A + 8*B)*Sin[x]^3))/(1 + Sin
[x])^4

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Maple [A]
time = 0.16, size = 115, normalized size = 1.53

method	result	size
risch	\(\frac {\frac {8 B \,{\mathrm e}^{4 i x}}{3}+\frac {8 i B \,{\mathrm e}^{3 i x}}{3}-\frac {12 A \,{\mathrm e}^{2 i x}}{5}-\frac {4 i A \,{\mathrm e}^{i x}}{5}+4 i A \,{\mathrm e}^{3 i x}-\frac {16 B \,{\mathrm e}^{2 i x}}{5}-\frac {16 i B \,{\mathrm e}^{i x}}{15}+\frac {4 A}{35}+\frac {16 B}{105}}{\left ({\mathrm e}^{i x}+i\right )^{7}}\)	\(80\)
default	\(-\frac {2 \left (18 A -10 B \right )}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-24 A +24 B}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{6}}-\frac {2 A}{\tan \left (\frac {x}{2}\right )+1}-\frac {-6 A +2 B}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {-32 A +24 B}{2 \left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {2 \left (36 A -32 B \right )}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {2 \left (8 A -8 B \right )}{7 \left (\tan \left (\frac {x}{2}\right )+1\right )^{7}}\)	\(115\)
norman	\(\frac {-2 A \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+\left (-\frac {318 A}{35}-\frac {362 B}{105}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\left (-\frac {102 A}{5}-\frac {98 B}{15}\right ) \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (-\frac {74 A}{5}-\frac {106 B}{15}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\left (-\frac {14 A}{5}-\frac {26 B}{15}\right ) \tan \left (\frac {x}{2}\right )+\left (-6 A -2 B \right ) \left (\tan ^{7}\left (\frac {x}{2}\right )\right )+\left (-14 A -\frac {10 B}{3}\right ) \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+\left (-18 A -\frac {22 B}{3}\right ) \left (\tan ^{5}\left (\frac {x}{2}\right )\right )-\frac {24 A}{35}-\frac {26 B}{105}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )^{7}}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sin(x))/(1+sin(x))^4,x,method=_RETURNVERBOSE)

[Out]

-2/3*(18*A-10*B)/(tan(1/2*x)+1)^3-1/3*(-24*A+24*B)/(tan(1/2*x)+1)^6-2*A/(tan(1/2*x)+1)-(-6*A+2*B)/(tan(1/2*x)+
1)^2-1/2*(-32*A+24*B)/(tan(1/2*x)+1)^4-2/5*(36*A-32*B)/(tan(1/2*x)+1)^5-2/7*(8*A-8*B)/(tan(1/2*x)+1)^7

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (67) = 134\).
time = 0.29, size = 309, normalized size = 4.12 \begin {gather*} -\frac {2 \, B {\left (\frac {91 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {168 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {280 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {175 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + 13\right )}}{105 \, {\left (\frac {7 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {21 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {35 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {21 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {7 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {\sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + 1\right )}} - \frac {2 \, A {\left (\frac {49 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {147 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {210 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {210 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {35 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 12\right )}}{35 \, {\left (\frac {7 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {21 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {35 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {21 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {7 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {\sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sin(x))/(1+sin(x))^4,x, algorithm="maxima")

[Out]

-2/105*B*(91*sin(x)/(cos(x) + 1) + 168*sin(x)^2/(cos(x) + 1)^2 + 280*sin(x)^3/(cos(x) + 1)^3 + 175*sin(x)^4/(c
os(x) + 1)^4 + 105*sin(x)^5/(cos(x) + 1)^5 + 13)/(7*sin(x)/(cos(x) + 1) + 21*sin(x)^2/(cos(x) + 1)^2 + 35*sin(
x)^3/(cos(x) + 1)^3 + 35*sin(x)^4/(cos(x) + 1)^4 + 21*sin(x)^5/(cos(x) + 1)^5 + 7*sin(x)^6/(cos(x) + 1)^6 + si
n(x)^7/(cos(x) + 1)^7 + 1) - 2/35*A*(49*sin(x)/(cos(x) + 1) + 147*sin(x)^2/(cos(x) + 1)^2 + 210*sin(x)^3/(cos(
x) + 1)^3 + 210*sin(x)^4/(cos(x) + 1)^4 + 105*sin(x)^5/(cos(x) + 1)^5 + 35*sin(x)^6/(cos(x) + 1)^6 + 12)/(7*si
n(x)/(cos(x) + 1) + 21*sin(x)^2/(cos(x) + 1)^2 + 35*sin(x)^3/(cos(x) + 1)^3 + 35*sin(x)^4/(cos(x) + 1)^4 + 21*
sin(x)^5/(cos(x) + 1)^5 + 7*sin(x)^6/(cos(x) + 1)^6 + sin(x)^7/(cos(x) + 1)^7 + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (67) = 134\).
time = 0.41, size = 150, normalized size = 2.00 \begin {gather*} \frac {2 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{4} + 8 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{3} - 9 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{2} - 15 \, {\left (4 \, A + 3 \, B\right )} \cos \left (x\right ) + {\left (2 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{3} - 6 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{2} - 15 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right ) + 15 \, A - 15 \, B\right )} \sin \left (x\right ) - 15 \, A + 15 \, B}{105 \, {\left (\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{3} + 4 \, \cos \left (x\right )^{2} - 4 \, \cos \left (x\right ) - 8\right )} \sin \left (x\right ) + 4 \, \cos \left (x\right ) + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sin(x))/(1+sin(x))^4,x, algorithm="fricas")

[Out]

1/105*(2*(3*A + 4*B)*cos(x)^4 + 8*(3*A + 4*B)*cos(x)^3 - 9*(3*A + 4*B)*cos(x)^2 - 15*(4*A + 3*B)*cos(x) + (2*(
3*A + 4*B)*cos(x)^3 - 6*(3*A + 4*B)*cos(x)^2 - 15*(3*A + 4*B)*cos(x) + 15*A - 15*B)*sin(x) - 15*A + 15*B)/(cos
(x)^4 - 3*cos(x)^3 - 8*cos(x)^2 - (cos(x)^3 + 4*cos(x)^2 - 4*cos(x) - 8)*sin(x) + 4*cos(x) + 8)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (75) = 150\).
time = 3.43, size = 889, normalized size = 11.85 \begin {gather*} - \frac {210 A \tan ^{6}{\left (\frac {x}{2} \right )}}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {630 A \tan ^{5}{\left (\frac {x}{2} \right )}}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {1260 A \tan ^{4}{\left (\frac {x}{2} \right )}}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {1260 A \tan ^{3}{\left (\frac {x}{2} \right )}}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {882 A \tan ^{2}{\left (\frac {x}{2} \right )}}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {294 A \tan {\left (\frac {x}{2} \right )}}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {72 A}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {210 B \tan ^{5}{\left (\frac {x}{2} \right )}}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {350 B \tan ^{4}{\left (\frac {x}{2} \right )}}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {560 B \tan ^{3}{\left (\frac {x}{2} \right )}}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {336 B \tan ^{2}{\left (\frac {x}{2} \right )}}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {182 B \tan {\left (\frac {x}{2} \right )}}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} - \frac {26 B}{105 \tan ^{7}{\left (\frac {x}{2} \right )} + 735 \tan ^{6}{\left (\frac {x}{2} \right )} + 2205 \tan ^{5}{\left (\frac {x}{2} \right )} + 3675 \tan ^{4}{\left (\frac {x}{2} \right )} + 3675 \tan ^{3}{\left (\frac {x}{2} \right )} + 2205 \tan ^{2}{\left (\frac {x}{2} \right )} + 735 \tan {\left (\frac {x}{2} \right )} + 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sin(x))/(1+sin(x))**4,x)

[Out]

-210*A*tan(x/2)**6/(105*tan(x/2)**7 + 735*tan(x/2)**6 + 2205*tan(x/2)**5 + 3675*tan(x/2)**4 + 3675*tan(x/2)**3
+ 2205*tan(x/2)**2 + 735*tan(x/2) + 105) - 630*A*tan(x/2)**5/(105*tan(x/2)**7 + 735*tan(x/2)**6 + 2205*tan(x/
2)**5 + 3675*tan(x/2)**4 + 3675*tan(x/2)**3 + 2205*tan(x/2)**2 + 735*tan(x/2) + 105) - 1260*A*tan(x/2)**4/(105
*tan(x/2)**7 + 735*tan(x/2)**6 + 2205*tan(x/2)**5 + 3675*tan(x/2)**4 + 3675*tan(x/2)**3 + 2205*tan(x/2)**2 + 7
35*tan(x/2) + 105) - 1260*A*tan(x/2)**3/(105*tan(x/2)**7 + 735*tan(x/2)**6 + 2205*tan(x/2)**5 + 3675*tan(x/2)*
*4 + 3675*tan(x/2)**3 + 2205*tan(x/2)**2 + 735*tan(x/2) + 105) - 882*A*tan(x/2)**2/(105*tan(x/2)**7 + 735*tan(
x/2)**6 + 2205*tan(x/2)**5 + 3675*tan(x/2)**4 + 3675*tan(x/2)**3 + 2205*tan(x/2)**2 + 735*tan(x/2) + 105) - 29
4*A*tan(x/2)/(105*tan(x/2)**7 + 735*tan(x/2)**6 + 2205*tan(x/2)**5 + 3675*tan(x/2)**4 + 3675*tan(x/2)**3 + 220
5*tan(x/2)**2 + 735*tan(x/2) + 105) - 72*A/(105*tan(x/2)**7 + 735*tan(x/2)**6 + 2205*tan(x/2)**5 + 3675*tan(x/
2)**4 + 3675*tan(x/2)**3 + 2205*tan(x/2)**2 + 735*tan(x/2) + 105) - 210*B*tan(x/2)**5/(105*tan(x/2)**7 + 735*t
an(x/2)**6 + 2205*tan(x/2)**5 + 3675*tan(x/2)**4 + 3675*tan(x/2)**3 + 2205*tan(x/2)**2 + 735*tan(x/2) + 105) -
350*B*tan(x/2)**4/(105*tan(x/2)**7 + 735*tan(x/2)**6 + 2205*tan(x/2)**5 + 3675*tan(x/2)**4 + 3675*tan(x/2)**3
+ 2205*tan(x/2)**2 + 735*tan(x/2) + 105) - 560*B*tan(x/2)**3/(105*tan(x/2)**7 + 735*tan(x/2)**6 + 2205*tan(x/
2)**5 + 3675*tan(x/2)**4 + 3675*tan(x/2)**3 + 2205*tan(x/2)**2 + 735*tan(x/2) + 105) - 336*B*tan(x/2)**2/(105*
tan(x/2)**7 + 735*tan(x/2)**6 + 2205*tan(x/2)**5 + 3675*tan(x/2)**4 + 3675*tan(x/2)**3 + 2205*tan(x/2)**2 + 73
5*tan(x/2) + 105) - 182*B*tan(x/2)/(105*tan(x/2)**7 + 735*tan(x/2)**6 + 2205*tan(x/2)**5 + 3675*tan(x/2)**4 +
3675*tan(x/2)**3 + 2205*tan(x/2)**2 + 735*tan(x/2) + 105) - 26*B/(105*tan(x/2)**7 + 735*tan(x/2)**6 + 2205*tan
(x/2)**5 + 3675*tan(x/2)**4 + 3675*tan(x/2)**3 + 2205*tan(x/2)**2 + 735*tan(x/2) + 105)

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Giac [A]
time = 0.44, size = 112, normalized size = 1.49 \begin {gather*} -\frac {2 \, {\left (105 \, A \tan \left (\frac {1}{2} \, x\right )^{6} + 315 \, A \tan \left (\frac {1}{2} \, x\right )^{5} + 105 \, B \tan \left (\frac {1}{2} \, x\right )^{5} + 630 \, A \tan \left (\frac {1}{2} \, x\right )^{4} + 175 \, B \tan \left (\frac {1}{2} \, x\right )^{4} + 630 \, A \tan \left (\frac {1}{2} \, x\right )^{3} + 280 \, B \tan \left (\frac {1}{2} \, x\right )^{3} + 441 \, A \tan \left (\frac {1}{2} \, x\right )^{2} + 168 \, B \tan \left (\frac {1}{2} \, x\right )^{2} + 147 \, A \tan \left (\frac {1}{2} \, x\right ) + 91 \, B \tan \left (\frac {1}{2} \, x\right ) + 36 \, A + 13 \, B\right )}}{105 \, {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sin(x))/(1+sin(x))^4,x, algorithm="giac")

[Out]

-2/105*(105*A*tan(1/2*x)^6 + 315*A*tan(1/2*x)^5 + 105*B*tan(1/2*x)^5 + 630*A*tan(1/2*x)^4 + 175*B*tan(1/2*x)^4
+ 630*A*tan(1/2*x)^3 + 280*B*tan(1/2*x)^3 + 441*A*tan(1/2*x)^2 + 168*B*tan(1/2*x)^2 + 147*A*tan(1/2*x) + 91*B
*tan(1/2*x) + 36*A + 13*B)/(tan(1/2*x) + 1)^7

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Mupad [B]
time = 7.07, size = 94, normalized size = 1.25 \begin {gather*} -\frac {2\,A\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+\left (6\,A+2\,B\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+\left (12\,A+\frac {10\,B}{3}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+\left (12\,A+\frac {16\,B}{3}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\left (\frac {42\,A}{5}+\frac {16\,B}{5}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\left (\frac {14\,A}{5}+\frac {26\,B}{15}\right )\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {24\,A}{35}+\frac {26\,B}{105}}{{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*sin(x))/(sin(x) + 1)^4,x)

[Out]

-((24*A)/35 + (26*B)/105 + 2*A*tan(x/2)^6 + tan(x/2)*((14*A)/5 + (26*B)/15) + tan(x/2)^5*(6*A + 2*B) + tan(x/2
)^4*(12*A + (10*B)/3) + tan(x/2)^3*(12*A + (16*B)/3) + tan(x/2)^2*((42*A)/5 + (16*B)/5))/(tan(x/2) + 1)^7

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