Optimal. Leaf size=363 \[ \frac {4 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{315 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4+279 a^2 b^2-10 b^4\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}} \]
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Rubi [A]
time = 0.84, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3926, 4189,
4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {2 \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{315 a d \sqrt {\sec (c+d x)}}+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4+279 a^2 b^2-10 b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^2 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {38 a b \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{63 d \sec ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 3926
Rule 3941
Rule 3943
Rule 4120
Rule 4189
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2}{9} \int \frac {\frac {19 a^2 b}{2}+\frac {1}{2} a \left (7 a^2+27 b^2\right ) \sec (c+d x)+\frac {3}{2} b \left (2 a^2+3 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {4 \int \frac {-\frac {1}{4} a^2 \left (49 a^2+75 b^2\right )-\frac {1}{4} a b \left (137 a^2+63 b^2\right ) \sec (c+d x)-19 a^2 b^2 \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{63 a}\\ &=\frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 \int \frac {\frac {3}{8} a^2 b \left (163 a^2+5 b^2\right )+\frac {1}{8} a^3 \left (147 a^2+605 b^2\right ) \sec (c+d x)+\frac {1}{4} a^2 b \left (49 a^2+75 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{315 a^2}\\ &=\frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}}-\frac {16 \int \frac {-\frac {3}{16} a^2 \left (147 a^4+279 a^2 b^2-10 b^4\right )-\frac {3}{16} a^3 b \left (261 a^2+155 b^2\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{945 a^3}\\ &=\frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}}+\frac {\left (147 a^4+279 a^2 b^2-10 b^4\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{315 a^2}+\frac {\left (2 b \left (57 a^4-62 a^2 b^2+5 b^4\right )\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 a^2}\\ &=\frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}}+\frac {\left (2 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{315 a^2 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4+279 a^2 b^2-10 b^4\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{315 a^2 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=\frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}}+\frac {\left (2 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{315 a^2 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4+279 a^2 b^2-10 b^4\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{315 a^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}\\ &=\frac {4 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{315 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4+279 a^2 b^2-10 b^4\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 1.73, size = 286, normalized size = 0.79 \begin {gather*} \frac {(a+b \sec (c+d x))^{5/2} \left (16 \left (147 a^5+147 a^4 b+279 a^3 b^2+279 a^2 b^3-10 a b^4-10 b^5\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )+32 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )+2 a \left (301 a^4+1984 a^2 b^2+40 b^4+4 a b \left (619 a^2+160 b^2\right ) \cos (c+d x)+8 \left (42 a^4+85 a^2 b^2\right ) \cos (2 (c+d x))+260 a^3 b \cos (3 (c+d x))+35 a^4 \cos (4 (c+d x))\right ) \sin (c+d x)\right )}{2520 a^2 d (b+a \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2787\) vs.
\(2(381)=762\).
time = 0.32, size = 2788, normalized size = 7.68
method | result | size |
default | \(\text {Expression too large to display}\) | \(2788\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.09, size = 541, normalized size = 1.49 \begin {gather*} \frac {\sqrt {2} {\left (-489 i \, a^{4} b + 93 i \, a^{2} b^{3} - 20 i \, b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (489 i \, a^{4} b - 93 i \, a^{2} b^{3} + 20 i \, b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (-147 i \, a^{5} - 279 i \, a^{3} b^{2} + 10 i \, a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (147 i \, a^{5} + 279 i \, a^{3} b^{2} - 10 i \, a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) + \frac {6 \, {\left (35 \, a^{5} \cos \left (d x + c\right )^{4} + 95 \, a^{4} b \cos \left (d x + c\right )^{3} + {\left (49 \, a^{5} + 75 \, a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (163 \, a^{4} b + 5 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{945 \, a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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