Optimal. Leaf size=154 \[ -\frac {2}{9 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {4}{15 a d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {16 \sqrt {a+a \sin (c+d x)}}{15 a^2 d e (e \cos (c+d x))^{3/2}}+\frac {32 (a+a \sin (c+d x))^{3/2}}{45 a^3 d e (e \cos (c+d x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.20, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750}
\begin {gather*} \frac {32 (a \sin (c+d x)+a)^{3/2}}{45 a^3 d e (e \cos (c+d x))^{3/2}}-\frac {16 \sqrt {a \sin (c+d x)+a}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac {4}{15 a d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac {2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2750
Rule 2751
Rubi steps
\begin {align*} \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx &=-\frac {2}{9 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}+\frac {2 \int \frac {1}{(e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}} \, dx}{3 a}\\ &=-\frac {2}{9 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {4}{15 a d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}+\frac {8 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx}{15 a^2}\\ &=-\frac {2}{9 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {4}{15 a d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {16 \sqrt {a+a \sin (c+d x)}}{15 a^2 d e (e \cos (c+d x))^{3/2}}+\frac {16 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx}{15 a^3}\\ &=-\frac {2}{9 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {4}{15 a d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {16 \sqrt {a+a \sin (c+d x)}}{15 a^2 d e (e \cos (c+d x))^{3/2}}+\frac {32 (a+a \sin (c+d x))^{3/2}}{45 a^3 d e (e \cos (c+d x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 66, normalized size = 0.43 \begin {gather*} -\frac {2 (7+12 \cos (2 (c+d x))-6 \sin (c+d x)+4 \sin (3 (c+d x)))}{45 d e (e \cos (c+d x))^{3/2} (a (1+\sin (c+d x)))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 70, normalized size = 0.45
method | result | size |
default | \(-\frac {2 \left (16 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+24 \left (\cos ^{2}\left (d x +c \right )\right )-10 \sin \left (d x +c \right )-5\right ) \cos \left (d x +c \right )}{45 d \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}} \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 335 vs.
\(2 (118) = 236\).
time = 0.55, size = 335, normalized size = 2.18 \begin {gather*} -\frac {2 \, {\left (19 \, \sqrt {a} + \frac {12 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {58 \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {116 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {116 \, \sqrt {a} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {58 \, \sqrt {a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {12 \, \sqrt {a} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {19 \, \sqrt {a} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4} e^{\left (-\frac {5}{2}\right )}}{45 \, {\left (a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 110, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (24 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) - 5\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{45 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} e^{\frac {5}{2}} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} e^{\frac {5}{2}} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{2} e^{\frac {5}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 [B]
time = 11.10, size = 230, normalized size = 1.49 \begin {gather*} -\frac {14\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-12\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+24\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+8\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\frac {225\,a^2\,d\,e^2\,\cos \left (c+d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}-\frac {45\,a^2\,d\,e^2\,\cos \left (3\,c+3\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}+45\,a^2\,d\,e^2\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________