Optimal. Leaf size=117 \[ -\frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{d^{3/2} f}+\frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 117, 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 = {2841, 21, 2854,
211} \begin {gather*} \frac {2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a^{3/2} \text {ArcTan}\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 )}{d^{3/2} f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 211
Rule 2841
Rule 2854
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{3/2}} \, dx &=\frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {(2 a) \int \frac {-\frac {1}{2} a (c+d)-\frac {1}{2} a (c+d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{d (c+d)}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {a \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{d}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\left (2 a^2\right ) \text {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 )}{d f}\\ &=-\frac {2 a^{3/2} \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 )}{d^{3/2} f}+\frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(377\) vs. \(2(117)=234\).
time = 5.95, size = 377, normalized size = 3.22 \begin {gather*} \frac {(a (1+\sin (e+f x)))^{3/2} \left (2 c \sqrt {d} \cos \left (\frac {1}{2} (e+f x)\right )-2 d^{3/2} \cos \left (\frac {1}{2} (e+f x)\right )-2 c \sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right )+2 d^{3/2} \sin \left (\frac {1}{2} (e+f x)\right )+2 (c+d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right ) \sqrt {c+d \sin (e+f x)}+(c+d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right ) \sqrt {c+d \sin (e+f x)}-c \log \left (\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\sqrt {c+d \sin (e+f x)}\right ) \sqrt {c+d \sin (e+f x)}-d \log \left (\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\sqrt {c+d \sin (e+f x)}\right ) \sqrt {c+d \sin (e+f x)}\right )}{d^{3/2} (c+d) f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {c+d \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6626\) vs.
\(2(101)=202\).
time = 9.23, size = 6627, normalized size = 56.64
method | result | size |
default | \(\text {Expression too large to display}\) | \(6627\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 439 vs.
\(2 (107) = 214\).
time = 0.68, size = 1351, normalized size = 11.55 \begin {gather*} \left [-\frac {{\left (a c^{2} + 2 \, a c d + a d^{2} - {\left (a c d + a d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (a c^{2} + a c d\right )} \cos \left (f x + e\right ) + {\left (a c^{2} + 2 \, a c d + a d^{2} + {\left (a c d + a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {-\frac {a}{d}} \log \left (\frac {128 \, a d^{4} \cos \left (f x + e\right )^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left (2 \, a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{4} - 32 \, {\left (5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, d^{4} \cos \left (f x + e\right )^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left (c d^{3} - d^{4}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - {\left (c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right )} \cos \left (f x + e\right ) + {\left (16 \, d^{4} \cos \left (f x + e\right )^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left (3 \, c d^{3} - 5 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {-\frac {a}{d}} + {\left (a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right )} \cos \left (f x + e\right ) + {\left (128 \, a d^{4} \cos \left (f x + e\right )^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left (a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} + 32 \, {\left (a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right ) + 8 \, {\left (a c - a d + {\left (a c - a d\right )} \cos \left (f x + e\right ) - {\left (a c - a d\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{4 \, {\left ({\left (c d^{2} + d^{3}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{2} d + c d^{2}\right )} f \cos \left (f x + e\right ) - {\left (c^{2} d + 2 \, c d^{2} + d^{3}\right )} f - {\left ({\left (c d^{2} + d^{3}\right )} f \cos \left (f x + e\right ) + {\left (c^{2} d + 2 \, c d^{2} + d^{3}\right )} f\right )} \sin \left (f x + e\right )\right )}}, -\frac {{\left (a c^{2} + 2 \, a c d + a d^{2} - {\left (a c d + a d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (a c^{2} + a c d\right )} \cos \left (f x + e\right ) + {\left (a c^{2} + 2 \, a c d + a d^{2} + {\left (a c d + a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {a}{d}} \arctan \left (\frac {{\left (8 \, d^{2} \cos \left (f x + e\right )^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left (c d - d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {\frac {a}{d}}}{4 \, {\left (2 \, a d^{2} \cos \left (f x + e\right )^{3} - {\left (3 \, a c d - a d^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (a c^{2} - a c d + 2 \, a d^{2}\right )} \cos \left (f x + e\right )\right )}}\right ) + 4 \, {\left (a c - a d + {\left (a c - a d\right )} \cos \left (f x + e\right ) - {\left (a c - a d\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{2 \, {\left ({\left (c d^{2} + d^{3}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{2} d + c d^{2}\right )} f \cos \left (f x + e\right ) - {\left (c^{2} d + 2 \, c d^{2} + d^{3}\right )} f - {\left ({\left (c d^{2} + d^{3}\right )} f \cos \left (f x + e\right ) + {\left (c^{2} d + 2 \, c d^{2} + d^{3}\right )} f\right )} \sin \left (f x + e\right )\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}{\left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________