Optimal. Leaf size=178 \[ -\frac {a^{5/2} \left (3 c^2-10 c d+19 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{4 d^{5/2} f}+\frac {3 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}{2 d f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.43, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2763, 2981, 2775, 205} \[ -\frac {a^{5/2} \left (3 c^2-10 c d+19 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{4 d^{5/2} f}+\frac {3 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}{2 d f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 2763
Rule 2775
Rule 2981
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx &=-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}+\frac {\int \frac {\sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a^2 (c+5 d)-\frac {3}{2} a^2 (c-3 d) \sin (e+f x)\right )}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 d}\\ &=\frac {3 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}+\frac {\left (a^2 \left (3 c^2-10 c d+19 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{8 d^2}\\ &=\frac {3 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}-\frac {\left (a^3 \left (3 c^2-10 c d+19 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{4 d^2 f}\\ &=-\frac {a^{5/2} \left (3 c^2-10 c d+19 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{4 d^{5/2} f}+\frac {3 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.80, size = 256, normalized size = 1.44 \[ \frac {(a (\sin (e+f x)+1))^{5/2} \left (\frac {\left (3 c^2-10 c d+19 d^2\right ) \left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{\sqrt {c+d \sin (e+f x)}}\right )-\log \left (\sqrt {c+d \sin (e+f x)}+\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{\sqrt {c+d \sin (e+f x)}}\right )\right )}{d^{5/2}}+\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (3 c-2 d \sin (e+f x)-11 d) \sqrt {c+d \sin (e+f x)}}{d^2}\right )}{8 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.08, size = 1219, normalized size = 6.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{\sqrt {c +d \sin \left (f x +e \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {d \sin \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________