3.4.12 \(\int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx\) [312]

Optimal. Leaf size=193 \[ -\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}-\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {128 (a+a \sin (c+d x))^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac {256 (a+a \sin (c+d x))^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}} \]

[Out]

-2/11/d/e/(e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c))^(3/2)+128/77*(a+a*sin(d*x+c))^(3/2)/a^3/d/e/(e*cos(d*x+c))^(5/
2)-256/385*(a+a*sin(d*x+c))^(5/2)/a^4/d/e/(e*cos(d*x+c))^(5/2)-16/77/a/d/e/(e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c
))^(1/2)-32/77*(a+a*sin(d*x+c))^(1/2)/a^2/d/e/(e*cos(d*x+c))^(5/2)

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Rubi [A]
time = 0.25, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {256 (a \sin (c+d x)+a)^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}}+\frac {128 (a \sin (c+d x)+a)^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac {32 \sqrt {a \sin (c+d x)+a}}{77 a^2 d e (e \cos (c+d x))^{5/2}}-\frac {16}{77 a d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^(3/2)),x]

[Out]

-2/(11*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2)) - 16/(77*a*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a
*Sin[c + d*x]]) - (32*Sqrt[a + a*Sin[c + d*x]])/(77*a^2*d*e*(e*Cos[c + d*x])^(5/2)) + (128*(a + a*Sin[c + d*x]
)^(3/2))/(77*a^3*d*e*(e*Cos[c + d*x])^(5/2)) - (256*(a + a*Sin[c + d*x])^(5/2))/(385*a^4*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 {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx &=-\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}+\frac {8 \int \frac {1}{(e \cos (c+d x))^{7/2} \sqrt {a+a \sin (c+d x)}} \, dx}{11 a}\\ &=-\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}+\frac {48 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx}{77 a^2}\\ &=-\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}-\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {64 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx}{77 a^3}\\ &=-\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}-\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {128 (a+a \sin (c+d x))^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac {128 \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx}{77 a^4}\\ &=-\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}-\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {128 (a+a \sin (c+d x))^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac {256 (a+a \sin (c+d x))^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 76, normalized size = 0.39 \begin {gather*} \frac {2 (45+8 \cos (2 (c+d x))-16 \cos (4 (c+d x))+104 \sin (c+d x)+48 \sin (3 (c+d x)))}{385 d e (e \cos (c+d x))^{5/2} (a (1+\sin (c+d x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^(3/2)),x]

[Out]

(2*(45 + 8*Cos[2*(c + d*x)] - 16*Cos[4*(c + d*x)] + 104*Sin[c + d*x] + 48*Sin[3*(c + d*x)]))/(385*d*e*(e*Cos[c
+ d*x])^(5/2)*(a*(1 + Sin[c + d*x]))^(3/2))

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Maple [A]
time = 0.19, size = 80, normalized size = 0.41

method	result	size
default	\(-\frac {2 \left (128 \left (\cos ^{4}\left (d x +c \right )\right )-192 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-144 \left (\cos ^{2}\left (d x +c \right )\right )-56 \sin \left (d x +c \right )-21\right ) \cos \left (d x +c \right )}{385 d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/385/d*(128*cos(d*x+c)^4-192*cos(d*x+c)^2*sin(d*x+c)-144*cos(d*x+c)^2-56*sin(d*x+c)-21)*cos(d*x+c)/(e*cos(d*
x+c))^(7/2)/(a*(1+sin(d*x+c)))^(3/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (148) = 296\).
time = 0.55, size = 404, normalized size = 2.09 \begin {gather*} \frac {2 \, {\left (37 \, \sqrt {a} + \frac {496 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {559 \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {544 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1526 \, \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1526 \, \sqrt {a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {544 \, \sqrt {a} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {559 \, \sqrt {a} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {496 \, \sqrt {a} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {37 \, \sqrt {a} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{5} e^{\left (-\frac {7}{2}\right )}}{385 \, {\left (a^{2} + \frac {5 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\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 {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

2/385*(37*sqrt(a) + 496*sqrt(a)*sin(d*x + c)/(cos(d*x + c) + 1) + 559*sqrt(a)*sin(d*x + c)^2/(cos(d*x + c) + 1
)^2 - 544*sqrt(a)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1526*sqrt(a)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 152
6*sqrt(a)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 544*sqrt(a)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 559*sqrt(a)*
sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 496*sqrt(a)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 37*sqrt(a)*sin(d*x + c
)^10/(cos(d*x + c) + 1)^10)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^5*e^(-7/2)/((a^2 + 5*a^2*sin(d*x + c)^2/
(cos(d*x + c) + 1)^2 + 10*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6
+ 5*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)*d*(sin(d*x + c)/(cos
(d*x + c) + 1) + 1)^(13/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2))

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Fricas [A]
time = 0.35, size = 120, normalized size = 0.62 \begin {gather*} \frac {2 \, {\left (128 \, \cos \left (d x + c\right )^{4} - 144 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (24 \, \cos \left (d x + c\right )^{2} + 7\right )} \sin \left (d x + c\right ) - 21\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{385 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} e^{\frac {7}{2}} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} e^{\frac {7}{2}} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3} e^{\frac {7}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

2/385*(128*cos(d*x + c)^4 - 144*cos(d*x + c)^2 - 8*(24*cos(d*x + c)^2 + 7)*sin(d*x + c) - 21)*sqrt(a*sin(d*x +
c) + a)*sqrt(cos(d*x + c))/(a^2*d*cos(d*x + c)^5*e^(7/2) - 2*a^2*d*cos(d*x + c)^3*e^(7/2)*sin(d*x + c) - 2*a^
2*d*cos(d*x + c)^3*e^(7/2))

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Sympy [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(1/(e*cos(d*x+c))**(7/2)/(a+a*sin(d*x+c))**(3/2),x)

[Out]

Timed out

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

[Out]

Timed out

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Mupad [B]
time = 11.65, size = 413, normalized size = 2.14 \begin {gather*} \frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}\,\left (\frac {288\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}}{77\,a^2\,d\,e^3}+\frac {256\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (2\,c+2\,d\,x\right )}{385\,a^2\,d\,e^3}-\frac {512\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (4\,c+4\,d\,x\right )}{385\,a^2\,d\,e^3}+\frac {1536\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (3\,c+3\,d\,x\right )}{385\,a^2\,d\,e^3}+\frac {3328\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (c+d\,x\right )}{385\,a^2\,d\,e^3}\right )}{10\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}+8\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (c+d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}+8\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}-2\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (4\,c+4\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}+8\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a + a*sin(c + d*x))^(1/2)*((288*exp(c*4i + d*x*4i))/(77*a^2*d*e^3) + (256*exp(c*4i + d*x*4i)*cos(2*c + 2*d*x
))/(385*a^2*d*e^3) - (512*exp(c*4i + d*x*4i)*cos(4*c + 4*d*x))/(385*a^2*d*e^3) + (1536*exp(c*4i + d*x*4i)*sin(
3*c + 3*d*x))/(385*a^2*d*e^3) + (3328*exp(c*4i + d*x*4i)*sin(c + d*x))/(385*a^2*d*e^3)))/(10*exp(c*4i + d*x*4i
)*(e*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2) + 8*exp(c*4i + d*x*4i)*sin(c + d*x)*(e*(exp(- c*1i
- d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2) + 8*exp(c*4i + d*x*4i)*cos(2*c + 2*d*x)*(e*(exp(- c*1i - d*x*1i)/2
+ exp(c*1i + d*x*1i)/2))^(1/2) - 2*exp(c*4i + d*x*4i)*cos(4*c + 4*d*x)*(e*(exp(- c*1i - d*x*1i)/2 + exp(c*1i
+ d*x*1i)/2))^(1/2) + 8*exp(c*4i + d*x*4i)*sin(3*c + 3*d*x)*(e*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)
)^(1/2))

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