∫ 1____________ (e cos(c+dx))3∕2(a+a sin(c+dx))5∕2 dx [319]

Optimal. Leaf size=154 \[ -\frac {2}{11 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac {12}{77 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a^2 d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^3 d e \sqrt {e \cos (c+d x)}} \]

-2/11/d/e/(a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(1/2)-12/77/a/d/e/(a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(1/2)-

16/77/a^2/d/e/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(1/2)+32/77*(a+a*sin(d*x+c))^(1/2)/a^3/d/e/(e*cos(d*x+c))^

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Rubi [A]
time = 0.20, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {32 \sqrt {a \sin (c+d x)+a}}{77 a^3 d e \sqrt {e \cos (c+d x)}}-\frac {16}{77 a^2 d e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}-\frac {12}{77 a d e (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}}-\frac {2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}} \end {gather*}

-2/(11*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2)) - 12/(77*a*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c +

d*x])^(3/2)) - 16/(77*a^2*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) + (32*Sqrt[a + a*Sin[c + d*x]])/

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^

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]

\begin {align*} \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx &=-\frac {2}{11 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}+\frac {6 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx}{11 a}\\ &=-\frac {2}{11 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac {12}{77 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}+\frac {24 \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}} \, dx}{77 a^2}\\ &=-\frac {2}{11 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac {12}{77 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a^2 d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {16 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx}{77 a^3}\\ &=-\frac {2}{11 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac {12}{77 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a^2 d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^3 d e \sqrt {e \cos (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 66, normalized size = 0.43 \begin {gather*} \frac {-10+52 \sin (c+d x)+80 \sin ^2(c+d x)+32 \sin ^3(c+d x)}{77 d e \sqrt {e \cos (c+d x)} (a (1+\sin (c+d x)))^{5/2}} \end {gather*}

(-10 + 52*Sin[c + d*x] + 80*Sin[c + d*x]^2 + 32*Sin[c + d*x]^3)/(77*d*e*Sqrt[e*Cos[c + d*x]]*(a*(1 + Sin[c + d

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method	result	size

default	\(-\frac {2 \left (16 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+40 \left (\cos ^{2}\left (d x +c \right )\right )-42 \sin \left (d x +c \right )-35\right ) \cos \left (d x +c \right )}{77 d \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}}}\)	\(70\)

-2/77/d*(16*cos(d*x+c)^2*sin(d*x+c)+40*cos(d*x+c)^2-42*sin(d*x+c)-35)*cos(d*x+c)/(e*cos(d*x+c))^(3/2)/(a*(1+si

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (118) = 236\).
time = 0.56, size = 335, normalized size = 2.18 \begin {gather*} -\frac {2 \, {\left (5 \, \sqrt {a} - \frac {52 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {150 \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {180 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {180 \, \sqrt {a} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {150 \, \sqrt {a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {52 \, \sqrt {a} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {5 \, \sqrt {a} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4} e^{\left (-\frac {3}{2}\right )}}{77 \, {\left (a^{3} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, 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 )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \end {gather*}

-2/77*(5*sqrt(a) - 52*sqrt(a)*sin(d*x + c)/(cos(d*x + c) + 1) - 150*sqrt(a)*sin(d*x + c)^2/(cos(d*x + c) + 1)^

2 - 180*sqrt(a)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 180*sqrt(a)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 150*sq

rt(a)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 52*sqrt(a)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 5*sqrt(a)*sin(d*x

+ c)^8/(cos(d*x + c) + 1)^8)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^4*e^(-3/2)/((a^3 + 4*a^3*sin(d*x + c)^

2/(cos(d*x + c) + 1)^2 + 6*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6

+ a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(-sin(d*x + c)/(cos

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Fricas [A]
time = 0.36, size = 124, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (40 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, \cos \left (d x + c\right )^{2} - 21\right )} \sin \left (d x + c\right ) - 35\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{77 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} e^{\frac {3}{2}} - 4 \, a^{3} d \cos \left (d x + c\right ) e^{\frac {3}{2}} + {\left (a^{3} d \cos \left (d x + c\right )^{3} e^{\frac {3}{2}} - 4 \, a^{3} d \cos \left (d x + c\right ) e^{\frac {3}{2}}\right )} \sin \left (d x + c\right )\right )}} \end {gather*}

2/77*(40*cos(d*x + c)^2 + 2*(8*cos(d*x + c)^2 - 21)*sin(d*x + c) - 35)*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x +

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

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

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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*}

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Mupad [B]
time = 11.17, size = 261, normalized size = 1.69 \begin {gather*} \frac {76\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+30\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-40\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-8\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\frac {385\,a^3\,d\,e\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}+\frac {1155\,a^3\,d\,e\,\sin \left (c+d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}-\frac {231\,a^3\,d\,e\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}-\frac {77\,a^3\,d\,e\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}} \end {gather*}

(76*sin(c + d*x)*(a + a*sin(c + d*x))^(1/2) + 30*(a + a*sin(c + d*x))^(1/2) - 40*cos(2*c + 2*d*x)*(a + a*sin(c

+ d*x))^(1/2) - 8*sin(3*c + 3*d*x)*(a + a*sin(c + d*x))^(1/2))/((385*a^3*d*e*((e*exp(- c*1i - d*x*1i))/2 + (e

*exp(c*1i + d*x*1i))/2)^(1/2))/2 + (1155*a^3*d*e*sin(c + d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*

1i))/2)^(1/2))/4 - (231*a^3*d*e*cos(2*c + 2*d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/2)

)/2 - (77*a^3*d*e*sin(3*c + 3*d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/2))/4)

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3.4.19 \(\int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx\) [319]