3.3.86 \(\int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx\) [286]

Optimal. Leaf size=74 \[ \frac {2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {4 (a+a \sin (c+d x))^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}} \]

[Out]

2*(a+a*sin(d*x+c))^(3/2)/d/e/(e*cos(d*x+c))^(5/2)-4/5*(a+a*sin(d*x+c))^(5/2)/a/d/e/(e*cos(d*x+c))^(5/2)

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Rubi [A]
time = 0.09, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750} \begin {gather*} \frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {4 (a \sin (c+d x)+a)^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(7/2),x]

[Out]

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

Rule 2750

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

Rule 2751

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

Rubi steps

\begin {align*} \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx &=\frac {2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {2 \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx}{a}\\ &=\frac {2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {4 (a+a \sin (c+d x))^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 72, normalized size = 0.97 \begin {gather*} -\frac {2 a \sqrt {a (1+\sin (c+d x))} (-3+2 \sin (c+d x))}{5 d e^3 \sqrt {e \cos (c+d x)} \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(7/2),x]

[Out]

(-2*a*Sqrt[a*(1 + Sin[c + d*x])]*(-3 + 2*Sin[c + d*x]))/(5*d*e^3*Sqrt[e*Cos[c + d*x]]*(Cos[(c + d*x)/2] - Sin[
(c + d*x)/2])^2)

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Maple [A]
time = 0.15, size = 44, normalized size = 0.59

method	result	size
default	\(-\frac {2 \left (2 \sin \left (d x +c \right )-3\right ) \cos \left (d x +c \right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}}}{5 d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/5/d*(2*sin(d*x+c)-3)*cos(d*x+c)*(a*(1+sin(d*x+c)))^(3/2)/(e*cos(d*x+c))^(7/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (58) = 116\).
time = 0.54, size = 189, normalized size = 2.55 \begin {gather*} \frac {2 \, {\left (3 \, a^{\frac {3}{2}} - \frac {4 \, a^{\frac {3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{\frac {3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3 \, a^{\frac {3}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2} e^{\left (-\frac {7}{2}\right )}}{5 \, d \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (\frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 65, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (2 \, a \sin \left (d x + c\right ) - 3 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{5 \, {\left (d \cos \left (d x + c\right ) e^{\frac {7}{2}} \sin \left (d x + c\right ) - d \cos \left (d x + c\right ) e^{\frac {7}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

2/5*(2*a*sin(d*x + c) - 3*a)*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c))/(d*cos(d*x + c)*e^(7/2)*sin(d*x + c)
- d*cos(d*x + c)*e^(7/2))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))**(3/2)/(e*cos(d*x+c))**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

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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((a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(7/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 5.98, size = 71, normalized size = 0.96 \begin {gather*} \frac {4\,a\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (5\,\sin \left (c+d\,x\right )+\cos \left (2\,c+2\,d\,x\right )-4\right )}{5\,d\,e^3\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (4\,\sin \left (c+d\,x\right )+\cos \left (2\,c+2\,d\,x\right )-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(7/2),x)

[Out]

(4*a*(a*(sin(c + d*x) + 1))^(1/2)*(5*sin(c + d*x) + cos(2*c + 2*d*x) - 4))/(5*d*e^3*(e*cos(c + d*x))^(1/2)*(4*
sin(c + d*x) + cos(2*c + 2*d*x) - 3))

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