3.298 \(\int \frac {1}{(d \cos (a+b x))^{7/2} \sqrt {c \sin (a+b x)}} \, dx\)

Optimal. Leaf size=75 \[ \frac {8 \sqrt {c \sin (a+b x)}}{5 b c d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sqrt {c \sin (a+b x)}}{5 b c d (d \cos (a+b x))^{5/2}} \]

[Out]

2/5*(c*sin(b*x+a))^(1/2)/b/c/d/(d*cos(b*x+a))^(5/2)+8/5*(c*sin(b*x+a))^(1/2)/b/c/d^3/(d*cos(b*x+a))^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2571, 2563} \[ \frac {8 \sqrt {c \sin (a+b x)}}{5 b c d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sqrt {c \sin (a+b x)}}{5 b c d (d \cos (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c*Sin[a + b*x]])/(5*b*c*d*(d*Cos[a + b*x])^(5/2)) + (8*Sqrt[c*Sin[a + b*x]])/(5*b*c*d^3*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 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 {1}{(d \cos (a+b x))^{7/2} \sqrt {c \sin (a+b x)}} \, dx &=\frac {2 \sqrt {c \sin (a+b x)}}{5 b c d (d \cos (a+b x))^{5/2}}+\frac {4 \int \frac {1}{(d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)}} \, dx}{5 d^2}\\ &=\frac {2 \sqrt {c \sin (a+b x)}}{5 b c d (d \cos (a+b x))^{5/2}}+\frac {8 \sqrt {c \sin (a+b x)}}{5 b c d^3 \sqrt {d \cos (a+b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 52, normalized size = 0.69 \[ \frac {2 (2 \cos (2 (a+b x))+3) \tan (a+b x)}{5 b d^2 \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3 + 2*Cos[2*(a + b*x)])*Tan[a + b*x])/(5*b*d^2*(d*Cos[a + b*x])^(3/2)*Sqrt[c*Sin[a + b*x]])

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fricas [A]  time = 0.50, size = 51, normalized size = 0.68 \[ \frac {2 \, \sqrt {d \cos \left (b x + a\right )} {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {c \sin \left (b x + a\right )}}{5 \, b c d^{4} \cos \left (b x + a\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}} \sqrt {c \sin \left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((d*cos(b*x + a))^(7/2)*sqrt(c*sin(b*x + a))), x)

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maple [A]  time = 0.11, size = 50, normalized size = 0.67 \[ \frac {2 \left (4 \left (\cos ^{2}\left (b x +a \right )\right )+1\right ) \sin \left (b x +a \right ) \cos \left (b x +a \right )}{5 b \left (d \cos \left (b x +a \right )\right )^{\frac {7}{2}} \sqrt {c \sin \left (b x +a \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5/b*(4*cos(b*x+a)^2+1)*sin(b*x+a)*cos(b*x+a)/(d*cos(b*x+a))^(7/2)/(c*sin(b*x+a))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}} \sqrt {c \sin \left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((d*cos(b*x + a))^(7/2)*sqrt(c*sin(b*x + a))), x)

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mupad [B]  time = 1.49, size = 77, normalized size = 1.03 \[ \frac {8\,\sqrt {c\,\sin \left (a+b\,x\right )}\,\left (5\,\cos \left (2\,a+2\,b\,x\right )+\cos \left (4\,a+4\,b\,x\right )+4\right )}{5\,b\,c\,d^3\,\sqrt {d\,\cos \left (a+b\,x\right )}\,\left (4\,\cos \left (2\,a+2\,b\,x\right )+\cos \left (4\,a+4\,b\,x\right )+3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*(c*sin(a + b*x))^(1/2)*(5*cos(2*a + 2*b*x) + cos(4*a + 4*b*x) + 4))/(5*b*c*d^3*(d*cos(a + b*x))^(1/2)*(4*co
s(2*a + 2*b*x) + cos(4*a + 4*b*x) + 3))

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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(1/(d*cos(b*x+a))**(7/2)/(c*sin(b*x+a))**(1/2),x)

[Out]

Timed out

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