3.275 \(\int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{15/2}} \, dx\)

Optimal. Leaf size=141 \[ -\frac {64 c \sqrt {c \sin (a+b x)}}{585 b d^7 \sqrt {d \cos (a+b x)}}-\frac {16 c \sqrt {c \sin (a+b x)}}{585 b d^5 (d \cos (a+b x))^{5/2}}-\frac {2 c \sqrt {c \sin (a+b x)}}{117 b d^3 (d \cos (a+b x))^{9/2}}+\frac {2 c \sqrt {c \sin (a+b x)}}{13 b d (d \cos (a+b x))^{13/2}} \]

[Out]

2/13*c*(c*sin(b*x+a))^(1/2)/b/d/(d*cos(b*x+a))^(13/2)-2/117*c*(c*sin(b*x+a))^(1/2)/b/d^3/(d*cos(b*x+a))^(9/2)-
16/585*c*(c*sin(b*x+a))^(1/2)/b/d^5/(d*cos(b*x+a))^(5/2)-64/585*c*(c*sin(b*x+a))^(1/2)/b/d^7/(d*cos(b*x+a))^(1
/2)

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Rubi [A]  time = 0.24, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2566, 2571, 2563} \[ -\frac {64 c \sqrt {c \sin (a+b x)}}{585 b d^7 \sqrt {d \cos (a+b x)}}-\frac {16 c \sqrt {c \sin (a+b x)}}{585 b d^5 (d \cos (a+b x))^{5/2}}-\frac {2 c \sqrt {c \sin (a+b x)}}{117 b d^3 (d \cos (a+b x))^{9/2}}+\frac {2 c \sqrt {c \sin (a+b x)}}{13 b d (d \cos (a+b x))^{13/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(15/2),x]

[Out]

(2*c*Sqrt[c*Sin[a + b*x]])/(13*b*d*(d*Cos[a + b*x])^(13/2)) - (2*c*Sqrt[c*Sin[a + b*x]])/(117*b*d^3*(d*Cos[a +
b*x])^(9/2)) - (16*c*Sqrt[c*Sin[a + b*x]])/(585*b*d^5*(d*Cos[a + b*x])^(5/2)) - (64*c*Sqrt[c*Sin[a + b*x]])/(
585*b*d^7*Sqrt[d*Cos[a + b*x]])

Rule 2563

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

Rule 2566

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(a*Sin[e
+ f*x])^(m - 1)*(b*Cos[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + Dist[(a^2*(m - 1))/(b^2*(n + 1)), Int[(a*Sin[e +
f*x])^(m - 2)*(b*Cos[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (Integ
ersQ[2*m, 2*n] || EqQ[m + n, 0])

Rule 2571

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[((b*Sin[e +
f*x])^(n + 1)*(a*Cos[e + f*x])^(m + 1))/(a*b*f*(m + 1)), x] + Dist[(m + n + 2)/(a^2*(m + 1)), Int[(b*Sin[e + f
*x])^n*(a*Cos[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]

Rubi steps

\begin {align*} \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{15/2}} \, dx &=\frac {2 c \sqrt {c \sin (a+b x)}}{13 b d (d \cos (a+b x))^{13/2}}-\frac {c^2 \int \frac {1}{(d \cos (a+b x))^{11/2} \sqrt {c \sin (a+b x)}} \, dx}{13 d^2}\\ &=\frac {2 c \sqrt {c \sin (a+b x)}}{13 b d (d \cos (a+b x))^{13/2}}-\frac {2 c \sqrt {c \sin (a+b x)}}{117 b d^3 (d \cos (a+b x))^{9/2}}-\frac {\left (8 c^2\right ) \int \frac {1}{(d \cos (a+b x))^{7/2} \sqrt {c \sin (a+b x)}} \, dx}{117 d^4}\\ &=\frac {2 c \sqrt {c \sin (a+b x)}}{13 b d (d \cos (a+b x))^{13/2}}-\frac {2 c \sqrt {c \sin (a+b x)}}{117 b d^3 (d \cos (a+b x))^{9/2}}-\frac {16 c \sqrt {c \sin (a+b x)}}{585 b d^5 (d \cos (a+b x))^{5/2}}-\frac {\left (32 c^2\right ) \int \frac {1}{(d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)}} \, dx}{585 d^6}\\ &=\frac {2 c \sqrt {c \sin (a+b x)}}{13 b d (d \cos (a+b x))^{13/2}}-\frac {2 c \sqrt {c \sin (a+b x)}}{117 b d^3 (d \cos (a+b x))^{9/2}}-\frac {16 c \sqrt {c \sin (a+b x)}}{585 b d^5 (d \cos (a+b x))^{5/2}}-\frac {64 c \sqrt {c \sin (a+b x)}}{585 b d^7 \sqrt {d \cos (a+b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 67, normalized size = 0.48 \[ \frac {2 (36 \cos (2 (a+b x))+4 \cos (4 (a+b x))+77) \sec ^7(a+b x) (c \sin (a+b x))^{5/2} \sqrt {d \cos (a+b x)}}{585 b c d^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(15/2),x]

[Out]

(2*Sqrt[d*Cos[a + b*x]]*(77 + 36*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)])*Sec[a + b*x]^7*(c*Sin[a + b*x])^(5/2))
/(585*b*c*d^8)

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fricas [A]  time = 0.67, size = 73, normalized size = 0.52 \[ -\frac {2 \, {\left (32 \, c \cos \left (b x + a\right )^{6} + 8 \, c \cos \left (b x + a\right )^{4} + 5 \, c \cos \left (b x + a\right )^{2} - 45 \, c\right )} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}{585 \, b d^{8} \cos \left (b x + a\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(15/2),x, algorithm="fricas")

[Out]

-2/585*(32*c*cos(b*x + a)^6 + 8*c*cos(b*x + a)^4 + 5*c*cos(b*x + a)^2 - 45*c)*sqrt(d*cos(b*x + a))*sqrt(c*sin(
b*x + a))/(b*d^8*cos(b*x + a)^7)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(15/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.16, size = 60, normalized size = 0.43 \[ \frac {2 \left (32 \left (\cos ^{4}\left (b x +a \right )\right )+40 \left (\cos ^{2}\left (b x +a \right )\right )+45\right ) \left (c \sin \left (b x +a \right )\right )^{\frac {3}{2}} \cos \left (b x +a \right ) \sin \left (b x +a \right )}{585 b \left (d \cos \left (b x +a \right )\right )^{\frac {15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(15/2),x)

[Out]

2/585/b*(32*cos(b*x+a)^4+40*cos(b*x+a)^2+45)*(c*sin(b*x+a))^(3/2)*cos(b*x+a)*sin(b*x+a)/(d*cos(b*x+a))^(15/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(15/2),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(3/2)/(d*cos(b*x + a))^(15/2), x)

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mupad [B]  time = 6.67, size = 193, normalized size = 1.37 \[ -\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\sqrt {c\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\,\left (-\frac {3776\,c\,{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}}{585\,b\,d^7}+\frac {2752\,c\,{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}\,\cos \left (2\,a+2\,b\,x\right )}{585\,b\,d^7}+\frac {896\,c\,{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}\,\cos \left (4\,a+4\,b\,x\right )}{585\,b\,d^7}+\frac {128\,c\,{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}\,\cos \left (6\,a+6\,b\,x\right )}{585\,b\,d^7}\right )}{64\,{\cos \left (a+b\,x\right )}^6\,\sqrt {d\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x))^(3/2)/(d*cos(a + b*x))^(15/2),x)

[Out]

-(exp(- a*6i - b*x*6i)*(c*((exp(- a*1i - b*x*1i)*1i)/2 - (exp(a*1i + b*x*1i)*1i)/2))^(1/2)*((2752*c*exp(a*6i +
b*x*6i)*cos(2*a + 2*b*x))/(585*b*d^7) - (3776*c*exp(a*6i + b*x*6i))/(585*b*d^7) + (896*c*exp(a*6i + b*x*6i)*c
os(4*a + 4*b*x))/(585*b*d^7) + (128*c*exp(a*6i + b*x*6i)*cos(6*a + 6*b*x))/(585*b*d^7)))/(64*cos(a + b*x)^6*(d
*(exp(- a*1i - b*x*1i)/2 + exp(a*1i + b*x*1i)/2))^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))**(3/2)/(d*cos(b*x+a))**(15/2),x)

[Out]

Timed out

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