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3.1479 \(\int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx\)

Optimal. Leaf size=312 \[ \frac {\sec (e+f x) (a \sin (e+f x)+b) \sqrt {a+b \sin (e+f x)}}{f \sqrt {d \sin (e+f x)}}-\frac {(a+b)^{3/2} \tan (e+f x) \sqrt {-\frac {a (\csc (e+f x)-1)}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{\sqrt {d} f}-\frac {b (a+b) (\sin (e+f x)+1) \tan (e+f x) \sqrt {-\frac {a (\csc (e+f x)-1)}{a+b}} \sqrt {\frac {a \csc (e+f x)+b}{b-a}} E\left (\sin ^{-1}\left (\sqrt {-\frac {b+a \csc (e+f x)}{a-b}}\right )|\frac {b-a}{a+b}\right )}{f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}}} \]

[Out]

sec(f*x+e)*(b+a*sin(f*x+e))*(a+b*sin(f*x+e))^(1/2)/f/(d*sin(f*x+e))^(1/2)-(a+b)^(3/2)*EllipticF(d^(1/2)*(a+b*s
in(f*x+e))^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2),((-a-b)/(a-b))^(1/2))*(-a*(-1+csc(f*x+e))/(a+b))^(1/2)*(a*(1
+csc(f*x+e))/(a-b))^(1/2)*tan(f*x+e)/f/d^(1/2)-b*(a+b)*EllipticE(((-b-a*csc(f*x+e))/(a-b))^(1/2),((-a+b)/(a+b)
)^(1/2))*(1+sin(f*x+e))*(-a*(-1+csc(f*x+e))/(a+b))^(1/2)*((b+a*csc(f*x+e))/(-a+b))^(1/2)*tan(f*x+e)/f/(a*(1+cs
c(f*x+e))/(a-b))^(1/2)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)

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Rubi [F]  time = 0.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]],x]

[Out]

Defer[Int][(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]], x]

Rubi steps

\begin {align*} \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx &=\int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx\\ \end {align*}

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Mathematica [A]  time = 22.91, size = 602, normalized size = 1.93 \[ \frac {\sqrt {\sin (e+f x)} \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right ) \sqrt {\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{2 \tan ^2\left (\frac {1}{2} (e+f x)\right )+2}} \sqrt {\frac {a \tan ^2\left (\frac {1}{2} (e+f x)\right )+a+2 b \tan \left (\frac {1}{2} (e+f x)\right )}{\tan ^2\left (\frac {1}{2} (e+f x)\right )+1}} \left (\frac {2 \sqrt {b^2-a^2} \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right ) \sqrt {\frac {a \left (a \tan ^2\left (\frac {1}{2} (e+f x)\right )+a+2 b \tan \left (\frac {1}{2} (e+f x)\right )\right )}{a^2-b^2}} \left (a \sqrt {\frac {a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b^2-a^2}-b}} \sqrt {-\frac {a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b^2-a^2}+b}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )+\sqrt {b^2-a^2}}{\sqrt {b^2-a^2}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-a^2}}{b+\sqrt {b^2-a^2}}\right )-b \tan \left (\frac {1}{2} (e+f x)\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {-b-a \tan \left (\frac {1}{2} (e+f x)\right )+\sqrt {b^2-a^2}}{\sqrt {b^2-a^2}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-a^2}}{\sqrt {b^2-a^2}-b}\right )\right )}{\sqrt {\frac {a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b^2-a^2}-b}} \left (a \tan ^2\left (\frac {1}{2} (e+f x)\right )+a+2 b \tan \left (\frac {1}{2} (e+f x)\right )\right )}-2 b \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f \left (\tan ^3\left (\frac {1}{2} (e+f x)\right )+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {d \sin (e+f x)}}+\frac {\tan (e+f x) (a+b \sin (e+f x))^{3/2}}{f \sqrt {d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]],x]

[Out]

(Sqrt[Sin[e + f*x]]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[Tan[(e + f*x)/2]/(2 + 2*Tan[(e + f*x)/2]^2)]*Sqrt[(a + 2*b*T
an[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)]*(-2*b*Tan[(e + f*x)/2]^2 + (2*Sqrt[-a^2 + b^
2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))/(a^2 - b^2)]*(-(b*Ellip
ticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])
/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/
2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + S
qrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt
[-a^2 + b^2])]*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))))/(f*Sqrt[d*Sin[e + f*x]]*(Tan[(e + f*x)/2]
+ Tan[(e + f*x)/2]^3)) + ((a + b*Sin[e + f*x])^(3/2)*Tan[e + f*x])/(f*Sqrt[d*Sin[e + f*x]])

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \sec \left (f x + e\right )^{2} \sin \left (f x + e\right ) + a \sec \left (f x + e\right )^{2}\right )} \sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right )}}{d \sin \left (f x + e\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral((b*sec(f*x + e)^2*sin(f*x + e) + a*sec(f*x + e)^2)*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))/(d*s
in(f*x + e)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{2}}{\sqrt {d \sin \left (f x + e\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^(3/2)*sec(f*x + e)^2/sqrt(d*sin(f*x + e)), x)

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maple [B]  time = 0.61, size = 2377, normalized size = 7.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x)

[Out]

-1/2/f*(2*(-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*
(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e
))/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-
a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*cos(f*x+e
)^2*(-a^2+b^2)^(1/2)*b^2-2*(-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/s
in(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)
*(a*(-1+cos(f*x+e))/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x
+e)+cos(f*x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))
^(1/2))*cos(f*x+e)^2*a^2*b+2*(-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(b+(-a^2+b^2)^(1/2))
/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-a^2+b^2)^(1/2)/sin(f*x+e))^(1/
2)*(a*(-1+cos(f*x+e))/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f
*x+e)+cos(f*x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2
))^(1/2))*cos(f*x+e)^2*b^3-(-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/s
in(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)
*(a*(-1+cos(f*x+e))/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticF((-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x
+e)+cos(f*x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))
^(1/2))*cos(f*x+e)^2*(-a^2+b^2)^(1/2)*a^2+2*(-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(b+(-
a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-a^2+b^2)^(1/2)/
sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((-(-(-a^2+b^2)^(1/2)*sin
(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/
(-a^2+b^2)^(1/2))^(1/2))*cos(f*x+e)*(-a^2+b^2)^(1/2)*b^2-2*(-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*
x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-
a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((-(-(-a^2
+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a
^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*cos(f*x+e)*a^2*b+2*(-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*
x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-
a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticE((-(-(-a^2
+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a
^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*cos(f*x+e)*b^3-(-(-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)
*a-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*(((-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(-a^2+
b^2)^(1/2)/sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticF((-(-(-a^2+b^2
)^(1/2)*sin(f*x+e)-b*sin(f*x+e)+cos(f*x+e)*a-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),1/2*2^(1/2)*((b+(-a^2+b
^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*cos(f*x+e)*(-a^2+b^2)^(1/2)*a^2+sin(f*x+e)*2^(1/2)*cos(f*x+e)*a*b^2+2^(1/2
)*cos(f*x+e)^2*a^2*b-sin(f*x+e)*2^(1/2)*a^3-sin(f*x+e)*2^(1/2)*a*b^2+2^(1/2)*cos(f*x+e)*a^2*b-2*2^(1/2)*a^2*b)
/cos(f*x+e)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)*2^(1/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^(3/2)*sec(f*x + e)^2/sqrt(d*sin(f*x + e)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\cos \left (e+f\,x\right )}^2\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(sec(f*x+e)**2*(a+b*sin(f*x+e))**(3/2)/(d*sin(f*x+e))**(1/2),x)

[Out]

Timed out

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