3.7.1 \(\int \frac {(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2}} \, dx\) [601]

Optimal. Leaf size=184 \[ -\frac {3 (c+d)^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} \sqrt {c-d} f}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c+7 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (a+a \sin (e+f x))^{3/2}} \]

[Out]

-3/32*(c+d)^2*arctanh(1/2*cos(f*x+e)*a^(1/2)*(c-d)^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)
)/a^(5/2)/f*2^(1/2)/(c-d)^(1/2)-1/4*(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))^(5/2)-1/16*(3*c
+7*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/a/f/(a+a*sin(f*x+e))^(3/2)

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Rubi [A]
time = 0.35, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2844, 3057, 12, 2861, 214} \begin {gather*} -\frac {3 (c+d)^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f \sqrt {c-d}}-\frac {(3 c+7 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a \sin (e+f x)+a)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^(5/2),x]

[Out]

(-3*(c + d)^2*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e +
f*x]])])/(16*Sqrt[2]*a^(5/2)*Sqrt[c - d]*f) - ((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*(a + a*Sin[
e + f*x])^(5/2)) - ((3*c + 7*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(16*a*f*(a + a*Sin[e + f*x])^(3/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2844

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

Rule 2861

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

Rule 3057

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

Rubi steps

\begin {align*} \int \frac {(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {\int \frac {-\frac {1}{2} a \left (3 c^2+6 c d-d^2\right )-a d (c+3 d) \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{4 a^2}\\ &=-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c+7 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {3 a^2 (c-d) (c+d)^2}{4 \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{8 a^4 (c-d)}\\ &=-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c+7 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (a+a \sin (e+f x))^{3/2}}+\frac {\left (3 (c+d)^2\right ) \int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{32 a^2}\\ &=-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c+7 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {\left (3 (c+d)^2\right ) \text {Subst}\left (\int \frac {1}{2 a^2-(a c-a d) x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{16 a f}\\ &=-\frac {3 (c+d)^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} \sqrt {c-d} f}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c+7 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (a+a \sin (e+f x))^{3/2}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(396\) vs. \(2(184)=368\).
time = 6.94, size = 396, normalized size = 2.15 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \left (-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x)) (7 c+3 d+(3 c+7 d) \sin (e+f x))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {3 (c+d)^2 \left (\log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )}{\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{2+2 \tan \left (\frac {1}{2} (e+f x)\right )}-\frac {-\frac {1}{2} (c-d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} (d+d \cos (e+f x)+c \sin (e+f x))}{\sqrt {c+d \sin (e+f x)}}}{c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}}\right )}{32 f (a (1+\sin (e+f x)))^{5/2} \sqrt {c+d \sin (e+f x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^(5/2),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*((-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])*(7*c
+ 3*d + (3*c + 7*d)*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (3*(c + d)^2*(Log[1 + Tan[(e + f*
x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e +
f*x)/2]]))/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((
1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt
[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]))))/(32*f*(a*(1 + S
in[e + f*x]))^(5/2)*Sqrt[c + d*Sin[e + f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3049\) vs. \(2(155)=310\).
time = 11.34, size = 3050, normalized size = 16.58

method	result	size
default	\(\text {Expression too large to display}\)	\(3050\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/64/f*(6*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*
c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*sin(f*x+e)*cos(f*x+e)
*c^2-12*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)*c^2+28*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*cos(
f*x+e)*d^2+12*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^3*c^2-28*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1
/2)*cos(f*x+e)^3*d^2-16*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)*c*d+16*((c+d*sin(f*x+e))/(cos(f*x+e
)+1))^(1/2)*cos(f*x+e)^3*c*d-12*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*
sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)
*c^2-12*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*c-
d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*d^2-28*((c+d*sin(f*x+e)
)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*cos(f*x+e)*c^2+12*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*cos(f*
x+e)*d^2-3*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)
*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*cos(f*x+e)^3*c^2-3*2
^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x
+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*cos(f*x+e)^3*d^2+9*2^(1/2)*ln(2*
((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*
x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*cos(f*x+e)^2*c^2+9*2^(1/2)*ln(2*((2*c-2*d)^(
1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f
*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*cos(f*x+e)^2*d^2-12*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2
)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/
(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*sin(f*x+e)*c^2-12*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f
*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)
+sin(f*x+e)))*(2*c-2*d)^(1/2)*sin(f*x+e)*d^2+6*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*
x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*
(2*c-2*d)^(1/2)*cos(f*x+e)*c^2+6*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)
*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2
)*cos(f*x+e)*d^2-24*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c
os(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*c*d+16*((c+
d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*cos(f*x+e)*c*d+6*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*si
n(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x
+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*sin(f*x+e)*cos(f*x+e)*d^2+18*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(
f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e
)+sin(f*x+e)))*(2*c-2*d)^(1/2)*cos(f*x+e)^2*c*d-24*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(co
s(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e
)))*(2*c-2*d)^(1/2)*sin(f*x+e)*c*d+12*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^
(1/2)*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)
^(1/2)*cos(f*x+e)*c*d+3*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+
e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*sin(f*x
+e)*cos(f*x+e)^2*c^2+3*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e
)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*sin(f*x+
e)*cos(f*x+e)^2*d^2-6*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)
+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*cos(f*x+e
)^3*c*d+6*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*
c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*sin(f*x+e)*cos(f*x+e)
^2*c*d+12*2^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*
c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*sin(f*x+e)*cos(f*x+e)
*c*d)*(c+d*sin(f*x+e))^(1/2)/((c+d*sin(f*x+e))/...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*sin(f*x + e) + c)^(3/2)/(a*sin(f*x + e) + a)^(5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (164) = 328\).
time = 0.58, size = 1364, normalized size = 7.41 \begin {gather*} \left [\frac {3 \, {\left ({\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right ) + {\left ({\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {2 \, a c - 2 \, a d} \log \left (\frac {{\left (a c^{2} - 14 \, a c d + 17 \, a d^{2}\right )} \cos \left (f x + e\right )^{3} - 4 \, a c^{2} - 8 \, a c d - 4 \, a d^{2} - {\left (13 \, a c^{2} - 22 \, a c d - 3 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (c - 3 \, d\right )} \cos \left (f x + e\right )^{2} - {\left (3 \, c - d\right )} \cos \left (f x + e\right ) + {\left ({\left (c - 3 \, d\right )} \cos \left (f x + e\right ) + 4 \, c - 4 \, d\right )} \sin \left (f x + e\right ) - 4 \, c + 4 \, d\right )} \sqrt {2 \, a c - 2 \, a d} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} - 2 \, {\left (9 \, a c^{2} - 14 \, a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right ) - {\left (4 \, a c^{2} + 8 \, a c d + 4 \, a d^{2} - {\left (a c^{2} - 14 \, a c d + 17 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (7 \, a c^{2} - 18 \, a c d + 7 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 4\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 4}\right ) + 8 \, {\left ({\left (3 \, c^{2} + 4 \, c d - 7 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left (7 \, c^{2} - 4 \, c d - 3 \, d^{2}\right )} \cos \left (f x + e\right ) - {\left (4 \, c^{2} - 8 \, c d + 4 \, d^{2} - {\left (3 \, c^{2} + 4 \, c d - 7 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{128 \, {\left ({\left (a^{3} c - a^{3} d\right )} f \cos \left (f x + e\right )^{3} + 3 \, {\left (a^{3} c - a^{3} d\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{3} c - a^{3} d\right )} f \cos \left (f x + e\right ) - 4 \, {\left (a^{3} c - a^{3} d\right )} f + {\left ({\left (a^{3} c - a^{3} d\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{3} c - a^{3} d\right )} f \cos \left (f x + e\right ) - 4 \, {\left (a^{3} c - a^{3} d\right )} f\right )} \sin \left (f x + e\right )\right )}}, -\frac {3 \, {\left ({\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right ) + {\left ({\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left (c^{2} + 2 \, c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {-2 \, a c + 2 \, a d} \arctan \left (\frac {\sqrt {-2 \, a c + 2 \, a d} \sqrt {a \sin \left (f x + e\right ) + a} {\left ({\left (c - 3 \, d\right )} \sin \left (f x + e\right ) - 3 \, c + d\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{4 \, {\left ({\left (a c d - a d^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (a c^{2} - a c d\right )} \cos \left (f x + e\right )\right )}}\right ) - 4 \, {\left ({\left (3 \, c^{2} + 4 \, c d - 7 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left (7 \, c^{2} - 4 \, c d - 3 \, d^{2}\right )} \cos \left (f x + e\right ) - {\left (4 \, c^{2} - 8 \, c d + 4 \, d^{2} - {\left (3 \, c^{2} + 4 \, c d - 7 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{64 \, {\left ({\left (a^{3} c - a^{3} d\right )} f \cos \left (f x + e\right )^{3} + 3 \, {\left (a^{3} c - a^{3} d\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{3} c - a^{3} d\right )} f \cos \left (f x + e\right ) - 4 \, {\left (a^{3} c - a^{3} d\right )} f + {\left ({\left (a^{3} c - a^{3} d\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{3} c - a^{3} d\right )} f \cos \left (f x + e\right ) - 4 \, {\left (a^{3} c - a^{3} d\right )} f\right )} \sin \left (f x + e\right )\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

[1/128*(3*((c^2 + 2*c*d + d^2)*cos(f*x + e)^3 + 3*(c^2 + 2*c*d + d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 -
2*(c^2 + 2*c*d + d^2)*cos(f*x + e) + ((c^2 + 2*c*d + d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 - 2*(c^2 + 2
*c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3
- 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 - 4*((c - 3*d)*cos(f*x + e)^2 -
(3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*s
qrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(f*x + e) - (4*a*c^2 +
8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^2 - 2*(7*a*c^2 - 18*a*c*d + 7*a*d^2)*cos(f*x +
e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*
cos(f*x + e) - 4)) + 8*((3*c^2 + 4*c*d - 7*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 - 4*c*d - 3*d^
2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - (3*c^2 + 4*c*d - 7*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x
+ e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c - a^3*d)*f*cos(f*x + e)^3 + 3*(a^3*c - a^3*d)*f*cos(f*x + e)^2 - 2
*(a^3*c - a^3*d)*f*cos(f*x + e) - 4*(a^3*c - a^3*d)*f + ((a^3*c - a^3*d)*f*cos(f*x + e)^2 - 2*(a^3*c - a^3*d)*
f*cos(f*x + e) - 4*(a^3*c - a^3*d)*f)*sin(f*x + e)), -1/64*(3*((c^2 + 2*c*d + d^2)*cos(f*x + e)^3 + 3*(c^2 + 2
*c*d + d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 - 2*(c^2 + 2*c*d + d^2)*cos(f*x + e) + ((c^2 + 2*c*d + d^2)
*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 - 2*(c^2 + 2*c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-2*a*c + 2*a*
d)*arctan(1/4*sqrt(-2*a*c + 2*a*d)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x
+ e) + c)/((a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) + (a*c^2 - a*c*d)*cos(f*x + e))) - 4*((3*c^2 + 4*c*d - 7*
d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 - 4*c*d - 3*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 -
(3*c^2 + 4*c*d - 7*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c
- a^3*d)*f*cos(f*x + e)^3 + 3*(a^3*c - a^3*d)*f*cos(f*x + e)^2 - 2*(a^3*c - a^3*d)*f*cos(f*x + e) - 4*(a^3*c
- a^3*d)*f + ((a^3*c - a^3*d)*f*cos(f*x + e)^2 - 2*(a^3*c - a^3*d)*f*cos(f*x + e) - 4*(a^3*c - a^3*d)*f)*sin(f
*x + e))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))**(3/2)/(a+a*sin(f*x+e))**(5/2),x)

[Out]

Integral((c + d*sin(e + f*x))**(3/2)/(a*(sin(e + f*x) + 1))**(5/2), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*sin(e + f*x))^(3/2)/(a + a*sin(e + f*x))^(5/2),x)

[Out]

int((c + d*sin(e + f*x))^(3/2)/(a + a*sin(e + f*x))^(5/2), x)

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